id	sid	tid	token	lemma	pos
easat-4026	1	1	edelweiss	edelweiss	PROPN
easat-4026	1	2	applied	apply	VERB
easat-4026	1	3	science	science	NOUN
easat-4026	1	4	and	and	CCONJ
easat-4026	1	5	technology	technology	NOUN
easat-4026	1	6	issn	issn	PROPN
easat-4026	1	7	:	:	PUNCT
easat-4026	1	8	2576	2576	NUM
easat-4026	1	9	-	-	SYM
easat-4026	1	10	8484	8484	NUM
easat-4026	1	11	vol	vol	NOUN
easat-4026	1	12	.	.	PROPN
easat-4026	1	13	8	8	NUM
easat-4026	1	14	,	,	PUNCT
easat-4026	1	15	no	no	INTJ
easat-4026	1	16	.	.	NOUN
easat-4026	1	17	6	6	NUM
easat-4026	1	18	,	,	PUNCT
easat-4026	1	19	9472	9472	NUM
easat-4026	1	20	-	-	SYM
easat-4026	1	21	9486	9486	NUM
easat-4026	1	22	2024	2024	NUM
easat-4026	1	23	publisher	publisher	NOUN
easat-4026	1	24	:	:	PUNCT
easat-4026	1	25	learning	learn	VERB
easat-4026	1	26	gate	gate	NOUN
easat-4026	1	27	doi	doi	PROPN
easat-4026	1	28	:	:	PUNCT
easat-4026	1	29	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	1	30	©	©	PROPN
easat-4026	1	31	2024	2024	NUM
easat-4026	1	32	by	by	ADP
easat-4026	1	33	the	the	DET
easat-4026	1	34	authors	author	NOUN
easat-4026	1	35	;	;	PUNCT
easat-4026	1	36	licensee	licensee	PROPN
easat-4026	1	37	learning	learning	NOUN
easat-4026	1	38	gate	gate	NOUN
easat-4026	1	39	©	©	PROPN
easat-4026	1	40	2024	2024	NUM
easat-4026	1	41	by	by	ADP
easat-4026	1	42	the	the	DET
easat-4026	1	43	authors	author	NOUN
easat-4026	1	44	;	;	PUNCT
easat-4026	1	45	licensee	licensee	PROPN
easat-4026	1	46	learning	learn	VERB
easat-4026	1	47	gate	gate	NOUN
easat-4026	1	48	*	*	PUNCT
easat-4026	1	49	correspondence	correspondence	NOUN
easat-4026	1	50	:	:	PUNCT
easat-4026	1	51	fatma.elzhraa6590@gmail.com	fatma.elzhraa6590@gmail.com	X
easat-4026	1	52	the	the	DET
easat-4026	1	53	excision	excision	NOUN
easat-4026	1	54	theory	theory	NOUN
easat-4026	1	55	for	for	ADP
easat-4026	1	56	homology	homology	NOUN
easat-4026	1	57	theory	theory	NOUN
easat-4026	1	58	through	through	ADP
easat-4026	1	59	a_∞-algebras	a_∞-algebras	ADP
easat-4026	1	60	faten	faten	ADJ
easat-4026	1	61	ragab	ragab	ADJ
easat-4026	1	62	karar1	karar1	PROPN
easat-4026	1	63	,	,	PUNCT
easat-4026	1	64	fatma	fatma	PROPN
easat-4026	1	65	elzhraa	elzhraa	PROPN
easat-4026	1	66	ahmed	ahme	VERB
easat-4026	1	67	mohammed2	mohammed2	PROPN
easat-4026	1	68	*	*	PROPN
easat-4026	1	69	,	,	PUNCT
easat-4026	1	70	a.	a.	PROPN
easat-4026	1	71	a.	a.	PROPN
easat-4026	1	72	el	el	PROPN
easat-4026	1	73	fattah3	fattah3	PROPN
easat-4026	2	1	1department	1department	NUM
easat-4026	2	2	of	of	ADP
easat-4026	2	3	mathematics	mathematic	NOUN
easat-4026	2	4	,	,	PUNCT
easat-4026	2	5	faculty	faculty	NOUN
easat-4026	2	6	of	of	ADP
easat-4026	2	7	science	science	NOUN
easat-4026	2	8	,	,	PUNCT
easat-4026	2	9	aswan	aswan	PROPN
easat-4026	2	10	university	university	PROPN
easat-4026	2	11	,	,	PUNCT
easat-4026	2	12	aswan	aswan	PROPN
easat-4026	2	13	,	,	PUNCT
easat-4026	2	14	egypt	egypt	PROPN
easat-4026	2	15	;	;	PUNCT
easat-4026	2	16	fatenragab2020@yahoo.com	fatenragab2020@yahoo.com	PROPN
easat-4026	2	17	(	(	PUNCT
easat-4026	2	18	f.r.k	f.r.k	PROPN
easat-4026	2	19	.	.	PUNCT
easat-4026	2	20	)	)	PUNCT
easat-4026	2	21	.	.	PUNCT
easat-4026	3	1	2,3mathematics	2,3mathematics	NUM
easat-4026	3	2	teacher	teacher	NOUN
easat-4026	3	3	in	in	ADP
easat-4026	3	4	secondary	secondary	ADJ
easat-4026	3	5	school	school	NOUN
easat-4026	3	6	,	,	PUNCT
easat-4026	3	7	ministry	ministry	PROPN
easat-4026	3	8	of	of	ADP
easat-4026	3	9	education	education	PROPN
easat-4026	3	10	;	;	PUNCT
easat-4026	3	11	fatma.elzhraa6590@gmail.com	fatma.elzhraa6590@gmail.com	PROPN
easat-4026	3	12	(	(	PUNCT
easat-4026	3	13	f.a.m	f.a.m	ADJ
easat-4026	3	14	.	.	PUNCT
easat-4026	3	15	)	)	PUNCT
easat-4026	3	16	ahmedfarrouk222@gmail.com	ahmedfarrouk222@gmail.com	X
easat-4026	3	17	(	(	PUNCT
easat-4026	3	18	a.a.e.f	a.a.e.f	PROPN
easat-4026	3	19	.	.	PROPN
easat-4026	3	20	)	)	PUNCT
easat-4026	3	21	.	.	PUNCT
easat-4026	4	1	abstract	abstract	ADV
easat-4026	4	2	:	:	PUNCT
easat-4026	5	1	this	this	DET
easat-4026	5	2	paper	paper	NOUN
easat-4026	5	3	investigated	investigate	VERB
easat-4026	5	4	a_∞-algebras	a_∞-algebras	ADP
easat-4026	5	5	,	,	PUNCT
easat-4026	5	6	which	which	PRON
easat-4026	5	7	are	be	AUX
easat-4026	5	8	generalizations	generalization	NOUN
easat-4026	5	9	of	of	ADP
easat-4026	5	10	associative	associative	ADJ
easat-4026	5	11	algebras	algebra	NOUN
easat-4026	5	12	that	that	PRON
easat-4026	5	13	incorporate	incorporate	VERB
easat-4026	5	14	higher	high	ADJ
easat-4026	5	15	homotopy	homotopy	NOUN
easat-4026	5	16	structures	structure	NOUN
easat-4026	5	17	.	.	PUNCT
easat-4026	6	1	we	we	PRON
easat-4026	6	2	began	begin	VERB
easat-4026	6	3	by	by	ADP
easat-4026	6	4	revisiting	revisit	VERB
easat-4026	6	5	the	the	DET
easat-4026	6	6	fundamental	fundamental	ADJ
easat-4026	6	7	definitions	definition	NOUN
easat-4026	6	8	and	and	CCONJ
easat-4026	6	9	properties	property	NOUN
easat-4026	6	10	of	of	ADP
easat-4026	6	11	a_∞-algebras	a_∞-algebras	ADP
easat-4026	6	12	and	and	CCONJ
easat-4026	6	13	their	their	PRON
easat-4026	6	14	associated	associated	ADJ
easat-4026	6	15	homological	homological	ADJ
easat-4026	6	16	theories	theory	NOUN
easat-4026	6	17	,	,	PUNCT
easat-4026	6	18	providing	provide	VERB
easat-4026	6	19	a	a	DET
easat-4026	6	20	solid	solid	ADJ
easat-4026	6	21	foundation	foundation	NOUN
easat-4026	6	22	for	for	ADP
easat-4026	6	23	understanding	understand	VERB
easat-4026	6	24	these	these	DET
easat-4026	6	25	complex	complex	ADJ
easat-4026	6	26	structures	structure	NOUN
easat-4026	6	27	.	.	PUNCT
easat-4026	7	1	the	the	DET
easat-4026	7	2	study	study	NOUN
easat-4026	7	3	included	include	VERB
easat-4026	7	4	an	an	DET
easat-4026	7	5	in	in	ADP
easat-4026	7	6	-	-	PUNCT
easat-4026	7	7	depth	depth	NOUN
easat-4026	7	8	analysis	analysis	NOUN
easat-4026	7	9	of	of	ADP
easat-4026	7	10	simplicial	simplicial	ADJ
easat-4026	7	11	homology	homology	NOUN
easat-4026	7	12	as	as	SCONJ
easat-4026	7	13	it	it	PRON
easat-4026	7	14	relates	relate	VERB
easat-4026	7	15	to	to	ADP
easat-4026	7	16	a_∞-algebras	a_∞-algebras	ADP
easat-4026	7	17	,	,	PUNCT
easat-4026	7	18	focusing	focus	VERB
easat-4026	7	19	on	on	ADP
easat-4026	7	20	significant	significant	ADJ
easat-4026	7	21	results	result	NOUN
easat-4026	7	22	,	,	PUNCT
easat-4026	7	23	particularly	particularly	ADV
easat-4026	7	24	those	those	DET
easat-4026	7	25	concerning	concern	VERB
easat-4026	7	26	excision	excision	NOUN
easat-4026	7	27	theory	theory	NOUN
easat-4026	7	28	.	.	PUNCT
easat-4026	8	1	in	in	ADP
easat-4026	8	2	this	this	DET
easat-4026	8	3	context	context	NOUN
easat-4026	8	4	,	,	PUNCT
easat-4026	8	5	we	we	PRON
easat-4026	8	6	introduced	introduce	VERB
easat-4026	8	7	new	new	ADJ
easat-4026	8	8	insights	insight	NOUN
easat-4026	8	9	into	into	ADP
easat-4026	8	10	the	the	DET
easat-4026	8	11	relationship	relationship	NOUN
easat-4026	8	12	between	between	ADP
easat-4026	8	13	bar	bar	NOUN
easat-4026	8	14	homology	homology	NOUN
easat-4026	8	15	and	and	CCONJ
easat-4026	8	16	simplicial	simplicial	ADJ
easat-4026	8	17	homology	homology	NOUN
easat-4026	8	18	,	,	PUNCT
easat-4026	8	19	presenting	present	VERB
easat-4026	8	20	a	a	DET
easat-4026	8	21	precise	precise	ADJ
easat-4026	8	22	sequence	sequence	NOUN
easat-4026	8	23	elucidating	elucidate	VERB
easat-4026	8	24	the	the	DET
easat-4026	8	25	interaction	interaction	NOUN
easat-4026	8	26	between	between	ADP
easat-4026	8	27	these	these	DET
easat-4026	8	28	two	two	NUM
easat-4026	8	29	homological	homological	ADJ
easat-4026	8	30	structures	structure	NOUN
easat-4026	8	31	.	.	PUNCT
easat-4026	9	1	within	within	ADP
easat-4026	9	2	this	this	DET
easat-4026	9	3	framework	framework	NOUN
easat-4026	9	4	,	,	PUNCT
easat-4026	9	5	we	we	PRON
easat-4026	9	6	provided	provide	VERB
easat-4026	9	7	proofs	proof	NOUN
easat-4026	9	8	for	for	ADP
easat-4026	9	9	key	key	ADJ
easat-4026	9	10	results	result	NOUN
easat-4026	9	11	,	,	PUNCT
easat-4026	9	12	such	such	ADJ
easat-4026	9	13	as	as	ADP
easat-4026	9	14	the	the	DET
easat-4026	9	15	quantitative	quantitative	ADJ
easat-4026	9	16	coherence	coherence	NOUN
easat-4026	9	17	of	of	ADP
easat-4026	9	18	certain	certain	ADJ
easat-4026	9	19	maps	map	NOUN
easat-4026	9	20	and	and	CCONJ
easat-4026	9	21	the	the	DET
easat-4026	9	22	interchanging	interchanging	ADJ
easat-4026	9	23	diagram	diagram	NOUN
easat-4026	9	24	that	that	PRON
easat-4026	9	25	connects	connect	VERB
easat-4026	9	26	different	different	ADJ
easat-4026	9	27	homological	homological	ADJ
easat-4026	9	28	categories	category	NOUN
easat-4026	9	29	.	.	PUNCT
easat-4026	10	1	we	we	PRON
easat-4026	10	2	address	address	VERB
easat-4026	10	3	the	the	DET
easat-4026	10	4	specific	specific	ADJ
easat-4026	10	5	failure	failure	NOUN
easat-4026	10	6	of	of	ADP
easat-4026	10	7	excision	excision	NOUN
easat-4026	10	8	properties	property	NOUN
easat-4026	10	9	and	and	CCONJ
easat-4026	10	10	its	its	PRON
easat-4026	10	11	implications	implication	NOUN
easat-4026	10	12	for	for	ADP
easat-4026	10	13	long	long	ADJ
easat-4026	10	14	exact	exact	ADJ
easat-4026	10	15	sequences	sequence	NOUN
easat-4026	10	16	in	in	ADP
easat-4026	10	17	both	both	CCONJ
easat-4026	10	18	homological	homological	ADJ
easat-4026	10	19	and	and	CCONJ
easat-4026	10	20	homotopical	homotopical	ADJ
easat-4026	10	21	contexts	contexts	NOUN
easat-4026	10	22	.	.	PUNCT
easat-4026	11	1	this	this	DET
easat-4026	11	2	paper	paper	NOUN
easat-4026	11	3	offered	offer	VERB
easat-4026	11	4	a	a	DET
easat-4026	11	5	comprehensive	comprehensive	ADJ
easat-4026	11	6	overview	overview	NOUN
easat-4026	11	7	of	of	ADP
easat-4026	11	8	current	current	ADJ
easat-4026	11	9	developments	development	NOUN
easat-4026	11	10	in	in	ADP
easat-4026	11	11	a_∞-algebra	a_∞-algebra	NOUN
easat-4026	11	12	theory	theory	NOUN
easat-4026	11	13	and	and	CCONJ
easat-4026	11	14	simplicial	simplicial	ADJ
easat-4026	11	15	cohomology	cohomology	NOUN
easat-4026	11	16	,	,	PUNCT
easat-4026	11	17	highlighting	highlight	VERB
easat-4026	11	18	classical	classical	ADJ
easat-4026	11	19	and	and	CCONJ
easat-4026	11	20	contemporary	contemporary	ADJ
easat-4026	11	21	insights	insight	NOUN
easat-4026	11	22	into	into	ADP
easat-4026	11	23	these	these	DET
easat-4026	11	24	sophisticated	sophisticated	ADJ
easat-4026	11	25	mathematical	mathematical	ADJ
easat-4026	11	26	structures	structure	NOUN
easat-4026	11	27	.	.	PUNCT
easat-4026	12	1	by	by	ADP
easat-4026	12	2	presenting	present	VERB
easat-4026	12	3	detailed	detailed	ADJ
easat-4026	12	4	definitions	definition	NOUN
easat-4026	12	5	,	,	PUNCT
easat-4026	12	6	examples	example	NOUN
easat-4026	12	7	,	,	PUNCT
easat-4026	12	8	and	and	CCONJ
easat-4026	12	9	theorems	theorem	NOUN
easat-4026	12	10	,	,	PUNCT
easat-4026	12	11	we	we	PRON
easat-4026	12	12	strive	strive	VERB
easat-4026	12	13	to	to	PART
easat-4026	12	14	contribute	contribute	VERB
easat-4026	12	15	to	to	ADP
easat-4026	12	16	a	a	DET
easat-4026	12	17	deeper	deep	ADJ
easat-4026	12	18	understanding	understanding	NOUN
easat-4026	12	19	of	of	ADP
easat-4026	12	20	homology	homology	NOUN
easat-4026	12	21	within	within	ADP
easat-4026	12	22	the	the	DET
easat-4026	12	23	framework	framework	NOUN
easat-4026	12	24	of	of	ADP
easat-4026	12	25	advanced	advanced	ADJ
easat-4026	12	26	algebraic	algebraic	ADJ
easat-4026	12	27	systems	system	NOUN
easat-4026	12	28	.	.	PUNCT
easat-4026	13	1	our	our	PRON
easat-4026	13	2	analysis	analysis	NOUN
easat-4026	13	3	sheds	shed	VERB
easat-4026	13	4	light	light	NOUN
easat-4026	13	5	on	on	ADP
easat-4026	13	6	existing	exist	VERB
easat-4026	13	7	theories	theory	NOUN
easat-4026	13	8	and	and	CCONJ
easat-4026	13	9	paves	pave	VERB
easat-4026	13	10	the	the	DET
easat-4026	13	11	way	way	NOUN
easat-4026	13	12	for	for	ADP
easat-4026	13	13	future	future	ADJ
easat-4026	13	14	research	research	NOUN
easat-4026	13	15	in	in	ADP
easat-4026	13	16	the	the	DET
easat-4026	13	17	field	field	NOUN
easat-4026	13	18	,	,	PUNCT
easat-4026	13	19	providing	provide	VERB
easat-4026	13	20	a	a	DET
easat-4026	13	21	valuable	valuable	ADJ
easat-4026	13	22	resource	resource	NOUN
easat-4026	13	23	for	for	ADP
easat-4026	13	24	mathematicians	mathematician	NOUN
easat-4026	13	25	interested	interested	ADJ
easat-4026	13	26	in	in	ADP
easat-4026	13	27	the	the	DET
easat-4026	13	28	interplay	interplay	NOUN
easat-4026	13	29	between	between	ADP
easat-4026	13	30	algebra	algebra	NOUN
easat-4026	13	31	and	and	CCONJ
easat-4026	13	32	topology	topology	NOUN
easat-4026	13	33	.	.	PUNCT
easat-4026	14	1	keywords	keyword	NOUN
easat-4026	14	2	:	:	PUNCT
easat-4026	14	3	a_∞-algebras	a_∞-algebras	ADP
easat-4026	14	4	,	,	PUNCT
easat-4026	14	5	excision	excision	NOUN
easat-4026	14	6	,	,	PUNCT
easat-4026	14	7	hochschild	hochschild	NOUN
easat-4026	14	8	,	,	PUNCT
easat-4026	14	9	homology	homology	NOUN
easat-4026	14	10	.	.	PUNCT
easat-4026	15	1	1	1	X
easat-4026	15	2	.	.	X
easat-4026	15	3	introduction	introduction	NOUN
easat-4026	15	4	as	as	SCONJ
easat-4026	15	5	demonstrated	demonstrate	VERB
easat-4026	15	6	by	by	ADP
easat-4026	15	7	riemann	riemann	PROPN
easat-4026	15	8	's	's	PART
easat-4026	15	9	solutions	solution	NOUN
easat-4026	15	10	to	to	ADP
easat-4026	15	11	problems	problem	NOUN
easat-4026	15	12	involving	involve	VERB
easat-4026	15	13	surface	surface	NOUN
easat-4026	15	14	connections	connection	NOUN
easat-4026	15	15	,	,	PUNCT
easat-4026	15	16	homology	homology	NOUN
easat-4026	15	17	is	be	AUX
easat-4026	15	18	pivotal	pivotal	ADJ
easat-4026	15	19	in	in	ADP
easat-4026	15	20	mathematical	mathematical	ADJ
easat-4026	15	21	investigations	investigation	NOUN
easat-4026	15	22	.	.	PUNCT
easat-4026	16	1	green	green	PROPN
easat-4026	16	2	's	's	PART
easat-4026	16	3	theorem	theorem	NOUN
easat-4026	16	4	,	,	PUNCT
easat-4026	16	5	which	which	PRON
easat-4026	16	6	relates	relate	VERB
easat-4026	16	7	line	line	NOUN
easat-4026	16	8	integrals	integral	NOUN
easat-4026	16	9	over	over	ADP
easat-4026	16	10	a	a	DET
easat-4026	16	11	closed	closed	ADJ
easat-4026	16	12	curve	curve	NOUN
easat-4026	16	13	to	to	ADP
easat-4026	16	14	double	double	ADJ
easat-4026	16	15	integrals	integral	NOUN
easat-4026	16	16	over	over	ADP
easat-4026	16	17	the	the	DET
easat-4026	16	18	enclosed	enclose	VERB
easat-4026	16	19	plane	plane	NOUN
easat-4026	16	20	region	region	NOUN
easat-4026	16	21	,	,	PUNCT
easat-4026	16	22	underpins	underpin	VERB
easat-4026	16	23	this	this	DET
easat-4026	16	24	principle	principle	NOUN
easat-4026	16	25	.	.	PUNCT
easat-4026	17	1	the	the	DET
easat-4026	17	2	theorem	theorem	NOUN
easat-4026	17	3	implies	imply	VERB
easat-4026	17	4	that	that	SCONJ
easat-4026	17	5	certain	certain	ADJ
easat-4026	17	6	integrals	integral	NOUN
easat-4026	17	7	will	will	AUX
easat-4026	17	8	yield	yield	VERB
easat-4026	17	9	identical	identical	ADJ
easat-4026	17	10	values	value	NOUN
easat-4026	17	11	for	for	ADP
easat-4026	17	12	any	any	DET
easat-4026	17	13	two	two	NUM
easat-4026	17	14	homologous	homologous	ADJ
easat-4026	17	15	curves	curve	NOUN
easat-4026	17	16	,	,	PUNCT
easat-4026	17	17	an	an	DET
easat-4026	17	18	idea	idea	NOUN
easat-4026	17	19	that	that	PRON
easat-4026	17	20	permeates	permeate	VERB
easat-4026	17	21	classical	classical	ADJ
easat-4026	17	22	vector	vector	NOUN
easat-4026	17	23	spaces	space	NOUN
easat-4026	17	24	and	and	CCONJ
easat-4026	17	25	theoretical	theoretical	ADJ
easat-4026	17	26	physics	physics	NOUN
easat-4026	17	27	.	.	PUNCT
easat-4026	18	1	simplicial	simplicial	ADJ
easat-4026	18	2	homology	homology	PROPN
easat-4026	18	3	,	,	PUNCT
easat-4026	18	4	a	a	DET
easat-4026	18	5	branch	branch	NOUN
easat-4026	18	6	of	of	ADP
easat-4026	18	7	algebraic	algebraic	ADJ
easat-4026	18	8	topology	topology	NOUN
easat-4026	18	9	,	,	PUNCT
easat-4026	18	10	addresses	address	VERB
easat-4026	18	11	the	the	DET
easat-4026	18	12	study	study	NOUN
easat-4026	18	13	of	of	ADP
easat-4026	18	14	homology	homology	NOUN
easat-4026	18	15	theory	theory	NOUN
easat-4026	18	16	in	in	ADP
easat-4026	18	17	associative	associative	ADJ
easat-4026	18	18	algebras	algebra	NOUN
easat-4026	18	19	over	over	ADP
easat-4026	18	20	a	a	DET
easat-4026	18	21	field	field	NOUN
easat-4026	18	22	.	.	PUNCT
easat-4026	19	1	gerhard	gerhard	PROPN
easat-4026	19	2	hochschild	hochschild	PROPN
easat-4026	19	3	introduced	introduce	VERB
easat-4026	19	4	this	this	DET
easat-4026	19	5	theory	theory	NOUN
easat-4026	19	6	[	[	X
easat-4026	19	7	1	1	NUM
easat-4026	19	8	]	]	PUNCT
easat-4026	19	9	,	,	PUNCT
easat-4026	19	10	initially	initially	ADV
easat-4026	19	11	focusing	focus	VERB
easat-4026	19	12	on	on	ADP
easat-4026	19	13	algebras	algebra	NOUN
easat-4026	19	14	over	over	ADP
easat-4026	19	15	a	a	DET
easat-4026	19	16	field	field	NOUN
easat-4026	19	17	.	.	PUNCT
easat-4026	20	1	henri	henri	PROPN
easat-4026	20	2	cartan	cartan	PROPN
easat-4026	20	3	and	and	CCONJ
easat-4026	20	4	samuel	samuel	PROPN
easat-4026	20	5	eilenberg	eilenberg	PROPN
easat-4026	21	1	[	[	X
easat-4026	21	2	2	2	NUM
easat-4026	21	3	]	]	PUNCT
easat-4026	21	4	later	later	ADV
easat-4026	21	5	extended	extend	VERB
easat-4026	21	6	this	this	DET
easat-4026	21	7	work	work	NOUN
easat-4026	21	8	to	to	ADP
easat-4026	21	9	more	more	ADV
easat-4026	21	10	generalized	generalized	ADJ
easat-4026	21	11	rings	ring	NOUN
easat-4026	21	12	.	.	PUNCT
easat-4026	22	1	in	in	ADP
easat-4026	22	2	[	[	X
easat-4026	22	3	3	3	NUM
easat-4026	22	4	]	]	PUNCT
easat-4026	22	5	,	,	PUNCT
easat-4026	22	6	stasheff	stasheff	NOUN
easat-4026	22	7	developed	develop	VERB
easat-4026	22	8	the	the	DET
easat-4026	22	9	concepts	concept	NOUN
easat-4026	22	10	of	of	ADP
easat-4026	22	11	𝒜∞-spaces	𝒜∞-spaces	PROPN
easat-4026	22	12	and	and	CCONJ
easat-4026	22	13	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	22	14	during	during	ADP
easat-4026	22	15	his	his	PRON
easat-4026	22	16	study	study	NOUN
easat-4026	22	17	of	of	ADP
easat-4026	22	18	topological	topological	ADJ
easat-4026	22	19	spaces	space	NOUN
easat-4026	22	20	.	.	PUNCT
easat-4026	23	1	these	these	DET
easat-4026	23	2	ideas	idea	NOUN
easat-4026	23	3	emerged	emerge	VERB
easat-4026	23	4	as	as	ADP
easat-4026	23	5	generalizations	generalization	NOUN
easat-4026	23	6	of	of	ADP
easat-4026	23	7	topological	topological	ADJ
easat-4026	23	8	groups	group	NOUN
easat-4026	23	9	while	while	SCONJ
easat-4026	23	10	maintaining	maintain	VERB
easat-4026	23	11	continuous	continuous	ADJ
easat-4026	23	12	,	,	PUNCT
easat-4026	23	13	albeit	albeit	SCONJ
easat-4026	23	14	non	non	ADJ
easat-4026	23	15	-	-	ADJ
easat-4026	23	16	associative	associative	ADJ
easat-4026	23	17	,	,	PUNCT
easat-4026	23	18	multiplication	multiplication	NOUN
easat-4026	23	19	.	.	PUNCT
easat-4026	24	1	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	24	2	are	be	AUX
easat-4026	24	3	essentially	essentially	ADV
easat-4026	24	4	chain	chain	NOUN
easat-4026	24	5	complexes	complex	NOUN
easat-4026	24	6	with	with	ADP
easat-4026	24	7	homotopy	homotopy	NOUN
easat-4026	24	8	associative	associative	ADJ
easat-4026	24	9	products	product	NOUN
easat-4026	24	10	,	,	PUNCT
easat-4026	24	11	satisfying	satisfy	VERB
easat-4026	24	12	higher	high	ADJ
easat-4026	24	13	homotopy	homotopy	NOUN
easat-4026	24	14	associativity	associativity	NOUN
easat-4026	24	15	conditions	condition	NOUN
easat-4026	24	16	.	.	PUNCT
easat-4026	25	1	a	a	DET
easat-4026	25	2	prime	prime	ADJ
easat-4026	25	3	example	example	NOUN
easat-4026	25	4	of	of	ADP
easat-4026	25	5	an	an	DET
easat-4026	25	6	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-4026	25	7	is	be	AUX
easat-4026	25	8	the	the	DET
easat-4026	25	9	singular	singular	ADJ
easat-4026	25	10	chain	chain	NOUN
easat-4026	25	11	complex	complex	NOUN
easat-4026	25	12	𝒞	𝒞	PROPN
easat-4026	25	13	●	●	NOUN
easat-4026	25	14	(𝒳	(𝒳	NOUN
easat-4026	25	15	)	)	PUNCT
easat-4026	25	16	,	,	PUNCT
easat-4026	25	17	which	which	PRON
easat-4026	25	18	illustrates	illustrate	VERB
easat-4026	25	19	their	their	PRON
easat-4026	25	20	homotopy	homotopy	NOUN
easat-4026	25	21	-	-	PUNCT
easat-4026	25	22	invariant	invariant	ADJ
easat-4026	25	23	nature	nature	NOUN
easat-4026	25	24	.	.	PUNCT
easat-4026	26	1	research	research	NOUN
easat-4026	26	2	into	into	ADP
easat-4026	26	3	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	26	4	has	have	AUX
easat-4026	26	5	advanced	advance	VERB
easat-4026	26	6	significantly	significantly	ADV
easat-4026	26	7	due	due	ADJ
easat-4026	26	8	to	to	ADP
easat-4026	26	9	contributions	contribution	NOUN
easat-4026	26	10	from	from	ADP
easat-4026	26	11	mathematicians	mathematician	NOUN
easat-4026	26	12	such	such	ADJ
easat-4026	26	13	as	as	ADP
easat-4026	26	14	kadeishvili	kadeishvili	NOUN
easat-4026	26	15	[	[	X
easat-4026	26	16	4	4	NUM
easat-4026	26	17	]	]	PUNCT
easat-4026	26	18	,	,	PUNCT
easat-4026	26	19	smirnov	smirnov	PROPN
easat-4026	26	20	[	[	X
easat-4026	26	21	5	5	NUM
easat-4026	26	22	]	]	PUNCT
easat-4026	26	23	,	,	PUNCT
easat-4026	26	24	and	and	CCONJ
easat-4026	26	25	prouté	prouté	X
easat-4026	27	1	[	[	X
easat-4026	27	2	6	6	NUM
easat-4026	27	3	]	]	PUNCT
easat-4026	27	4	.	.	PUNCT
easat-4026	28	1	j.	j.	PROPN
easat-4026	28	2	huebschmann	huebschmann	PROPN
easat-4026	28	3	highlighted	highlight	VERB
easat-4026	28	4	the	the	DET
easat-4026	28	5	relevance	relevance	NOUN
easat-4026	28	6	of	of	ADP
easat-4026	28	7	homological	homological	ADJ
easat-4026	28	8	perturbation	perturbation	NOUN
easat-4026	28	9	theory	theory	NOUN
easat-4026	28	10	and	and	CCONJ
easat-4026	28	11	𝒜∞-structures	𝒜∞-structure	NOUN
easat-4026	28	12	in	in	ADP
easat-4026	28	13	homological	homological	ADJ
easat-4026	28	14	algebra	algebra	NOUN
easat-4026	28	15	,	,	PUNCT
easat-4026	28	16	especially	especially	ADV
easat-4026	28	17	within	within	ADP
easat-4026	28	18	9473	9473	NUM
easat-4026	28	19	edelweiss	edelweiss	PROPN
easat-4026	28	20	applied	apply	VERB
easat-4026	28	21	science	science	NOUN
easat-4026	28	22	and	and	CCONJ
easat-4026	28	23	technology	technology	NOUN
easat-4026	28	24	issn	issn	PROPN
easat-4026	28	25	:	:	PUNCT
easat-4026	28	26	2576	2576	NUM
easat-4026	28	27	-	-	SYM
easat-4026	28	28	8484	8484	NUM
easat-4026	28	29	vol	vol	NOUN
easat-4026	28	30	.	.	PROPN
easat-4026	29	1	8	8	NUM
easat-4026	29	2	,	,	PUNCT
easat-4026	29	3	no	no	INTJ
easat-4026	29	4	.	.	NOUN
easat-4026	30	1	6	6	NUM
easat-4026	30	2	:	:	SYM
easat-4026	30	3	9472	9472	NUM
easat-4026	30	4	-	-	SYM
easat-4026	30	5	9486	9486	NUM
easat-4026	30	6	,	,	PUNCT
easat-4026	30	7	2024	2024	NUM
easat-4026	30	8	doi	doi	NOUN
easat-4026	30	9	:	:	PUNCT
easat-4026	30	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	30	11	©	©	PROPN
easat-4026	30	12	2024	2024	NUM
easat-4026	30	13	by	by	ADP
easat-4026	30	14	the	the	DET
easat-4026	30	15	authors	author	NOUN
easat-4026	30	16	;	;	PUNCT
easat-4026	30	17	licensee	licensee	PROPN
easat-4026	30	18	learning	learn	VERB
easat-4026	30	19	gate	gate	NOUN
easat-4026	30	20	topological	topological	ADJ
easat-4026	30	21	contexts	context	NOUN
easat-4026	31	1	[	[	X
easat-4026	31	2	7	7	NUM
easat-4026	31	3	]	]	PUNCT
easat-4026	31	4	.	.	PUNCT
easat-4026	32	1	further	further	ADJ
easat-4026	32	2	advancements	advancement	NOUN
easat-4026	32	3	were	be	AUX
easat-4026	32	4	made	make	VERB
easat-4026	32	5	by	by	ADP
easat-4026	32	6	john	john	PROPN
easat-4026	32	7	d.	d.	PROPN
easat-4026	32	8	s.	s.	PROPN
easat-4026	32	9	jones	jones	PROPN
easat-4026	32	10	and	and	CCONJ
easat-4026	32	11	e.	e.	PROPN
easat-4026	32	12	getzler	getzler	NOUN
easat-4026	33	1	[	[	X
easat-4026	33	2	8	8	NUM
easat-4026	33	3	]	]	PUNCT
easat-4026	33	4	.	.	PUNCT
easat-4026	34	1	kenji	kenji	PROPN
easat-4026	34	2	fukaya	fukaya	PROPN
easat-4026	34	3	's	's	PART
easat-4026	34	4	exploration	exploration	NOUN
easat-4026	34	5	of	of	ADP
easat-4026	34	6	𝒜∞-categories	𝒜∞-categorie	NOUN
easat-4026	34	7	[	[	X
easat-4026	34	8	9	9	NUM
easat-4026	34	9	]	]	PUNCT
easat-4026	34	10	and	and	CCONJ
easat-4026	34	11	kontsevich	kontsevich	PROPN
easat-4026	34	12	's	's	PART
easat-4026	34	13	influential	influential	ADJ
easat-4026	34	14	1994	1994	NUM
easat-4026	34	15	lecture	lecture	NOUN
easat-4026	34	16	on	on	ADP
easat-4026	34	17	categorical	categorical	ADJ
easat-4026	34	18	mirror	mirror	NOUN
easat-4026	34	19	symmetry	symmetry	NOUN
easat-4026	34	20	contributed	contribute	VERB
easat-4026	34	21	significantly	significantly	ADV
easat-4026	34	22	to	to	ADP
easat-4026	34	23	the	the	DET
easat-4026	34	24	field	field	NOUN
easat-4026	34	25	's	's	PART
easat-4026	34	26	development	development	NOUN
easat-4026	34	27	[	[	X
easat-4026	34	28	10	10	NUM
easat-4026	34	29	]	]	PUNCT
easat-4026	34	30	.	.	PUNCT
easat-4026	35	1	in	in	ADP
easat-4026	35	2	[	[	X
easat-4026	35	3	11	11	NUM
easat-4026	35	4	]	]	PUNCT
easat-4026	35	5	,	,	PUNCT
easat-4026	35	6	keller	keller	PROPN
easat-4026	35	7	integrated	integrate	VERB
easat-4026	35	8	𝒜∞-language	𝒜∞-language	NOUN
easat-4026	35	9	into	into	ADP
easat-4026	35	10	ring	ring	NOUN
easat-4026	35	11	theory	theory	NOUN
easat-4026	35	12	and	and	CCONJ
easat-4026	35	13	representation	representation	NOUN
easat-4026	35	14	theory	theory	NOUN
easat-4026	35	15	,	,	PUNCT
easat-4026	35	16	showing	show	VERB
easat-4026	35	17	that	that	SCONJ
easat-4026	35	18	the	the	DET
easat-4026	35	19	derived	derived	ADJ
easat-4026	35	20	category	category	NOUN
easat-4026	35	21	of	of	ADP
easat-4026	35	22	any	any	DET
easat-4026	35	23	grothendieck	grothendieck	NOUN
easat-4026	35	24	category	category	NOUN
easat-4026	35	25	with	with	ADP
easat-4026	35	26	a	a	DET
easat-4026	35	27	compact	compact	ADJ
easat-4026	35	28	generator	generator	NOUN
easat-4026	35	29	is	be	AUX
easat-4026	35	30	equivalent	equivalent	ADJ
easat-4026	35	31	to	to	ADP
easat-4026	35	32	the	the	DET
easat-4026	35	33	derived	derive	VERB
easat-4026	35	34	category	category	NOUN
easat-4026	35	35	of	of	ADP
easat-4026	35	36	an	an	DET
easat-4026	35	37	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-4026	35	38	.	.	PUNCT
easat-4026	36	1	additional	additional	ADJ
easat-4026	36	2	studies	study	NOUN
easat-4026	36	3	by	by	ADP
easat-4026	36	4	p.	p.	PROPN
easat-4026	36	5	seidel	seidel	PROPN
easat-4026	36	6	on	on	ADP
easat-4026	36	7	𝒜∞-structures	𝒜∞-structure	NOUN
easat-4026	36	8	concerning	concern	VERB
easat-4026	36	9	lefschetz	lefschetz	ADJ
easat-4026	36	10	fibrations	fibration	NOUN
easat-4026	36	11	[	[	X
easat-4026	36	12	12	12	NUM
easat-4026	36	13	]	]	PUNCT
easat-4026	36	14	,	,	PUNCT
easat-4026	36	15	and	and	CCONJ
easat-4026	36	16	research	research	NOUN
easat-4026	36	17	by	by	ADP
easat-4026	36	18	alaa	alaa	PROPN
easat-4026	36	19	h.	h.	PROPN
easat-4026	36	20	and	and	CCONJ
easat-4026	36	21	y.	y.	PROPN
easat-4026	36	22	gouda	gouda	NOUN
easat-4026	36	23	on	on	ADP
easat-4026	36	24	simplicial	simplicial	ADJ
easat-4026	36	25	cohomology	cohomology	NOUN
easat-4026	36	26	for	for	ADP
easat-4026	36	27	𝒜∞algebras	𝒜∞algebras	PRON
easat-4026	37	1	[	[	X
easat-4026	37	2	13,14	13,14	X
easat-4026	37	3	]	]	PUNCT
easat-4026	37	4	expanded	expand	VERB
easat-4026	37	5	our	our	PRON
easat-4026	37	6	understanding	understanding	NOUN
easat-4026	37	7	of	of	ADP
easat-4026	37	8	these	these	DET
easat-4026	37	9	structures	structure	NOUN
easat-4026	37	10	.	.	PUNCT
easat-4026	38	1	the	the	DET
easat-4026	38	2	lack	lack	NOUN
easat-4026	38	3	of	of	ADP
easat-4026	38	4	excision	excision	NOUN
easat-4026	38	5	in	in	ADP
easat-4026	38	6	specific	specific	ADJ
easat-4026	38	7	contexts	context	NOUN
easat-4026	38	8	leads	lead	VERB
easat-4026	38	9	to	to	ADP
easat-4026	38	10	long	long	ADJ
easat-4026	38	11	exact	exact	ADJ
easat-4026	38	12	sequences	sequence	NOUN
easat-4026	38	13	in	in	ADP
easat-4026	38	14	homological	homological	ADJ
easat-4026	38	15	categories	category	NOUN
easat-4026	38	16	but	but	CCONJ
easat-4026	38	17	not	not	PART
easat-4026	38	18	in	in	ADP
easat-4026	38	19	homotopic	homotopic	ADJ
easat-4026	38	20	categories	category	NOUN
easat-4026	38	21	.	.	PUNCT
easat-4026	39	1	this	this	DET
easat-4026	39	2	paper	paper	NOUN
easat-4026	39	3	presents	present	VERB
easat-4026	39	4	various	various	ADJ
easat-4026	39	5	definitions	definition	NOUN
easat-4026	39	6	of	of	ADP
easat-4026	39	7	homology	homology	NOUN
easat-4026	39	8	for	for	ADP
easat-4026	39	9	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	39	10	,	,	PUNCT
easat-4026	39	11	examining	examine	VERB
easat-4026	39	12	the	the	DET
easat-4026	39	13	simplicial	simplicial	ADJ
easat-4026	39	14	homology	homology	NOUN
easat-4026	39	15	of	of	ADP
easat-4026	39	16	these	these	DET
easat-4026	39	17	structures	structure	NOUN
easat-4026	39	18	and	and	CCONJ
easat-4026	39	19	providing	provide	VERB
easat-4026	39	20	reliable	reliable	ADJ
easat-4026	39	21	results	result	NOUN
easat-4026	39	22	for	for	ADP
easat-4026	39	23	their	their	PRON
easat-4026	39	24	excision	excision	NOUN
easat-4026	39	25	theory	theory	NOUN
easat-4026	39	26	.	.	PUNCT
easat-4026	40	1	we	we	PRON
easat-4026	40	2	explored	explore	VERB
easat-4026	40	3	the	the	DET
easat-4026	40	4	excision	excision	NOUN
easat-4026	40	5	theory	theory	NOUN
easat-4026	40	6	of	of	ADP
easat-4026	40	7	simplicial	simplicial	ADJ
easat-4026	40	8	homology	homology	NOUN
easat-4026	40	9	for	for	ADP
easat-4026	40	10	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	40	11	,	,	PUNCT
easat-4026	40	12	establishing	establish	VERB
easat-4026	40	13	the	the	DET
easat-4026	40	14	relationship	relationship	NOUN
easat-4026	40	15	between	between	ADP
easat-4026	40	16	bar	bar	NOUN
easat-4026	40	17	homology	homology	NOUN
easat-4026	40	18	ℋ𝐵𝑛(ℐ	ℋ𝐵𝑛(ℐ	NOUN
easat-4026	40	19	)	)	PUNCT
easat-4026	40	20	and	and	CCONJ
easat-4026	40	21	simplicial	simplicial	ADJ
easat-4026	40	22	homology	homology	NOUN
easat-4026	40	23	ℋℋ𝑛(ℐ	ℋℋ𝑛(ℐ	NOUN
easat-4026	40	24	)	)	PUNCT
easat-4026	40	25	through	through	ADP
easat-4026	40	26	an	an	DET
easat-4026	40	27	exact	exact	ADJ
easat-4026	40	28	sequence	sequence	NOUN
easat-4026	40	29	:	:	PUNCT
easat-4026	40	30	…	…	PUNCT
easat-4026	40	31	←	←	PROPN
easat-4026	40	32	𝐻𝑛−1(ℐ	𝐻𝑛−1(ℐ	PROPN
easat-4026	40	33	)	)	PUNCT
easat-4026	40	34	←	←	PROPN
easat-4026	40	35	ℋ𝐵𝑛−1(ℐ	ℋ𝐵𝑛−1(ℐ	NUM
easat-4026	40	36	)	)	PUNCT
easat-4026	40	37	←	←	PROPN
easat-4026	40	38	ℋℋ𝑛(ℐ	ℋℋ𝑛(ℐ	NOUN
easat-4026	40	39	)	)	PUNCT
easat-4026	40	40	←	←	PROPN
easat-4026	40	41	𝐻𝑛(ℐ	𝐻𝑛(ℐ	PROPN
easat-4026	40	42	)	)	PUNCT
easat-4026	40	43	←	←	PROPN
easat-4026	40	44	ℋ𝐵𝑛(ℐ	ℋ𝐵𝑛(ℐ	PROPN
easat-4026	40	45	)	)	PUNCT
easat-4026	40	46	←	←	PROPN
easat-4026	40	47	ℋℋ𝑛+1(ℐ	ℋℋ𝑛+1(ℐ	PROPN
easat-4026	40	48	)	)	PUNCT
easat-4026	40	49	←	←	PROPN
easat-4026	40	50	…	…	PUNCT
easat-4026	40	51	..	..	PUNCT
easat-4026	40	52	additionally	additionally	ADV
easat-4026	40	53	,	,	PUNCT
easat-4026	40	54	we	we	PRON
easat-4026	40	55	proved	prove	VERB
easat-4026	40	56	that	that	SCONJ
easat-4026	40	57	the	the	DET
easat-4026	40	58	quasi	quasi	NOUN
easat-4026	40	59	-	-	NOUN
easat-4026	40	60	isomorphisms	isomorphism	NOUN
easat-4026	40	61	for	for	ADP
easat-4026	40	62	the	the	DET
easat-4026	40	63	following	follow	VERB
easat-4026	40	64	maps	map	NOUN
easat-4026	40	65	:	:	PUNCT
easat-4026	40	66	𝑖	𝑖	NUM
easat-4026	40	67	:	:	PUNCT
easat-4026	40	68	(	(	PUNCT
easat-4026	40	69	ℛ	ℛ	PROPN
easat-4026	40	70	⊗	⊗	PROPN
easat-4026	40	71	ℐ⨂∗	ℐ⨂∗	X
easat-4026	40	72	,	,	PUNCT
easat-4026	40	73	𝜌∗)⊗	𝜌∗)⊗	PUNCT
easat-4026	40	74	𝒢	𝒢	PROPN
easat-4026	40	75	↪	↪	PROPN
easat-4026	40	76	(	(	PUNCT
easat-4026	40	77	ℛ	ℛ	NOUN
easat-4026	40	78	⊗𝒜⨂∗	⊗𝒜⨂∗	NOUN
easat-4026	40	79	,	,	PUNCT
easat-4026	40	80	𝜌∗)⊗	𝜌∗)⊗	PROPN
easat-4026	40	81	𝒢	𝒢	NOUN
easat-4026	40	82	,	,	PUNCT
easat-4026	40	83	𝑖′	𝑖′	NUM
easat-4026	40	84	:	:	PUNCT
easat-4026	40	85	(	(	PUNCT
easat-4026	40	86	ℛ	ℛ	PROPN
easat-4026	40	87	⊗	⊗	PROPN
easat-4026	40	88	ℐ⨂∗	ℐ⨂∗	NOUN
easat-4026	40	89	,	,	PUNCT
easat-4026	40	90	𝜌∗	𝜌∗	NOUN
easat-4026	40	91	′)⊗	′)⊗	NOUN
easat-4026	40	92	𝒢	𝒢	PROPN
easat-4026	40	93	↪	↪	PROPN
easat-4026	40	94	(	(	PUNCT
easat-4026	40	95	ℛ	ℛ	NOUN
easat-4026	40	96	⊗𝒜⨂∗	⊗𝒜⨂∗	NOUN
easat-4026	40	97	,	,	PUNCT
easat-4026	40	98	𝜌∗	𝜌∗	NOUN
easat-4026	40	99	′)⊗	′)⊗	NOUN
easat-4026	40	100	𝒢	𝒢	PROPN
easat-4026	40	101	,	,	PUNCT
easat-4026	40	102	are	be	AUX
easat-4026	40	103	quasi	quasi	NOUN
easat-4026	40	104	-	-	NOUN
easat-4026	40	105	isomorphisms	isomorphism	NOUN
easat-4026	40	106	,	,	PUNCT
easat-4026	40	107	as	as	SCONJ
easat-4026	40	108	are	be	AUX
easat-4026	40	109	the	the	DET
easat-4026	40	110	two	two	NUM
easat-4026	40	111	maps	map	NOUN
easat-4026	40	112	:	:	PUNCT
easat-4026	40	113	𝜋	𝜋	X
easat-4026	40	114	:	:	PUNCT
easat-4026	40	115	(	(	PUNCT
easat-4026	40	116	ℬ	ℬ	NOUN
easat-4026	40	117	⊗𝒜⊗∗	⊗𝒜⊗∗	NUM
easat-4026	40	118	,	,	PUNCT
easat-4026	40	119	𝜌∗)⊗	𝜌∗)⊗	PUNCT
easat-4026	41	1	𝒢	𝒢	NOUN
easat-4026	41	2	→	→	SYM
easat-4026	41	3	(	(	PUNCT
easat-4026	41	4	ℬ	ℬ	NOUN
easat-4026	41	5	⊗ℬ⊗∗	⊗ℬ⊗∗	NOUN
easat-4026	41	6	,	,	PUNCT
easat-4026	41	7	𝜌∗)⊗	𝜌∗)⊗	PUNCT
easat-4026	41	8	𝒢	𝒢	NOUN
easat-4026	41	9	,	,	PUNCT
easat-4026	41	10	𝜋′	𝜋′	NUM
easat-4026	41	11	:	:	PUNCT
easat-4026	41	12	(	(	PUNCT
easat-4026	41	13	ℬ	ℬ	NOUN
easat-4026	41	14	⊗𝒜⊗∗	⊗𝒜⊗∗	NUM
easat-4026	41	15	,	,	PUNCT
easat-4026	41	16	𝜌′∗)⊗	𝜌′∗)⊗	PROPN
easat-4026	41	17	𝒢	𝒢	PROPN
easat-4026	41	18	→	→	SYM
easat-4026	41	19	(	(	PUNCT
easat-4026	41	20	ℬ	ℬ	NOUN
easat-4026	41	21	⊗ℬ⊗∗	⊗ℬ⊗∗	NOUN
easat-4026	41	22	,	,	PUNCT
easat-4026	41	23	𝜌′∗)⊗	𝜌′∗)⊗	PROPN
easat-4026	41	24	𝒢	𝒢	PROPN
easat-4026	41	25	,	,	PUNCT
easat-4026	41	26	these	these	DET
easat-4026	41	27	results	result	NOUN
easat-4026	41	28	enable	enable	VERB
easat-4026	41	29	us	we	PRON
easat-4026	41	30	to	to	PART
easat-4026	41	31	construct	construct	VERB
easat-4026	41	32	a	a	DET
easat-4026	41	33	commutative	commutative	ADJ
easat-4026	41	34	diagram	diagram	NOUN
easat-4026	41	35	,	,	PUNCT
easat-4026	41	36	elucidating	elucidate	VERB
easat-4026	41	37	the	the	DET
easat-4026	41	38	intricate	intricate	ADJ
easat-4026	41	39	relationships	relationship	NOUN
easat-4026	41	40	between	between	ADP
easat-4026	41	41	these	these	DET
easat-4026	41	42	homological	homological	ADJ
easat-4026	41	43	structures	structure	NOUN
easat-4026	41	44	:	:	PUNCT
easat-4026	41	45	0	0	NUM
easat-4026	41	46	0	0	NUM
easat-4026	41	47	→	→	SYM
easat-4026	41	48	→	→	X
easat-4026	41	49	(	(	PUNCT
easat-4026	41	50	ℐ	ℐ	PRON
easat-4026	41	51	⊗𝒜⨂	⊗𝒜⨂	PROPN
easat-4026	41	52	∗	∗	NOUN
easat-4026	41	53	,	,	PUNCT
easat-4026	41	54	𝜌∗	𝜌∗	NOUN
easat-4026	41	55	)	)	PUNCT
easat-4026	41	56	⊗	⊗	PROPN
easat-4026	41	57	𝒢	𝒢	PROPN
easat-4026	41	58	|𝑗	|𝑗	NOUN
easat-4026	41	59	𝑘𝑒𝑟	𝑘𝑒𝑟	NOUN
easat-4026	41	60	(	(	PUNCT
easat-4026	41	61	𝜋	𝜋	NOUN
easat-4026	41	62	)	)	PUNCT
easat-4026	41	63	→	→	SYM
easat-4026	41	64	→	→	SYM
easat-4026	41	65	(	(	PUNCT
easat-4026	41	66	𝒜	𝒜	NOUN
easat-4026	41	67	⊗𝒜⨂	⊗𝒜⨂	PROPN
easat-4026	41	68	∗	∗	NOUN
easat-4026	41	69	,	,	PUNCT
easat-4026	41	70	𝜌∗	𝜌∗	NOUN
easat-4026	41	71	)	)	PUNCT
easat-4026	41	72	⊗	⊗	NOUN
easat-4026	42	1	𝒢	𝒢	NOUN
easat-4026	42	2	|	|	NOUN
easat-4026	42	3	=	=	SYM
easat-4026	42	4	(	(	PUNCT
easat-4026	42	5	𝒜	𝒜	NOUN
easat-4026	42	6	⊗𝒜⨂	⊗𝒜⨂	PROPN
easat-4026	42	7	∗	∗	NOUN
easat-4026	42	8	,	,	PUNCT
easat-4026	42	9	𝜌∗	𝜌∗	NOUN
easat-4026	42	10	)	)	PUNCT
easat-4026	42	11	⊗	⊗	PROPN
easat-4026	42	12	𝒢	𝒢	PROPN
easat-4026	42	13	→	→	SYM
easat-4026	42	14	𝜋	𝜋	X
easat-4026	42	15	→	→	PUNCT
easat-4026	42	16	(	(	PUNCT
easat-4026	42	17	ℬ	ℬ	SYM
easat-4026	42	18	⊗𝒜⨂	⊗𝒜⨂	PROPN
easat-4026	42	19	∗	∗	NOUN
easat-4026	42	20	,	,	PUNCT
easat-4026	42	21	𝜌∗	𝜌∗	NOUN
easat-4026	42	22	)	)	PUNCT
easat-4026	42	23	⊗	⊗	NOUN
easat-4026	43	1	𝒢	𝒢	NOUN
easat-4026	43	2	|𝜋1	|𝜋1	NOUN
easat-4026	43	3	(	(	PUNCT
easat-4026	43	4	ℬ	ℬ	PROPN
easat-4026	43	5	⊗𝒜⨂∗	⊗𝒜⨂∗	NOUN
easat-4026	43	6	,	,	PUNCT
easat-4026	43	7	𝜌∗	𝜌∗	NOUN
easat-4026	43	8	)	)	PUNCT
easat-4026	43	9	⊗	⊗	PROPN
easat-4026	43	10	𝒢	𝒢	NOUN
easat-4026	43	11	→	→	SYM
easat-4026	43	12	→	→	SYM
easat-4026	43	13	0	0	NUM
easat-4026	43	14	0	0	NUM
easat-4026	43	15	.	.	PUNCT
easat-4026	44	1	2	2	X
easat-4026	44	2	.	.	X
easat-4026	44	3	homology	homology	NOUN
easat-4026	44	4	theory	theory	NOUN
easat-4026	44	5	of	of	ADP
easat-4026	44	6	𝓐∞-algebras	𝓐∞-algebras	PROPN
easat-4026	44	7	in	in	ADP
easat-4026	44	8	this	this	DET
easat-4026	44	9	section	section	NOUN
easat-4026	44	10	,	,	PUNCT
easat-4026	44	11	we	we	PRON
easat-4026	44	12	will	will	AUX
easat-4026	44	13	explore	explore	VERB
easat-4026	44	14	the	the	DET
easat-4026	44	15	fundamentals	fundamental	NOUN
easat-4026	44	16	and	and	CCONJ
easat-4026	44	17	definitions	definition	NOUN
easat-4026	44	18	related	relate	VERB
easat-4026	44	19	to	to	ADP
easat-4026	44	20	the	the	DET
easat-4026	44	21	homology	homology	NOUN
easat-4026	44	22	theory	theory	NOUN
easat-4026	44	23	of	of	ADP
easat-4026	44	24	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	44	25	.	.	PUNCT
easat-4026	45	1	we	we	PRON
easat-4026	45	2	begin	begin	VERB
easat-4026	45	3	by	by	ADP
easat-4026	45	4	introducing	introduce	VERB
easat-4026	45	5	the	the	DET
easat-4026	45	6	basic	basic	ADJ
easat-4026	45	7	definitions	definition	NOUN
easat-4026	45	8	and	and	CCONJ
easat-4026	45	9	concepts	concept	NOUN
easat-4026	45	10	of	of	ADP
easat-4026	45	11	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	45	12	.	.	PUNCT
easat-4026	46	1	following	follow	VERB
easat-4026	46	2	this	this	PRON
easat-4026	46	3	,	,	PUNCT
easat-4026	46	4	we	we	PRON
easat-4026	46	5	will	will	AUX
easat-4026	46	6	present	present	VERB
easat-4026	46	7	the	the	DET
easat-4026	46	8	simple	simple	ADJ
easat-4026	46	9	homology	homology	NOUN
easat-4026	46	10	of	of	ADP
easat-4026	46	11	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	46	12	.	.	PUNCT
easat-4026	47	1	firstly	firstly	ADV
easat-4026	47	2	,	,	PUNCT
easat-4026	47	3	we	we	PRON
easat-4026	47	4	define	define	VERB
easat-4026	47	5	an	an	DET
easat-4026	47	6	algebra	algebra	NOUN
easat-4026	47	7	over	over	ADP
easat-4026	47	8	the	the	DET
easat-4026	47	9	field	field	NOUN
easat-4026	47	10	ℛ	ℛ	PROPN
easat-4026	47	11	,	,	PUNCT
easat-4026	47	12	focusing	focus	VERB
easat-4026	47	13	on	on	ADP
easat-4026	47	14	properties	property	NOUN
easat-4026	47	15	.	.	PUNCT
easat-4026	48	1	2.1	2.1	NUM
easat-4026	48	2	.	.	PUNCT
easat-4026	48	3	definition	definition	NOUN
easat-4026	48	4	[	[	X
easat-4026	48	5	2	2	X
easat-4026	48	6	]	]	PUNCT
easat-4026	48	7	over	over	ADP
easat-4026	48	8	the	the	DET
easat-4026	48	9	field	field	NOUN
easat-4026	48	10	ℛ	ℛ	PROPN
easat-4026	48	11	,	,	PUNCT
easat-4026	48	12	algebra	algebra	NOUN
easat-4026	48	13	is	be	AUX
easat-4026	48	14	represented	represent	VERB
easat-4026	48	15	by	by	ADP
easat-4026	48	16	the	the	DET
easat-4026	48	17	linear	linear	PROPN
easat-4026	48	18	vector	vector	NOUN
easat-4026	48	19	space	space	NOUN
easat-4026	48	20	𝒳	𝒳	PROPN
easat-4026	48	21	,	,	PUNCT
easat-4026	48	22	which	which	PRON
easat-4026	48	23	has	have	VERB
easat-4026	48	24	the	the	DET
easat-4026	48	25	multiplication	multiplication	NOUN
easat-4026	48	26	function	function	VERB
easat-4026	48	27	𝒯:𝒳	𝒯:𝒳	X
easat-4026	48	28	×𝒳	×𝒳	PROPN
easat-4026	48	29	→	→	SYM
easat-4026	48	30	𝒳	𝒳	PROPN
easat-4026	48	31	,	,	PUNCT
easat-4026	48	32	(	(	PUNCT
easat-4026	48	33	𝓋,𝓊	𝓋,𝓊	NOUN
easat-4026	48	34	)	)	PUNCT
easat-4026	48	35	↦	↦	VERB
easat-4026	48	36	𝓋𝓊	𝓋𝓊	ADP
easat-4026	48	37	,	,	PUNCT
easat-4026	48	38	that	that	PRON
easat-4026	48	39	which	which	PRON
easat-4026	48	40	𝒯	𝒯	PROPN
easat-4026	48	41	is	be	AUX
easat-4026	48	42	distributed	distribute	VERB
easat-4026	48	43	and	and	CCONJ
easat-4026	48	44	linear	linear	ADJ
easat-4026	48	45	in	in	ADP
easat-4026	48	46	the	the	DET
easat-4026	48	47	two	two	NUM
easat-4026	48	48	variables	variable	NOUN
easat-4026	48	49	.	.	PUNCT
easat-4026	49	1	that	that	PRON
easat-4026	49	2	applies	apply	VERB
easat-4026	49	3	for	for	ADP
easat-4026	49	4	all	all	PRON
easat-4026	49	5	𝓋	𝓋	NOUN
easat-4026	49	6	,	,	PUNCT
easat-4026	49	7	𝓊,𝓌	𝓊,𝓌	NOUN
easat-4026	49	8	∈	∈	PROPN
easat-4026	49	9	𝒳	𝒳	PROPN
easat-4026	49	10	,	,	PUNCT
easat-4026	49	11	𝛼	𝛼	PROPN
easat-4026	49	12	∈	∈	NOUN
easat-4026	49	13	ℛ	ℛ	ADJ
easat-4026	49	14	▪	▪	ADJ
easat-4026	49	15	𝓌(𝓋	𝓌(𝓋	PROPN
easat-4026	49	16	+	+	SYM
easat-4026	49	17	𝓊	𝓊	X
easat-4026	49	18	)	)	PUNCT
easat-4026	49	19	=	=	SYM
easat-4026	50	1	𝓌𝓊	𝓌𝓊	PROPN
easat-4026	51	1	+	+	NOUN
easat-4026	51	2	𝓌𝓊.	𝓌𝓊.	ADJ
easat-4026	51	3	▪	▪	ADV
easat-4026	51	4	(	(	PUNCT
easat-4026	51	5	𝓋	𝓋	NOUN
easat-4026	51	6	+	+	NOUN
easat-4026	51	7	𝓊)𝓌	𝓊)𝓌	NOUN
easat-4026	51	8	=	=	NOUN
easat-4026	51	9	𝓋𝓌	𝓋𝓌	ADP
easat-4026	51	10	+	+	PROPN
easat-4026	51	11	𝓊𝓌.	𝓊𝓌.	X
easat-4026	51	12	▪	▪	X
easat-4026	51	13	𝛼(𝓋𝓊	𝛼(𝓋𝓊	NOUN
easat-4026	51	14	)	)	PUNCT
easat-4026	51	15	=	=	SYM
easat-4026	51	16	(	(	PUNCT
easat-4026	51	17	𝛼𝓋)𝓊	𝛼𝓋)𝓊	PROPN
easat-4026	51	18	=	=	SYM
easat-4026	51	19	𝓋(𝛼𝓊	𝓋(𝛼𝓊	PROPN
easat-4026	51	20	)	)	PUNCT
easat-4026	51	21	.	.	PUNCT
easat-4026	52	1	following	follow	VERB
easat-4026	52	2	,	,	PUNCT
easat-4026	52	3	we	we	PRON
easat-4026	52	4	define	define	VERB
easat-4026	52	5	graded	grade	VERB
easat-4026	52	6	vector	vector	NOUN
easat-4026	52	7	spaces	space	NOUN
easat-4026	52	8	where	where	SCONJ
easat-4026	52	9	we	we	PRON
easat-4026	52	10	specify	specify	VERB
easat-4026	52	11	the	the	DET
easat-4026	52	12	vector	vector	NOUN
easat-4026	52	13	space	space	NOUN
easat-4026	52	14	𝒳	𝒳	PROPN
easat-4026	52	15	and	and	CCONJ
easat-4026	52	16	related	relate	VERB
easat-4026	52	17	graded	grade	VERB
easat-4026	52	18	vector	vector	NOUN
easat-4026	52	19	spaces	space	NOUN
easat-4026	52	20	𝒳𝚤.	𝒳𝚤.	VERB
easat-4026	52	21	we	we	PRON
easat-4026	52	22	describe	describe	VERB
easat-4026	52	23	how	how	SCONJ
easat-4026	52	24	to	to	PART
easat-4026	52	25	handle	handle	VERB
easat-4026	52	26	homogeneous	homogeneous	ADJ
easat-4026	52	27	elements	element	NOUN
easat-4026	52	28	in	in	ADP
easat-4026	52	29	these	these	DET
easat-4026	52	30	spaces	space	NOUN
easat-4026	52	31	.	.	PUNCT
easat-4026	53	1	2.2	2.2	NUM
easat-4026	53	2	.	.	PUNCT
easat-4026	53	3	definition	definition	NOUN
easat-4026	53	4	[	[	X
easat-4026	53	5	3	3	X
easat-4026	53	6	]	]	PUNCT
easat-4026	53	7	suppose	suppose	VERB
easat-4026	53	8	ℐ	ℐ	PRON
easat-4026	53	9	denotes	denote	VERB
easat-4026	53	10	the	the	DET
easat-4026	53	11	set	set	NOUN
easat-4026	53	12	of	of	ADP
easat-4026	53	13	index	index	NOUN
easat-4026	53	14	.	.	PUNCT
easat-4026	54	1	the	the	DET
easat-4026	54	2	vector	vector	NOUN
easat-4026	54	3	space	space	NOUN
easat-4026	54	4	𝒳	𝒳	PROPN
easat-4026	54	5	that	that	PRON
easat-4026	54	6	has	have	VERB
easat-4026	54	7	the	the	DET
easat-4026	54	8	grade	grade	NOUN
easat-4026	54	9	ℐ	ℐ	PROPN
easat-4026	54	10	,	,	PUNCT
easat-4026	54	11	known	know	VERB
easat-4026	54	12	as	as	ADP
easat-4026	54	13	the	the	DET
easat-4026	54	14	ℐ-graded	ℐ-graded	ADJ
easat-4026	54	15	vector	vector	NOUN
easat-4026	54	16	space	space	NOUN
easat-4026	54	17	,	,	PUNCT
easat-4026	54	18	takes	take	VERB
easat-4026	54	19	the	the	DET
easat-4026	54	20	following	follow	VERB
easat-4026	54	21	form	form	NOUN
easat-4026	54	22	𝒳	𝒳	PROPN
easat-4026	54	23	=	=	SYM
easat-4026	54	24	⨁	⨁	PROPN
easat-4026	54	25	𝚤	𝚤	ADP
easat-4026	54	26	∈	∈	PROPN
easat-4026	54	27	ℐ	ℐ	PROPN
easat-4026	54	28	𝒳𝚤	𝒳𝚤	PROPN
easat-4026	54	29	,	,	PUNCT
easat-4026	54	30	hence	hence	ADV
easat-4026	54	31	for	for	ADP
easat-4026	54	32	each	each	DET
easat-4026	54	33	𝚤	𝚤	NOUN
easat-4026	54	34	,	,	PUNCT
easat-4026	54	35	and	and	CCONJ
easat-4026	54	36	then	then	ADV
easat-4026	54	37	𝒳𝚤	𝒳𝚤	PROPN
easat-4026	54	38	would	would	AUX
easat-4026	54	39	be	be	AUX
easat-4026	54	40	a	a	DET
easat-4026	54	41	vector	vector	NOUN
easat-4026	54	42	9474	9474	NUM
easat-4026	54	43	edelweiss	edelweiss	PROPN
easat-4026	54	44	applied	apply	VERB
easat-4026	54	45	science	science	NOUN
easat-4026	54	46	and	and	CCONJ
easat-4026	54	47	technology	technology	NOUN
easat-4026	54	48	issn	issn	PROPN
easat-4026	54	49	:	:	PUNCT
easat-4026	54	50	2576	2576	NUM
easat-4026	54	51	-	-	SYM
easat-4026	54	52	8484	8484	NUM
easat-4026	54	53	vol	vol	NOUN
easat-4026	54	54	.	.	PROPN
easat-4026	54	55	8	8	NUM
easat-4026	54	56	,	,	PUNCT
easat-4026	54	57	no	no	INTJ
easat-4026	54	58	.	.	NOUN
easat-4026	55	1	6	6	NUM
easat-4026	55	2	:	:	SYM
easat-4026	55	3	9472	9472	NUM
easat-4026	55	4	-	-	SYM
easat-4026	55	5	9486	9486	NUM
easat-4026	55	6	,	,	PUNCT
easat-4026	55	7	2024	2024	NUM
easat-4026	55	8	doi	doi	NOUN
easat-4026	55	9	:	:	PUNCT
easat-4026	55	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	55	11	©	©	PROPN
easat-4026	55	12	2024	2024	NUM
easat-4026	55	13	by	by	ADP
easat-4026	55	14	the	the	DET
easat-4026	55	15	authors	author	NOUN
easat-4026	55	16	;	;	PUNCT
easat-4026	55	17	licensee	licensee	PROPN
easat-4026	55	18	learning	learn	VERB
easat-4026	55	19	gate	gate	NOUN
easat-4026	55	20	space	space	NOUN
easat-4026	55	21	.	.	PUNCT
easat-4026	56	1	the	the	DET
easat-4026	56	2	elements	element	NOUN
easat-4026	56	3	𝓍	𝓍	X
easat-4026	56	4	∈	∈	PROPN
easat-4026	57	1	𝒳𝚤	𝒳𝚤	NOUN
easat-4026	57	2	are	be	AUX
easat-4026	57	3	hence	hence	ADV
easat-4026	57	4	known	know	VERB
easat-4026	57	5	as	as	ADP
easat-4026	57	6	homogeneous	homogeneous	ADJ
easat-4026	57	7	elements	element	NOUN
easat-4026	57	8	with	with	ADP
easat-4026	57	9	degree	degree	NOUN
easat-4026	57	10	𝚤	𝚤	PROPN
easat-4026	57	11	and	and	CCONJ
easat-4026	57	12	denoted	denote	VERB
easat-4026	57	13	by	by	ADP
easat-4026	57	14	𝑑𝑒𝑔	𝑑𝑒𝑔	PROPN
easat-4026	57	15	𝓍	𝓍	X
easat-4026	57	16	=	=	PUNCT
easat-4026	57	17	𝚤	𝚤	PROPN
easat-4026	57	18	or	or	CCONJ
easat-4026	57	19	|𝓍|	|𝓍|	X
easat-4026	57	20	=	=	PUNCT
easat-4026	57	21	𝚤.	𝚤.	PROPN
easat-4026	57	22	next	next	ADV
easat-4026	57	23	,	,	PUNCT
easat-4026	57	24	we	we	PRON
easat-4026	57	25	introduce	introduce	VERB
easat-4026	57	26	the	the	DET
easat-4026	57	27	tensor	tensor	NOUN
easat-4026	57	28	product	product	NOUN
easat-4026	57	29	of	of	ADP
easat-4026	57	30	vector	vector	NOUN
easat-4026	57	31	spaces	space	NOUN
easat-4026	57	32	𝒳	𝒳	PROPN
easat-4026	57	33	and	and	CCONJ
easat-4026	57	34	𝒴	𝒴	PROPN
easat-4026	57	35	over	over	ADP
easat-4026	57	36	a	a	DET
easat-4026	57	37	field	field	NOUN
easat-4026	57	38	𝔉.	𝔉.	NOUN
easat-4026	57	39	we	we	PRON
easat-4026	57	40	define	define	VERB
easat-4026	57	41	the	the	DET
easat-4026	57	42	bilinear	bilinear	NOUN
easat-4026	57	43	map	map	NOUN
easat-4026	57	44	and	and	CCONJ
easat-4026	57	45	discuss	discuss	VERB
easat-4026	57	46	the	the	DET
easat-4026	57	47	properties	property	NOUN
easat-4026	57	48	of	of	ADP
easat-4026	57	49	these	these	DET
easat-4026	57	50	tensor	tensor	NOUN
easat-4026	57	51	spaces	space	NOUN
easat-4026	57	52	.	.	PUNCT
easat-4026	58	1	2.3	2.3	NUM
easat-4026	58	2	.	.	PUNCT
easat-4026	58	3	definition	definition	NOUN
easat-4026	58	4	[	[	X
easat-4026	58	5	3	3	X
easat-4026	58	6	]	]	PUNCT
easat-4026	58	7	if	if	SCONJ
easat-4026	58	8	the	the	DET
easat-4026	58	9	vector	vector	NOUN
easat-4026	58	10	spaces	space	VERB
easat-4026	58	11	𝒳,𝒴	𝒳,𝒴	VERB
easat-4026	58	12	over	over	ADP
easat-4026	58	13	a	a	DET
easat-4026	58	14	field	field	NOUN
easat-4026	58	15	𝔉	𝔉	NOUN
easat-4026	58	16	with	with	ADP
easat-4026	58	17	elements	element	NOUN
easat-4026	58	18	{	{	PUNCT
easat-4026	58	19	𝓍𝚤}𝚤∈ℐ	𝓍𝚤}𝚤∈ℐ	PROPN
easat-4026	58	20	and	and	CCONJ
easat-4026	58	21	{	{	PUNCT
easat-4026	58	22	𝓎ℓ}ℓ∈ℒ	𝓎ℓ}ℓ∈ℒ	PROPN
easat-4026	58	23	;	;	PUNCT
easat-4026	58	24	then	then	ADV
easat-4026	58	25	,	,	PUNCT
easat-4026	58	26	the	the	DET
easat-4026	58	27	tensor	tensor	NOUN
easat-4026	58	28	product	product	NOUN
easat-4026	58	29	𝒳	𝒳	PROPN
easat-4026	58	30	⊗𝒴	⊗𝒴	NOUN
easat-4026	58	31	is	be	AUX
easat-4026	58	32	defined	define	VERB
easat-4026	58	33	as	as	ADP
easat-4026	58	34	a	a	DET
easat-4026	58	35	vector	vector	NOUN
easat-4026	58	36	space	space	NOUN
easat-4026	58	37	over	over	ADP
easat-4026	58	38	𝔉	𝔉	PROPN
easat-4026	58	39	with	with	ADP
easat-4026	58	40	the	the	DET
easat-4026	58	41	symbols	symbol	NOUN
easat-4026	58	42	{	{	PUNCT
easat-4026	58	43	𝓍𝚤	𝓍𝚤	NOUN
easat-4026	58	44	⊗𝓎ℓ	⊗𝓎ℓ	NOUN
easat-4026	58	45	,	,	PUNCT
easat-4026	58	46	∀𝚤	∀𝚤	PROPN
easat-4026	58	47	∈	∈	PROPN
easat-4026	58	48	ℐ	ℐ	PROPN
easat-4026	58	49	,	,	PUNCT
easat-4026	58	50	ℓ	ℓ	PROPN
easat-4026	58	51	∈	∈	PROPN
easat-4026	58	52	ℒ	ℒ	PROPN
easat-4026	58	53	}	}	PUNCT
easat-4026	58	54	as	as	ADP
easat-4026	58	55	its	its	PRON
easat-4026	58	56	basis	basis	NOUN
easat-4026	58	57	.	.	PUNCT
easat-4026	59	1	additionally	additionally	ADV
easat-4026	59	2	,	,	PUNCT
easat-4026	59	3	we	we	PRON
easat-4026	59	4	define	define	VERB
easat-4026	59	5	the	the	DET
easat-4026	59	6	bilinear	bilinear	NOUN
easat-4026	59	7	map	map	NOUN
easat-4026	59	8	𝒳	𝒳	PROPN
easat-4026	59	9	×	×	PROPN
easat-4026	59	10	𝒴	𝒴	PROPN
easat-4026	59	11	→	→	SYM
easat-4026	59	12	𝒳⊗𝒴	𝒳⊗𝒴	PROPN
easat-4026	59	13	as	as	ADP
easat-4026	59	14	the	the	DET
easat-4026	59	15	two	two	NUM
easat-4026	59	16	vectors	vector	NOUN
easat-4026	59	17	'	'	PART
easat-4026	59	18	combined	combine	VERB
easat-4026	59	19	tensor	tensor	NOUN
easat-4026	59	20	product	product	NOUN
easat-4026	59	21	𝓍	𝓍	X
easat-4026	59	22	=	=	PUNCT
easat-4026	59	23	∑	∑	PUNCT
easat-4026	59	24	𝒶𝚤𝓍𝚤𝚤	𝒶𝚤𝓍𝚤𝚤	PROPN
easat-4026	59	25	and	and	CCONJ
easat-4026	59	26	𝓎	𝓎	X
easat-4026	59	27	=	=	SYM
easat-4026	59	28	∑	∑	PROPN
easat-4026	59	29	𝒷ℓ𝓎ℓℓ	𝒷ℓ𝓎ℓℓ	PROPN
easat-4026	59	30	,	,	PUNCT
easat-4026	59	31	which	which	PRON
easat-4026	59	32	is	be	AUX
easat-4026	59	33	provided	provide	VERB
easat-4026	59	34	by	by	ADP
easat-4026	59	35	:	:	PUNCT
easat-4026	59	36	𝒳⊗𝒴	𝒳⊗𝒴	PROPN
easat-4026	59	37	=	=	SYM
easat-4026	59	38	(	(	PUNCT
easat-4026	59	39	∑	∑	PROPN
easat-4026	59	40	𝒶𝚤𝓍𝚤𝚤	𝒶𝚤𝓍𝚤𝚤	PROPN
easat-4026	59	41	)	)	PUNCT
easat-4026	60	1	⊗	⊗	PROPN
easat-4026	60	2	(	(	PUNCT
easat-4026	60	3	∑	∑	INTJ
easat-4026	60	4	𝒷ℓ𝓎ℓℓ	𝒷ℓ𝓎ℓℓ	PROPN
easat-4026	60	5	)	)	PUNCT
easat-4026	60	6	=	=	PUNCT
easat-4026	61	1	∑	∑	PUNCT
easat-4026	61	2	𝒶𝚤𝒷ℓ(𝓍𝚤	𝒶𝚤𝒷ℓ(𝓍𝚤	NOUN
easat-4026	61	3	⊗𝓎ℓ𝚤,ℓ	⊗𝓎ℓ𝚤,ℓ	ADJ
easat-4026	61	4	)	)	PUNCT
easat-4026	61	5	.	.	PUNCT
easat-4026	62	1	in	in	ADP
easat-4026	62	2	the	the	DET
easat-4026	62	3	following	follow	VERB
easat-4026	62	4	proposition	proposition	NOUN
easat-4026	62	5	,	,	PUNCT
easat-4026	62	6	we	we	PRON
easat-4026	62	7	will	will	AUX
easat-4026	62	8	prove	prove	VERB
easat-4026	62	9	the	the	DET
easat-4026	62	10	existence	existence	NOUN
easat-4026	62	11	of	of	ADP
easat-4026	62	12	a	a	DET
easat-4026	62	13	unique	unique	ADJ
easat-4026	62	14	linear	linear	NOUN
easat-4026	62	15	map	map	NOUN
easat-4026	62	16	between	between	ADP
easat-4026	62	17	the	the	DET
easat-4026	62	18	tensor	tensor	NOUN
easat-4026	62	19	products	product	NOUN
easat-4026	62	20	of	of	ADP
easat-4026	62	21	vector	vector	NOUN
easat-4026	62	22	spaces	space	NOUN
easat-4026	62	23	.	.	PUNCT
easat-4026	63	1	we	we	PRON
easat-4026	63	2	will	will	AUX
easat-4026	63	3	demonstrate	demonstrate	VERB
easat-4026	63	4	how	how	SCONJ
easat-4026	63	5	to	to	PART
easat-4026	63	6	establish	establish	VERB
easat-4026	63	7	this	this	DET
easat-4026	63	8	map	map	NOUN
easat-4026	63	9	and	and	CCONJ
easat-4026	63	10	its	its	PRON
easat-4026	63	11	distinctive	distinctive	ADJ
easat-4026	63	12	properties	property	NOUN
easat-4026	63	13	.	.	PUNCT
easat-4026	64	1	2.4	2.4	NUM
easat-4026	64	2	.	.	PUNCT
easat-4026	64	3	proposition	proposition	NOUN
easat-4026	64	4	assume	assume	VERB
easat-4026	64	5	that	that	SCONJ
easat-4026	64	6	ℳ	ℳ	PROPN
easat-4026	64	7	is	be	AUX
easat-4026	64	8	the	the	DET
easat-4026	64	9	vector	vector	NOUN
easat-4026	64	10	space	space	NOUN
easat-4026	64	11	.	.	PUNCT
easat-4026	65	1	a	a	DET
easat-4026	65	2	unique	unique	ADJ
easat-4026	65	3	linear	linear	NOUN
easat-4026	65	4	map	map	NOUN
easat-4026	65	5	ℜ′:𝒳	ℜ′:𝒳	PROPN
easat-4026	65	6	⊗𝒴	⊗𝒴	PROPN
easat-4026	65	7	→ℳ	→ℳ	PROPN
easat-4026	65	8	exists	exist	VERB
easat-4026	65	9	for	for	ADP
easat-4026	65	10	the	the	DET
easat-4026	65	11	bilinear	bilinear	NOUN
easat-4026	65	12	map	map	NOUN
easat-4026	65	13	ℜ:𝒳	ℜ:𝒳	PROPN
easat-4026	65	14	×	×	PROPN
easat-4026	65	15	𝒴	𝒴	PROPN
easat-4026	65	16	→ℳ	→ℳ	NOUN
easat-4026	65	17	,	,	PUNCT
easat-4026	65	18	such	such	ADJ
easat-4026	65	19	as	as	ADP
easat-4026	65	20	ℜ	ℜ	NOUN
easat-4026	65	21	=	=	PUNCT
easat-4026	65	22	ℜ′	ℜ′	PROPN
easat-4026	65	23	∘	∘	NUM
easat-4026	65	24	𝜙	𝜙	PROPN
easat-4026	65	25	and	and	CCONJ
easat-4026	65	26	𝜙	𝜙	PROPN
easat-4026	65	27	is	be	AUX
easat-4026	65	28	the	the	DET
easat-4026	65	29	normal	normal	ADJ
easat-4026	65	30	incorporation	incorporation	NOUN
easat-4026	65	31	of	of	ADP
easat-4026	65	32	𝒳	𝒳	NOUN
easat-4026	65	33	×𝒴	×𝒴	NOUN
easat-4026	65	34	in	in	ADP
easat-4026	65	35	𝒳	𝒳	PROPN
easat-4026	65	36	⊗𝒴.	⊗𝒴.	NUM
easat-4026	65	37	additionally	additionally	ADV
easat-4026	65	38	,	,	PUNCT
easat-4026	65	39	the	the	DET
easat-4026	65	40	universal	universal	ADJ
easat-4026	65	41	characteristic	characteristic	NOUN
easat-4026	65	42	is	be	AUX
easat-4026	65	43	satisfied	satisfy	VERB
easat-4026	65	44	by	by	ADP
easat-4026	65	45	the	the	DET
easat-4026	65	46	unique	unique	ADJ
easat-4026	65	47	isomorphism	isomorphism	NOUN
easat-4026	65	48	in	in	ADP
easat-4026	65	49	a	a	DET
easat-4026	65	50	vector	vector	NOUN
easat-4026	65	51	space	space	NOUN
easat-4026	65	52	with	with	ADP
easat-4026	65	53	a	a	DET
easat-4026	65	54	bilinear	bilinear	NOUN
easat-4026	65	55	map	map	NOUN
easat-4026	65	56	.	.	PUNCT
easat-4026	66	1	proof	proof	NOUN
easat-4026	66	2	:	:	PUNCT
easat-4026	66	3	the	the	DET
easat-4026	66	4	relation	relation	NOUN
easat-4026	66	5	ℜ′(𝓍𝚤	ℜ′(𝓍𝚤	NOUN
easat-4026	66	6	⊗𝓎ℓ	⊗𝓎ℓ	NOUN
easat-4026	66	7	)	)	PUNCT
easat-4026	66	8	=	=	SYM
easat-4026	67	1	𝛽(𝓍𝚤	𝛽(𝓍𝚤	PROPN
easat-4026	67	2	,	,	PUNCT
easat-4026	67	3	𝓎ℓ	𝓎ℓ	NOUN
easat-4026	67	4	)	)	PUNCT
easat-4026	67	5	is	be	AUX
easat-4026	67	6	a	a	DET
easat-4026	67	7	basic	basic	ADJ
easat-4026	67	8	𝒳⊗𝒴	𝒳⊗𝒴	NOUN
easat-4026	67	9	in	in	ADP
easat-4026	67	10	ℜ′.	ℜ′.	ADP
easat-4026	67	11	this	this	DET
easat-4026	67	12	map	map	NOUN
easat-4026	67	13	is	be	AUX
easat-4026	67	14	unique	unique	ADJ
easat-4026	67	15	because	because	SCONJ
easat-4026	67	16	it	it	PRON
easat-4026	67	17	satisfies	satisfy	VERB
easat-4026	67	18	the	the	DET
easat-4026	67	19	requirement	requirement	NOUN
easat-4026	67	20	for	for	ADP
easat-4026	67	21	ℜ	ℜ	NOUN
easat-4026	67	22	=	=	PUNCT
easat-4026	67	23	ℜ′	ℜ′	PROPN
easat-4026	67	24	∘	∘	PROPN
easat-4026	67	25	𝜙.	𝜙.	NOUN
easat-4026	67	26	now	now	ADV
easat-4026	67	27	,	,	PUNCT
easat-4026	67	28	let	let	VERB
easat-4026	67	29	𝒩	𝒩	PROPN
easat-4026	67	30	be	be	AUX
easat-4026	67	31	the	the	DET
easat-4026	67	32	vector	vector	NOUN
easat-4026	67	33	space	space	NOUN
easat-4026	67	34	with	with	ADP
easat-4026	67	35	the	the	DET
easat-4026	67	36	bilinear	bilinear	NOUN
easat-4026	67	37	map	map	NOUN
easat-4026	67	38	𝜔:𝒳	𝜔:𝒳	PROPN
easat-4026	67	39	×	×	PROPN
easat-4026	67	40	𝒴	𝒴	PROPN
easat-4026	67	41	→	→	SYM
easat-4026	67	42	𝒩.	𝒩.	PROPN
easat-4026	67	43	so	so	ADV
easat-4026	67	44	,	,	PUNCT
easat-4026	67	45	for	for	ADP
easat-4026	67	46	each	each	DET
easat-4026	67	47	bilinear	bilinear	NOUN
easat-4026	67	48	map	map	NOUN
easat-4026	68	1	ℜ:𝒳	ℜ:𝒳	PROPN
easat-4026	68	2	×	×	PROPN
easat-4026	68	3	𝒴	𝒴	PROPN
easat-4026	68	4	→ℳ	→ℳ	NOUN
easat-4026	68	5	,	,	PUNCT
easat-4026	68	6	there	there	PRON
easat-4026	68	7	is	be	VERB
easat-4026	68	8	a	a	DET
easat-4026	68	9	unique	unique	ADJ
easat-4026	68	10	linear	linear	NOUN
easat-4026	68	11	map	map	NOUN
easat-4026	68	12	ℜ′	ℜ′	PROPN
easat-4026	68	13	,	,	PUNCT
easat-4026	68	14	for	for	ADP
easat-4026	68	15	instance	instance	NOUN
easat-4026	68	16	ℜ	ℜ	PROPN
easat-4026	68	17	=	=	PUNCT
easat-4026	68	18	ℜ′	ℜ′	PROPN
easat-4026	68	19	∘	∘	PROPN
easat-4026	68	20	𝜙.	𝜙.	NOUN
easat-4026	68	21	suppose	suppose	VERB
easat-4026	68	22	that	that	SCONJ
easat-4026	68	23	ℳ	ℳ	PROPN
easat-4026	68	24	=	=	SYM
easat-4026	68	25	𝒳⊗𝒴	𝒳⊗𝒴	PROPN
easat-4026	68	26	and	and	CCONJ
easat-4026	68	27	ℜ	ℜ	PROPN
easat-4026	68	28	is	be	AUX
easat-4026	68	29	the	the	DET
easat-4026	68	30	map	map	NOUN
easat-4026	68	31	(	(	PUNCT
easat-4026	68	32	𝓍	𝓍	X
easat-4026	68	33	,	,	PUNCT
easat-4026	68	34	𝓎	𝓎	NUM
easat-4026	68	35	)	)	PUNCT
easat-4026	68	36	⟼	⟼	PRON
easat-4026	68	37	𝓍⊗𝓎.	𝓍⊗𝓎.	NOUN
easat-4026	68	38	therefore	therefore	ADV
easat-4026	68	39	,	,	PUNCT
easat-4026	68	40	through	through	ADP
easat-4026	68	41	the	the	DET
easat-4026	68	42	general	general	ADJ
easat-4026	68	43	property	property	NOUN
easat-4026	68	44	of	of	ADP
easat-4026	68	45	𝒩	𝒩	PROPN
easat-4026	68	46	,	,	PUNCT
easat-4026	68	47	a	a	DET
easat-4026	68	48	unique	unique	ADJ
easat-4026	68	49	linear	linear	NOUN
easat-4026	68	50	map	map	NOUN
easat-4026	68	51	ℜ′:𝒩	ℜ′:𝒩	NOUN
easat-4026	68	52	→	→	SYM
easat-4026	68	53	𝒳⊗𝒴	𝒳⊗𝒴	NOUN
easat-4026	68	54	exists	exist	VERB
easat-4026	68	55	as	as	ADP
easat-4026	68	56	ℜ	ℜ	ADV
easat-4026	68	57	=	=	PUNCT
easat-4026	68	58	ℜ′	ℜ′	PROPN
easat-4026	68	59	∘	∘	PROPN
easat-4026	68	60	𝜙.	𝜙.	NOUN
easat-4026	68	61	likewise	likewise	ADV
easat-4026	68	62	,	,	PUNCT
easat-4026	68	63	through	through	ADP
easat-4026	68	64	the	the	DET
easat-4026	68	65	general	general	ADJ
easat-4026	68	66	property	property	NOUN
easat-4026	68	67	of	of	ADP
easat-4026	68	68	𝒳	𝒳	PROPN
easat-4026	68	69	⊗𝒴	⊗𝒴	NOUN
easat-4026	68	70	,	,	PUNCT
easat-4026	68	71	there	there	PRON
easat-4026	68	72	exists	exist	VERB
easat-4026	68	73	the	the	DET
easat-4026	68	74	unique	unique	ADJ
easat-4026	68	75	linear	linear	NOUN
easat-4026	68	76	map	map	NOUN
easat-4026	68	77	ℰ′:𝒳	ℰ′:𝒳	NOUN
easat-4026	68	78	⊗𝒴	⊗𝒴	NOUN
easat-4026	68	79	→	→	SYM
easat-4026	68	80	𝒩	𝒩	PROPN
easat-4026	68	81	as	as	ADP
easat-4026	68	82	𝜔	𝜔	NOUN
easat-4026	68	83	=	=	NUM
easat-4026	68	84	ℰ′	ℰ′	PROPN
easat-4026	69	1	∘	∘	NUM
easat-4026	69	2	𝜙	𝜙	NOUN
easat-4026	70	1	so	so	ADV
easat-4026	70	2	𝜙	𝜙	NOUN
easat-4026	70	3	=	=	PUNCT
easat-4026	70	4	(	(	PUNCT
easat-4026	70	5	ℜ′	ℜ′	PROPN
easat-4026	70	6	∘	∘	PROPN
easat-4026	70	7	ℰ′	ℰ′	PROPN
easat-4026	70	8	)	)	PUNCT
easat-4026	70	9	∘	∘	PROPN
easat-4026	70	10	𝜙.	𝜙.	NOUN
easat-4026	70	11	therefore	therefore	ADV
easat-4026	70	12	,	,	PUNCT
easat-4026	70	13	ℜ′	ℜ′	PROPN
easat-4026	70	14	∘	∘	PROPN
easat-4026	70	15	ℰ′	ℰ′	PROPN
easat-4026	70	16	=	=	PUNCT
easat-4026	70	17	𝑖𝑑	𝑖𝑑	PROPN
easat-4026	70	18	and	and	CCONJ
easat-4026	70	19	ℜ′	ℜ′	PROPN
easat-4026	70	20	is	be	AUX
easat-4026	70	21	unique	unique	ADJ
easat-4026	70	22	isomorphism	isomorphism	NOUN
easat-4026	70	23	ℜ′:𝒩	ℜ′:𝒩	NOUN
easat-4026	70	24	∼	∼	NOUN
easat-4026	70	25	→	→	SYM
easat-4026	70	26	𝒳⊗𝒴.	𝒳⊗𝒴.	VERB
easat-4026	70	27	in	in	ADP
easat-4026	70	28	the	the	DET
easat-4026	70	29	following	following	NOUN
easat-4026	70	30	,	,	PUNCT
easat-4026	70	31	we	we	PRON
easat-4026	70	32	define	define	VERB
easat-4026	70	33	a	a	DET
easat-4026	70	34	graded	grade	VERB
easat-4026	70	35	algebra	algebra	NOUN
easat-4026	70	36	over	over	ADP
easat-4026	70	37	a	a	DET
easat-4026	70	38	field	field	NOUN
easat-4026	70	39	ℛ	ℛ	NOUN
easat-4026	70	40	and	and	CCONJ
easat-4026	70	41	how	how	SCONJ
easat-4026	70	42	to	to	PART
easat-4026	70	43	deal	deal	VERB
easat-4026	70	44	with	with	ADP
easat-4026	70	45	the	the	DET
easat-4026	70	46	resulting	result	VERB
easat-4026	70	47	multiplicity	multiplicity	NOUN
easat-4026	70	48	of	of	ADP
easat-4026	70	49	multiplication	multiplication	NOUN
easat-4026	70	50	maps	map	NOUN
easat-4026	70	51	in	in	ADP
easat-4026	70	52	this	this	DET
easat-4026	70	53	type	type	NOUN
easat-4026	70	54	of	of	ADP
easat-4026	70	55	algebra	algebra	NOUN
easat-4026	70	56	.	.	PUNCT
easat-4026	71	1	2.5	2.5	NUM
easat-4026	71	2	.	.	PUNCT
easat-4026	72	1	definition	definition	NOUN
easat-4026	72	2	[	[	X
easat-4026	72	3	14	14	NUM
easat-4026	72	4	]	]	PUNCT
easat-4026	72	5	suppose	suppose	VERB
easat-4026	72	6	ℳ	ℳ	PROPN
easat-4026	72	7	is	be	AUX
easat-4026	72	8	the	the	DET
easat-4026	72	9	graded	grade	VERB
easat-4026	72	10	vector	vector	NOUN
easat-4026	72	11	space	space	NOUN
easat-4026	72	12	on	on	ADP
easat-4026	72	13	the	the	DET
easat-4026	72	14	field	field	NOUN
easat-4026	72	15	ℛ	ℛ	PROPN
easat-4026	72	16	,	,	PUNCT
easat-4026	72	17	then	then	ADV
easat-4026	72	18	the	the	DET
easat-4026	72	19	graded	grade	VERB
easat-4026	72	20	algebra	algebra	NOUN
easat-4026	72	21	on	on	ADP
easat-4026	72	22	ℛ	ℛ	PROPN
easat-4026	72	23	is	be	AUX
easat-4026	72	24	defined	define	VERB
easat-4026	72	25	as	as	ADP
easat-4026	72	26	the	the	DET
easat-4026	72	27	algebra	algebra	NOUN
easat-4026	72	28	ℳ	ℳ	NOUN
easat-4026	72	29	such	such	ADJ
easat-4026	72	30	as	as	ADP
easat-4026	72	31	ℳ	ℳ	PROPN
easat-4026	72	32	=	=	SYM
easat-4026	72	33	⊕	⊕	PROPN
easat-4026	72	34	𝚤	𝚤	ADP
easat-4026	72	35	∈	∈	PROPN
easat-4026	72	36	ℐ	ℐ	PROPN
easat-4026	72	37	ℳ𝚤	ℳ𝚤	PROPN
easat-4026	72	38	,	,	PUNCT
easat-4026	72	39	and	and	CCONJ
easat-4026	72	40	the	the	DET
easat-4026	72	41	multiplication	multiplication	NOUN
easat-4026	72	42	map	map	NOUN
easat-4026	72	43	for	for	ADP
easat-4026	72	44	all	all	DET
easat-4026	72	45	𝓂,𝓃	𝓂,𝓃	ADP
easat-4026	72	46	∈	∈	NOUN
easat-4026	72	47	ℳ	ℳ	PROPN
easat-4026	72	48	is	be	AUX
easat-4026	72	49	given	give	VERB
easat-4026	72	50	by	by	ADP
easat-4026	72	51	:	:	PUNCT
easat-4026	72	52	𝑑𝑒𝑔	𝑑𝑒𝑔	PROPN
easat-4026	72	53	𝓂𝓃	𝓂𝓃	ADP
easat-4026	73	1	=	=	PUNCT
easat-4026	73	2	𝑑𝑒𝑔	𝑑𝑒𝑔	PROPN
easat-4026	73	3	𝓂	𝓂	PROPN
easat-4026	73	4	+	+	CCONJ
easat-4026	73	5	𝑑𝑒𝑔	𝑑𝑒𝑔	PROPN
easat-4026	73	6	𝓃.	𝓃.	NOUN
easat-4026	73	7	after	after	ADP
easat-4026	73	8	that	that	PRON
easat-4026	73	9	,	,	PUNCT
easat-4026	73	10	we	we	PRON
easat-4026	73	11	will	will	AUX
easat-4026	73	12	introduce	introduce	VERB
easat-4026	73	13	the	the	DET
easat-4026	73	14	construction	construction	NOUN
easat-4026	73	15	of	of	ADP
easat-4026	73	16	a	a	DET
easat-4026	73	17	tensor	tensor	NOUN
easat-4026	73	18	algebra	algebra	NOUN
easat-4026	73	19	from	from	ADP
easat-4026	73	20	a	a	DET
easat-4026	73	21	vector	vector	NOUN
easat-4026	73	22	space	space	NOUN
easat-4026	73	23	ℳ	ℳ	PROPN
easat-4026	73	24	,	,	PUNCT
easat-4026	73	25	explaining	explain	VERB
easat-4026	73	26	how	how	SCONJ
easat-4026	73	27	this	this	DET
easat-4026	73	28	construction	construction	NOUN
easat-4026	73	29	produces	produce	VERB
easat-4026	73	30	a	a	DET
easat-4026	73	31	graded	grade	VERB
easat-4026	73	32	algebra	algebra	NOUN
easat-4026	73	33	defined	define	VERB
easat-4026	73	34	by	by	ADP
easat-4026	73	35	the	the	DET
easat-4026	73	36	tensor	tensor	NOUN
easat-4026	73	37	product	product	NOUN
easat-4026	73	38	.	.	PUNCT
easat-4026	74	1	2.6	2.6	NUM
easat-4026	74	2	.	.	PUNCT
easat-4026	74	3	definition	definition	NOUN
easat-4026	74	4	[	[	X
easat-4026	74	5	15	15	NUM
easat-4026	74	6	]	]	X
easat-4026	74	7	if	if	SCONJ
easat-4026	74	8	ℛ	ℛ	PROPN
easat-4026	74	9	is	be	AUX
easat-4026	74	10	a	a	DET
easat-4026	74	11	field	field	NOUN
easat-4026	74	12	,	,	PUNCT
easat-4026	74	13	then	then	ADV
easat-4026	74	14	the	the	DET
easat-4026	74	15	vector	vector	NOUN
easat-4026	74	16	space	space	NOUN
easat-4026	74	17	ℳ	ℳ	PROPN
easat-4026	74	18	provides	provide	VERB
easat-4026	74	19	the	the	DET
easat-4026	74	20	tensor	tensor	NOUN
easat-4026	74	21	algebra	algebra	NOUN
easat-4026	74	22	of	of	ADP
easat-4026	74	23	ℳ	ℳ	PROPN
easat-4026	74	24	by	by	ADP
easat-4026	74	25	:	:	PUNCT
easat-4026	74	26	𝑇(ℳ	𝑇(ℳ	NOUN
easat-4026	74	27	)	)	PUNCT
easat-4026	74	28	=	=	SYM
easat-4026	75	1	ℛ⊕ℳ⊕ℳ⊗2⊕ℳ⊗3⊕	ℛ⊕ℳ⊕ℳ⊗2⊕ℳ⊗3⊕	PROPN
easat-4026	75	2	…	…	PUNCT
easat-4026	75	3	the	the	DET
easat-4026	75	4	tensor	tensor	NOUN
easat-4026	75	5	product	product	NOUN
easat-4026	75	6	's	's	PART
easat-4026	75	7	presumed	presume	VERB
easat-4026	75	8	multiplication	multiplication	NOUN
easat-4026	75	9	appears	appear	VERB
easat-4026	75	10	as	as	ADP
easat-4026	75	11	a	a	DET
easat-4026	75	12	graded	grade	VERB
easat-4026	75	13	algebra	algebra	NOUN
easat-4026	75	14	.	.	PUNCT
easat-4026	76	1	in	in	ADP
easat-4026	76	2	the	the	DET
easat-4026	76	3	following	follow	VERB
easat-4026	76	4	definition	definition	NOUN
easat-4026	76	5	,	,	PUNCT
easat-4026	76	6	we	we	PRON
easat-4026	76	7	define	define	VERB
easat-4026	76	8	linear	linear	ADJ
easat-4026	76	9	morphisms	morphism	NOUN
easat-4026	76	10	between	between	ADP
easat-4026	76	11	graded	grade	VERB
easat-4026	76	12	algebras	algebra	NOUN
easat-4026	76	13	using	use	VERB
easat-4026	76	14	the	the	DET
easat-4026	76	15	tensor	tensor	NOUN
easat-4026	76	16	product	product	NOUN
easat-4026	76	17	and	and	CCONJ
easat-4026	76	18	describe	describe	VERB
easat-4026	76	19	the	the	DET
easat-4026	76	20	degree	degree	NOUN
easat-4026	76	21	-	-	PUNCT
easat-4026	76	22	related	relate	VERB
easat-4026	76	23	properties	property	NOUN
easat-4026	76	24	of	of	ADP
easat-4026	76	25	these	these	DET
easat-4026	76	26	morphisms	morphism	NOUN
easat-4026	76	27	.	.	PUNCT
easat-4026	77	1	9475	9475	NUM
easat-4026	77	2	edelweiss	edelweiss	PROPN
easat-4026	77	3	applied	apply	VERB
easat-4026	77	4	science	science	NOUN
easat-4026	77	5	and	and	CCONJ
easat-4026	77	6	technology	technology	NOUN
easat-4026	77	7	issn	issn	PROPN
easat-4026	77	8	:	:	PUNCT
easat-4026	77	9	2576	2576	NUM
easat-4026	77	10	-	-	SYM
easat-4026	77	11	8484	8484	NUM
easat-4026	77	12	vol	vol	NOUN
easat-4026	77	13	.	.	PROPN
easat-4026	77	14	8	8	NUM
easat-4026	77	15	,	,	PUNCT
easat-4026	77	16	no	no	INTJ
easat-4026	77	17	.	.	NOUN
easat-4026	78	1	6	6	NUM
easat-4026	78	2	:	:	SYM
easat-4026	78	3	9472	9472	NUM
easat-4026	78	4	-	-	SYM
easat-4026	78	5	9486	9486	NUM
easat-4026	78	6	,	,	PUNCT
easat-4026	78	7	2024	2024	NUM
easat-4026	78	8	doi	doi	NOUN
easat-4026	78	9	:	:	PUNCT
easat-4026	78	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	78	11	©	©	PROPN
easat-4026	78	12	2024	2024	NUM
easat-4026	78	13	by	by	ADP
easat-4026	78	14	the	the	DET
easat-4026	78	15	authors	author	NOUN
easat-4026	78	16	;	;	PUNCT
easat-4026	78	17	licensee	licensee	PROPN
easat-4026	78	18	learning	learning	NOUN
easat-4026	78	19	gate	gate	NOUN
easat-4026	78	20	2.7	2.7	NUM
easat-4026	78	21	.	.	PUNCT
easat-4026	79	1	definition	definition	NOUN
easat-4026	79	2	[	[	X
easat-4026	79	3	15	15	NUM
easat-4026	79	4	]	]	PUNCT
easat-4026	79	5	suppose	suppose	VERB
easat-4026	79	6	that	that	SCONJ
easat-4026	79	7	ℳ	ℳ	NOUN
easat-4026	79	8	=	=	NOUN
easat-4026	79	9	⊗𝑛∈𝑍	⊗𝑛∈𝑍	NOUN
easat-4026	79	10	ℳ𝑛	ℳ𝑛	NOUN
easat-4026	79	11	is	be	AUX
easat-4026	79	12	the	the	DET
easat-4026	79	13	ℤ-graded	ℤ-graded	ADJ
easat-4026	79	14	vector	vector	NOUN
easat-4026	79	15	space	space	NOUN
easat-4026	79	16	.	.	PUNCT
easat-4026	80	1	then	then	ADV
easat-4026	80	2	,	,	PUNCT
easat-4026	80	3	the	the	DET
easat-4026	80	4	homeomorphisms	homeomorphism	NOUN
easat-4026	80	5	of	of	ADP
easat-4026	80	6	vector	vector	NOUN
easat-4026	80	7	spaces	space	NOUN
easat-4026	80	8	are	be	AUX
easat-4026	80	9	linear	linear	PROPN
easat-4026	80	10	maps	map	NOUN
easat-4026	80	11	𝑓:ℳ⊗𝑛	𝑓:ℳ⊗𝑛	ADV
easat-4026	80	12	→ℳ.	→ℳ.	PUNCT
easat-4026	80	13	since	since	SCONJ
easat-4026	80	14	the	the	DET
easat-4026	80	15	degree	degree	NOUN
easat-4026	80	16	of	of	ADP
easat-4026	80	17	𝑓	𝑓	PRON
easat-4026	80	18	,	,	PUNCT
easat-4026	80	19	|𝑓|	|𝑓|	NOUN
easat-4026	80	20	is	be	AUX
easat-4026	80	21	given	give	VERB
easat-4026	80	22	by	by	ADP
easat-4026	80	23	𝚤	𝚤	PRON
easat-4026	80	24	,	,	PUNCT
easat-4026	80	25	when	when	SCONJ
easat-4026	80	26	𝑑𝑒𝑔	𝑑𝑒𝑔	PROPN
easat-4026	80	27	𝑓(𝓂	𝑓(𝓂	PROPN
easat-4026	80	28	)	)	PUNCT
easat-4026	81	1	=	=	PRON
easat-4026	81	2	(	(	PUNCT
easat-4026	81	3	𝑑𝑒𝑔	𝑑𝑒𝑔	PROPN
easat-4026	81	4	𝓂	𝓂	NOUN
easat-4026	81	5	)	)	PUNCT
easat-4026	82	1	+	+	CCONJ
easat-4026	82	2	𝚤	𝚤	PROPN
easat-4026	82	3	for	for	ADP
easat-4026	82	4	all	all	DET
easat-4026	82	5	𝓂	𝓂	NOUN
easat-4026	82	6	∈	∈	PROPN
easat-4026	82	7	ℳ	ℳ	PROPN
easat-4026	82	8	.	.	PUNCT
easat-4026	83	1	obviously	obviously	ADV
easat-4026	83	2	,	,	PUNCT
easat-4026	83	3	|𝑓(𝓂1⊗𝓂2⊗	|𝑓(𝓂1⊗𝓂2⊗	X
easat-4026	83	4	·	·	PUNCT
easat-4026	84	1	·	·	PUNCT
easat-4026	84	2	·	·	PUNCT
easat-4026	84	3	⊗𝓂𝑛)|	⊗𝓂𝑛)|	PROPN
easat-4026	84	4	=	=	SYM
easat-4026	84	5	(	(	PUNCT
easat-4026	84	6	|𝓂1|	|𝓂1|	NOUN
easat-4026	84	7	+	+	CCONJ
easat-4026	84	8	|𝓂2|	|𝓂2|	X
easat-4026	84	9	+	+	X
easat-4026	84	10	·	·	PUNCT
easat-4026	84	11	·	·	PUNCT
easat-4026	84	12	·	·	PUNCT
easat-4026	84	13	+	+	NUM
easat-4026	84	14	|𝓂𝑛|	|𝓂𝑛|	PROPN
easat-4026	84	15	)	)	PUNCT
easat-4026	85	1	+	+	CCONJ
easat-4026	85	2	𝚤.	𝚤.	ADJ
easat-4026	85	3	we	we	PRON
easat-4026	85	4	will	will	AUX
easat-4026	85	5	provide	provide	VERB
easat-4026	85	6	an	an	DET
easat-4026	85	7	illustrative	illustrative	ADJ
easat-4026	85	8	example	example	NOUN
easat-4026	85	9	of	of	ADP
easat-4026	85	10	a	a	DET
easat-4026	85	11	graded	grade	VERB
easat-4026	85	12	algebra	algebra	NOUN
easat-4026	85	13	using	use	VERB
easat-4026	85	14	a	a	DET
easat-4026	85	15	complex	complex	ADJ
easat-4026	85	16	sequence	sequence	NOUN
easat-4026	85	17	over	over	ADP
easat-4026	85	18	a	a	DET
easat-4026	85	19	specific	specific	ADJ
easat-4026	85	20	field	field	NOUN
easat-4026	85	21	and	and	CCONJ
easat-4026	85	22	explaine	explaine	VERB
easat-4026	85	23	how	how	SCONJ
easat-4026	85	24	to	to	PART
easat-4026	85	25	explore	explore	VERB
easat-4026	85	26	homology	homology	NOUN
easat-4026	85	27	through	through	ADP
easat-4026	85	28	this	this	DET
easat-4026	85	29	example	example	NOUN
easat-4026	85	30	.	.	PUNCT
easat-4026	86	1	2.8	2.8	NUM
easat-4026	86	2	.	.	PUNCT
easat-4026	86	3	example	example	NOUN
easat-4026	86	4	assume	assume	VERB
easat-4026	86	5	the	the	DET
easat-4026	86	6	complex	complex	NOUN
easat-4026	86	7	over	over	ADP
easat-4026	86	8	a	a	DET
easat-4026	86	9	field	field	NOUN
easat-4026	86	10	known	know	VERB
easat-4026	86	11	as	as	ADP
easat-4026	86	12	:	:	PUNCT
easat-4026	86	13	𝑉•	𝑉•	NOUN
easat-4026	86	14	:	:	PUNCT
easat-4026	86	15	…	…	PUNCT
easat-4026	86	16	→	→	SYM
easat-4026	86	17	𝑉𝑛+1	𝑉𝑛+1	X
easat-4026	86	18	𝒹𝑛+1	𝒹𝑛+1	X
easat-4026	86	19	→	→	SYM
easat-4026	86	20	𝑉𝑛	𝑉𝑛	NOUN
easat-4026	86	21	𝒹𝑛	𝒹𝑛	ADJ
easat-4026	86	22	→	→	SYM
easat-4026	86	23	𝑉𝑛−1	𝑉𝑛−1	NOUN
easat-4026	86	24	→	→	SYM
easat-4026	86	25	…	…	PUNCT
easat-4026	86	26	,	,	PUNCT
easat-4026	86	27	hence	hence	ADV
easat-4026	86	28	the	the	DET
easat-4026	86	29	modules	module	NOUN
easat-4026	86	30	𝑉𝑛	𝑉𝑛	PRON
easat-4026	86	31	seem	seem	VERB
easat-4026	86	32	to	to	PART
easat-4026	86	33	be	be	AUX
easat-4026	86	34	,	,	PUNCT
easat-4026	86	35	in	in	ADP
easat-4026	86	36	fact	fact	NOUN
easat-4026	86	37	,	,	PUNCT
easat-4026	86	38	vector	vector	NOUN
easat-4026	86	39	spaces	space	NOUN
easat-4026	86	40	.	.	PUNCT
easat-4026	87	1	so	so	ADV
easat-4026	87	2	𝑉	𝑉	PROPN
easat-4026	87	3	=	=	PRON
easat-4026	87	4	⨁𝑛𝑉𝑛	⨁𝑛𝑉𝑛	PROPN
easat-4026	87	5	is	be	AUX
easat-4026	87	6	known	know	VERB
easat-4026	87	7	as	as	ADP
easat-4026	87	8	the	the	DET
easat-4026	87	9	graded	grade	VERB
easat-4026	87	10	vector	vector	NOUN
easat-4026	87	11	space	space	NOUN
easat-4026	87	12	,	,	PUNCT
easat-4026	87	13	where	where	SCONJ
easat-4026	87	14	𝒹:𝑉⊗1	𝒹:𝑉⊗1	ADJ
easat-4026	87	15	→	→	SYM
easat-4026	87	16	𝑉	𝑉	PROPN
easat-4026	87	17	have	have	VERB
easat-4026	87	18	a	a	DET
easat-4026	87	19	degree	degree	NOUN
easat-4026	87	20	−1	−1	NOUN
easat-4026	87	21	.	.	PUNCT
easat-4026	88	1	following	follow	VERB
easat-4026	88	2	,	,	PUNCT
easat-4026	88	3	we	we	PRON
easat-4026	88	4	define	define	VERB
easat-4026	88	5	morphisms	morphism	NOUN
easat-4026	88	6	between	between	ADP
easat-4026	88	7	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	88	8	and	and	CCONJ
easat-4026	88	9	explain	explain	VERB
easat-4026	88	10	the	the	DET
easat-4026	88	11	conditions	condition	NOUN
easat-4026	88	12	and	and	CCONJ
easat-4026	88	13	properties	property	NOUN
easat-4026	88	14	necessary	necessary	ADJ
easat-4026	88	15	for	for	ADP
easat-4026	88	16	these	these	DET
easat-4026	88	17	morphisms	morphism	NOUN
easat-4026	88	18	.	.	PUNCT
easat-4026	89	1	2.9	2.9	NUM
easat-4026	89	2	.	.	PUNCT
easat-4026	89	3	definition	definition	NOUN
easat-4026	89	4	[	[	X
easat-4026	89	5	13	13	NUM
easat-4026	89	6	]	]	PUNCT
easat-4026	89	7	by	by	ADP
easat-4026	89	8	assuming	assume	VERB
easat-4026	89	9	that	that	SCONJ
easat-4026	89	10	𝛽	𝛽	NOUN
easat-4026	89	11	,	,	PUNCT
easat-4026	89	12	𝛾	𝛾	PROPN
easat-4026	89	13	are	be	AUX
easat-4026	89	14	two	two	NUM
easat-4026	89	15	linear	linear	ADJ
easat-4026	89	16	maps	map	NOUN
easat-4026	89	17	of	of	ADP
easat-4026	89	18	the	the	DET
easat-4026	89	19	graded	grade	VERB
easat-4026	89	20	vector	vector	NOUN
easat-4026	89	21	spaces	space	NOUN
easat-4026	89	22	𝒳,𝒴	𝒳,𝒴	VERB
easat-4026	89	23	respectively	respectively	ADV
easat-4026	89	24	,	,	PUNCT
easat-4026	89	25	such	such	ADJ
easat-4026	89	26	that	that	SCONJ
easat-4026	89	27	𝛽:𝒳	𝛽:𝒳	PROPN
easat-4026	89	28	→	→	SYM
easat-4026	89	29	𝒳	𝒳	PROPN
easat-4026	89	30	,	,	PUNCT
easat-4026	89	31	𝛾	𝛾	PROPN
easat-4026	89	32	:	:	PUNCT
easat-4026	89	33	𝒴	𝒴	PROPN
easat-4026	89	34	→	→	SYM
easat-4026	89	35	𝒴.	𝒴.	PROPN
easat-4026	89	36	next	next	ADV
easat-4026	89	37	,	,	PUNCT
easat-4026	89	38	the	the	DET
easat-4026	89	39	linear	linear	PROPN
easat-4026	89	40	maps	map	NOUN
easat-4026	89	41	'	'	PART
easat-4026	89	42	tensor	tensor	NOUN
easat-4026	89	43	product	product	NOUN
easat-4026	89	44	𝛽	𝛽	NOUN
easat-4026	89	45	and	and	CCONJ
easat-4026	89	46	𝛾	𝛾	PROPN
easat-4026	89	47	is	be	AUX
easat-4026	89	48	denoted	denote	VERB
easat-4026	89	49	by	by	ADP
easat-4026	89	50	:	:	PUNCT
easat-4026	89	51	𝛽	𝛽	PROPN
easat-4026	89	52	⊗	⊗	PROPN
easat-4026	89	53	𝛾	𝛾	ADP
easat-4026	89	54	∶	∶	NOUN
easat-4026	89	55	𝒳	𝒳	ADP
easat-4026	89	56	⊗𝒴	⊗𝒴	NOUN
easat-4026	89	57	→	→	SYM
easat-4026	89	58	𝒳⊗𝒴	𝒳⊗𝒴	PROPN
easat-4026	89	59	(	(	PUNCT
easat-4026	89	60	𝓍	𝓍	X
easat-4026	89	61	⊗𝓎	⊗𝓎	NOUN
easat-4026	89	62	)	)	PUNCT
easat-4026	90	1	⟼	⟼	PROPN
easat-4026	90	2	(	(	PUNCT
easat-4026	90	3	−1)|𝛾||𝓎|(𝛽(𝓍)⊗	−1)|𝛾||𝓎|(𝛽(𝓍)⊗	NOUN
easat-4026	90	4	𝛾(𝓎	𝛾(𝓎	PROPN
easat-4026	90	5	)	)	PUNCT
easat-4026	90	6	)	)	PUNCT
easat-4026	90	7	.	.	PUNCT
easat-4026	91	1	(	(	PUNCT
easat-4026	91	2	1	1	X
easat-4026	91	3	)	)	PUNCT
easat-4026	91	4	remind	remind	VERB
easat-4026	91	5	that	that	SCONJ
easat-4026	91	6	the	the	DET
easat-4026	91	7	degree	degree	NOUN
easat-4026	91	8	of	of	ADP
easat-4026	91	9	(	(	PUNCT
easat-4026	91	10	𝛽	𝛽	PROPN
easat-4026	91	11	⊗	⊗	PROPN
easat-4026	91	12	𝛾	𝛾	NOUN
easat-4026	91	13	)	)	PUNCT
easat-4026	91	14	is	be	AUX
easat-4026	91	15	|𝛽|	|𝛽|	ADJ
easat-4026	92	1	+	+	CCONJ
easat-4026	92	2	|𝛾|	|𝛾|	PROPN
easat-4026	92	3	.	.	PUNCT
easat-4026	92	4	remark	remark	PROPN
easat-4026	92	5	:	:	PUNCT
easat-4026	92	6	the	the	DET
easat-4026	92	7	koszul	koszul	ADJ
easat-4026	92	8	sign	sign	NOUN
easat-4026	92	9	rule	rule	NOUN
easat-4026	92	10	states	state	NOUN
easat-4026	92	11	that	that	SCONJ
easat-4026	92	12	if	if	SCONJ
easat-4026	92	13	two	two	NUM
easat-4026	92	14	symbols	symbol	NOUN
easat-4026	92	15	'	'	PART
easat-4026	92	16	positions	position	NOUN
easat-4026	92	17	𝓈	𝓈	X
easat-4026	92	18	and	and	CCONJ
easat-4026	92	19	𝓇	𝓇	X
easat-4026	92	20	are	be	AUX
easat-4026	92	21	switched	switch	VERB
easat-4026	92	22	,	,	PUNCT
easat-4026	92	23	the	the	DET
easat-4026	92	24	outcome	outcome	NOUN
easat-4026	92	25	is	be	AUX
easat-4026	92	26	multiplied	multiply	VERB
easat-4026	92	27	by	by	ADP
easat-4026	92	28	(	(	PUNCT
easat-4026	92	29	−1)|𝓈||𝓇|	−1)|𝓈||𝓇|	PROPN
easat-4026	92	30	,	,	PUNCT
easat-4026	92	31	which	which	PRON
easat-4026	92	32	is	be	AUX
easat-4026	92	33	the	the	DET
easat-4026	92	34	actual	actual	ADJ
easat-4026	92	35	cause	cause	NOUN
easat-4026	92	36	of	of	ADP
easat-4026	92	37	the	the	DET
easat-4026	92	38	change	change	NOUN
easat-4026	92	39	in	in	ADP
easat-4026	92	40	sign	sign	NOUN
easat-4026	92	41	in	in	ADP
easat-4026	92	42	the	the	DET
easat-4026	92	43	previous	previous	ADJ
easat-4026	92	44	expression	expression	NOUN
easat-4026	92	45	.	.	PUNCT
easat-4026	93	1	in	in	ADP
easat-4026	93	2	the	the	DET
easat-4026	93	3	expression	expression	NOUN
easat-4026	93	4	above	above	ADV
easat-4026	93	5	,	,	PUNCT
easat-4026	93	6	it	it	PRON
easat-4026	93	7	is	be	AUX
easat-4026	93	8	applied	apply	VERB
easat-4026	93	9	as	as	ADP
easat-4026	93	10	(	(	PUNCT
easat-4026	93	11	𝛽	𝛽	PROPN
easat-4026	93	12	⊗	⊗	PROPN
easat-4026	93	13	𝛾)(𝓍	𝛾)(𝓍	NOUN
easat-4026	93	14	⊗𝓎	⊗𝓎	NOUN
easat-4026	93	15	)	)	PUNCT
easat-4026	94	1	=	=	SYM
easat-4026	94	2	(	(	PUNCT
easat-4026	94	3	−1)|𝛾||𝓎|(𝛽(𝓍)⊗	−1)|𝛾||𝓎|(𝛽(𝓍)⊗	NOUN
easat-4026	94	4	𝛾(𝓎	𝛾(𝓎	PROPN
easat-4026	94	5	)	)	PUNCT
easat-4026	94	6	)	)	PUNCT
easat-4026	94	7	,	,	PUNCT
easat-4026	94	8	where	where	SCONJ
easat-4026	94	9	the	the	DET
easat-4026	94	10	morphisms	morphism	NOUN
easat-4026	94	11	acting	act	VERB
easat-4026	94	12	on	on	ADP
easat-4026	94	13	the	the	DET
easat-4026	94	14	elements	element	NOUN
easat-4026	94	15	are	be	AUX
easat-4026	94	16	represented	represent	VERB
easat-4026	94	17	by	by	ADP
easat-4026	94	18	the	the	DET
easat-4026	94	19	symbols	symbol	NOUN
easat-4026	94	20	𝓍	𝓍	PROPN
easat-4026	94	21	and	and	CCONJ
easat-4026	94	22	𝓎	𝓎	X
easat-4026	94	23	,	,	PUNCT
easat-4026	94	24	which	which	PRON
easat-4026	94	25	are	be	AUX
easat-4026	94	26	swapped	swap	VERB
easat-4026	94	27	.	.	PUNCT
easat-4026	95	1	in	in	ADP
easat-4026	95	2	the	the	DET
easat-4026	95	3	following	following	NOUN
easat-4026	95	4	,	,	PUNCT
easat-4026	95	5	we	we	PRON
easat-4026	95	6	describe	describe	VERB
easat-4026	95	7	the	the	DET
easat-4026	95	8	concept	concept	NOUN
easat-4026	95	9	of	of	ADP
easat-4026	95	10	differential	differential	NOUN
easat-4026	95	11	graded	grade	VERB
easat-4026	95	12	algebras	algebra	NOUN
easat-4026	95	13	(	(	PUNCT
easat-4026	95	14	dgas	dgas	PROPN
easat-4026	95	15	)	)	PUNCT
easat-4026	95	16	and	and	CCONJ
easat-4026	95	17	how	how	SCONJ
easat-4026	95	18	to	to	PART
easat-4026	95	19	define	define	VERB
easat-4026	95	20	them	they	PRON
easat-4026	95	21	using	use	VERB
easat-4026	95	22	the	the	DET
easat-4026	95	23	chain	chain	NOUN
easat-4026	95	24	complex	complex	ADJ
easat-4026	95	25	maps	map	NOUN
easat-4026	95	26	and	and	CCONJ
easat-4026	95	27	leibniz	leibniz	PROPN
easat-4026	95	28	rules	rule	NOUN
easat-4026	95	29	.	.	PUNCT
easat-4026	96	1	2.10	2.10	NUM
easat-4026	96	2	.	.	PUNCT
easat-4026	96	3	definition	definition	NOUN
easat-4026	96	4	[	[	X
easat-4026	96	5	15	15	NUM
easat-4026	96	6	]	]	X
easat-4026	96	7	the	the	DET
easat-4026	96	8	differential	differential	NOUN
easat-4026	96	9	graded	grade	VERB
easat-4026	96	10	algebra	algebra	NOUN
easat-4026	96	11	(	(	PUNCT
easat-4026	96	12	dga	dga	NOUN
easat-4026	96	13	)	)	PUNCT
easat-4026	96	14	is	be	AUX
easat-4026	96	15	the	the	DET
easat-4026	96	16	complex	complex	ADJ
easat-4026	96	17	ℳ	ℳ	NOUN
easat-4026	96	18	with	with	ADP
easat-4026	96	19	the	the	DET
easat-4026	96	20	degree	degree	NOUN
easat-4026	96	21	+1	+1	PROPN
easat-4026	96	22	chain	chain	NOUN
easat-4026	96	23	map	map	NOUN
easat-4026	96	24	𝒹:ℳ⊗ℳ	𝒹:ℳ⊗ℳ	NOUN
easat-4026	96	25	→	→	SYM
easat-4026	96	26	ℳ	ℳ	VERB
easat-4026	96	27	so	so	ADV
easat-4026	96	28	that	that	DET
easat-4026	96	29	𝒹2	𝒹2	NOUN
easat-4026	96	30	=	=	SYM
easat-4026	96	31	0	0	NUM
easat-4026	96	32	,	,	PUNCT
easat-4026	96	33	which	which	PRON
easat-4026	96	34	is	be	AUX
easat-4026	96	35	unital	unital	ADJ
easat-4026	96	36	and	and	CCONJ
easat-4026	96	37	associative	associative	ADJ
easat-4026	96	38	.	.	PUNCT
easat-4026	97	1	the	the	DET
easat-4026	97	2	differential	differential	ADJ
easat-4026	97	3	graded	grade	VERB
easat-4026	97	4	algebra	algebra	NOUN
easat-4026	97	5	is	be	AUX
easat-4026	97	6	simply	simply	ADV
easat-4026	97	7	a	a	DET
easat-4026	97	8	differential	differential	NOUN
easat-4026	97	9	𝒹	𝒹	NOUN
easat-4026	97	10	and	and	CCONJ
easat-4026	97	11	a	a	DET
easat-4026	97	12	graded	grade	VERB
easat-4026	97	13	module	module	NOUN
easat-4026	97	14	equivalent	equivalent	NOUN
easat-4026	97	15	.	.	PUNCT
easat-4026	98	1	the	the	DET
easat-4026	98	2	graded	grade	VERB
easat-4026	98	3	leibniz	leibniz	PROPN
easat-4026	98	4	rule	rule	NOUN
easat-4026	98	5	for	for	ADP
easat-4026	98	6	all	all	DET
easat-4026	98	7	𝓂,𝓃	𝓂,𝓃	ADP
easat-4026	98	8	∈	∈	NOUN
easat-4026	98	9	ℳ	ℳ	PROPN
easat-4026	98	10	is	be	AUX
easat-4026	98	11	denoted	denote	VERB
easat-4026	98	12	by	by	ADP
easat-4026	98	13	:	:	PUNCT
easat-4026	98	14	𝒹(𝓂𝓃	𝒹(𝓂𝓃	NUM
easat-4026	98	15	)	)	PUNCT
easat-4026	98	16	=	=	SYM
easat-4026	98	17	𝒹(𝓂)𝓃	𝒹(𝓂)𝓃	PROPN
easat-4026	98	18	+	+	CCONJ
easat-4026	98	19	(	(	PUNCT
easat-4026	98	20	−1)|𝓂|𝒹(𝓃	−1)|𝓂|𝒹(𝓃	PROPN
easat-4026	98	21	)	)	PUNCT
easat-4026	98	22	.	.	PUNCT
easat-4026	99	1	(	(	PUNCT
easat-4026	99	2	2	2	X
easat-4026	99	3	)	)	PUNCT
easat-4026	99	4	remark	remark	NOUN
easat-4026	99	5	:	:	PUNCT
easat-4026	99	6	(	(	PUNCT
easat-4026	99	7	1	1	X
easat-4026	99	8	)	)	PUNCT
easat-4026	99	9	if	if	SCONJ
easat-4026	99	10	𝓂𝓃	𝓂𝓃	NOUN
easat-4026	99	11	=	=	SYM
easat-4026	99	12	(	(	PUNCT
easat-4026	99	13	−1)|𝓂||𝓃|𝓃𝓂	−1)|𝓂||𝓃|𝓃𝓂	PROPN
easat-4026	99	14	,	,	PUNCT
easat-4026	99	15	for	for	ADP
easat-4026	99	16	all	all	DET
easat-4026	99	17	𝓂,𝓃	𝓂,𝓃	ADP
easat-4026	99	18	∈	∈	PROPN
easat-4026	99	19	ℳ	ℳ	PROPN
easat-4026	99	20	,	,	PUNCT
easat-4026	99	21	then	then	ADV
easat-4026	99	22	the	the	DET
easat-4026	99	23	differential	differential	NOUN
easat-4026	99	24	graded	grade	VERB
easat-4026	99	25	algebra	algebra	NOUN
easat-4026	99	26	ℳ	ℳ	PROPN
easat-4026	99	27	is	be	AUX
easat-4026	99	28	regarded	regard	VERB
easat-4026	99	29	as	as	ADP
easat-4026	99	30	commutative	commutative	ADJ
easat-4026	99	31	.	.	PUNCT
easat-4026	100	1	if	if	SCONJ
easat-4026	100	2	1/2	1/2	NUM
easat-4026	100	3	∈	∈	NOUN
easat-4026	100	4	𝑘	𝑘	NOUN
easat-4026	100	5	,	,	PUNCT
easat-4026	100	6	it	it	PRON
easat-4026	100	7	is	be	AUX
easat-4026	100	8	implied	imply	VERB
easat-4026	100	9	that	that	SCONJ
easat-4026	100	10	𝓂2	𝓂2	PROPN
easat-4026	100	11	=	=	SYM
easat-4026	100	12	0	0	NUM
easat-4026	100	13	holds	hold	VERB
easat-4026	100	14	true	true	ADJ
easat-4026	100	15	if	if	SCONJ
easat-4026	100	16	𝓂	𝓂	PROPN
easat-4026	100	17	would	would	AUX
easat-4026	100	18	have	have	VERB
easat-4026	100	19	an	an	DET
easat-4026	100	20	odd	odd	ADJ
easat-4026	100	21	degree	degree	NOUN
easat-4026	100	22	.	.	PUNCT
easat-4026	101	1	when	when	SCONJ
easat-4026	101	2	ℳ	ℳ	PROPN
easat-4026	101	3	is	be	AUX
easat-4026	101	4	the	the	DET
easat-4026	101	5	commutative	commutative	ADJ
easat-4026	101	6	graded	grade	VERB
easat-4026	101	7	algebra	algebra	NOUN
easat-4026	101	8	,	,	PUNCT
easat-4026	101	9	therefore	therefore	ADV
easat-4026	101	10	the	the	DET
easat-4026	101	11	left	left	ADJ
easat-4026	101	12	ℳ-module	ℳ-module	PROPN
easat-4026	101	13	,	,	PUNCT
easat-4026	101	14	and	and	CCONJ
easat-4026	101	15	automatically	automatically	ADV
easat-4026	101	16	𝑅	𝑅	PROPN
easat-4026	101	17	is	be	AUX
easat-4026	101	18	a	a	DET
easat-4026	101	19	right	right	ADJ
easat-4026	101	20	ℳ-module	ℳ-module	PROPN
easat-4026	101	21	,	,	PUNCT
easat-4026	101	22	by	by	ADP
easat-4026	101	23	the	the	DET
easat-4026	101	24	formula	formula	NOUN
easat-4026	101	25	𝑟𝓂	𝑟𝓂	NOUN
easat-4026	101	26	=	=	SYM
easat-4026	101	27	(	(	PUNCT
easat-4026	101	28	−1)|𝑟||𝓂|𝓂𝑟.	−1)|𝑟||𝓂|𝓂𝑟.	PROPN
easat-4026	101	29	(	(	PUNCT
easat-4026	101	30	2	2	NUM
easat-4026	101	31	)	)	PUNCT
easat-4026	101	32	similar	similar	ADJ
easat-4026	101	33	definitions	definition	NOUN
easat-4026	101	34	are	be	AUX
easat-4026	101	35	provided	provide	VERB
easat-4026	101	36	for	for	ADP
easat-4026	101	37	the	the	DET
easat-4026	101	38	terms	term	NOUN
easat-4026	101	39	differential	differential	VERB
easat-4026	101	40	graded	grade	VERB
easat-4026	101	41	algebra	algebra	NOUN
easat-4026	101	42	and	and	CCONJ
easat-4026	101	43	derivation	derivation	NOUN
easat-4026	101	44	.	.	PUNCT
easat-4026	102	1	here	here	ADV
easat-4026	102	2	,	,	PUNCT
easat-4026	102	3	we	we	PRON
easat-4026	102	4	will	will	AUX
easat-4026	102	5	explain	explain	VERB
easat-4026	102	6	how	how	SCONJ
easat-4026	102	7	to	to	PART
easat-4026	102	8	handle	handle	VERB
easat-4026	102	9	left	leave	VERB
easat-4026	102	10	models	model	NOUN
easat-4026	102	11	of	of	ADP
easat-4026	102	12	differential	differential	NOUN
easat-4026	102	13	graded	grade	VERB
easat-4026	102	14	algebras	algebra	NOUN
easat-4026	102	15	and	and	CCONJ
easat-4026	102	16	use	use	VERB
easat-4026	102	17	rules	rule	NOUN
easat-4026	102	18	for	for	ADP
easat-4026	102	19	dealing	deal	VERB
easat-4026	102	20	with	with	ADP
easat-4026	102	21	derivatives	derivative	NOUN
easat-4026	102	22	in	in	ADP
easat-4026	102	23	these	these	DET
easat-4026	102	24	models	model	NOUN
easat-4026	102	25	.	.	PUNCT
easat-4026	103	1	9476	9476	NUM
easat-4026	103	2	edelweiss	edelweiss	PROPN
easat-4026	103	3	applied	apply	VERB
easat-4026	103	4	science	science	NOUN
easat-4026	103	5	and	and	CCONJ
easat-4026	103	6	technology	technology	NOUN
easat-4026	103	7	issn	issn	PROPN
easat-4026	103	8	:	:	PUNCT
easat-4026	103	9	2576	2576	NUM
easat-4026	103	10	-	-	SYM
easat-4026	103	11	8484	8484	NUM
easat-4026	103	12	vol	vol	NOUN
easat-4026	103	13	.	.	PROPN
easat-4026	103	14	8	8	NUM
easat-4026	103	15	,	,	PUNCT
easat-4026	103	16	no	no	INTJ
easat-4026	103	17	.	.	NOUN
easat-4026	104	1	6	6	NUM
easat-4026	104	2	:	:	SYM
easat-4026	104	3	9472	9472	NUM
easat-4026	104	4	-	-	SYM
easat-4026	104	5	9486	9486	NUM
easat-4026	104	6	,	,	PUNCT
easat-4026	104	7	2024	2024	NUM
easat-4026	104	8	doi	doi	NOUN
easat-4026	104	9	:	:	PUNCT
easat-4026	104	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	104	11	©	©	PROPN
easat-4026	104	12	2024	2024	NUM
easat-4026	104	13	by	by	ADP
easat-4026	104	14	the	the	DET
easat-4026	104	15	authors	author	NOUN
easat-4026	104	16	;	;	PUNCT
easat-4026	104	17	licensee	licensee	PROPN
easat-4026	104	18	learning	learning	NOUN
easat-4026	104	19	gate	gate	NOUN
easat-4026	104	20	2.11	2.11	NUM
easat-4026	104	21	.	.	PUNCT
easat-4026	105	1	definition	definition	NOUN
easat-4026	105	2	[	[	X
easat-4026	105	3	16	16	NUM
easat-4026	105	4	]	]	PUNCT
easat-4026	105	5	assume	assume	VERB
easat-4026	105	6	that	that	SCONJ
easat-4026	105	7	the	the	DET
easat-4026	105	8	differential	differential	NOUN
easat-4026	105	9	graded	grade	VERB
easat-4026	105	10	algebra	algebra	NOUN
easat-4026	105	11	is	be	AUX
easat-4026	105	12	(	(	PUNCT
easat-4026	105	13	ℳ,𝒹	ℳ,𝒹	NUM
easat-4026	105	14	)	)	PUNCT
easat-4026	105	15	.	.	PUNCT
easat-4026	106	1	in	in	ADP
easat-4026	106	2	addition	addition	NOUN
easat-4026	106	3	,	,	PUNCT
easat-4026	106	4	let	let	VERB
easat-4026	106	5	𝑅	𝑅	PROPN
easat-4026	106	6	be	be	AUX
easat-4026	106	7	the	the	DET
easat-4026	106	8	left	leave	VERB
easat-4026	106	9	graded	grade	VERB
easat-4026	106	10	ℳmodule	ℳmodule	PROPN
easat-4026	106	11	and	and	CCONJ
easat-4026	106	12	𝒹𝑅	𝒹𝑅	PROPN
easat-4026	106	13	be	be	VERB
easat-4026	106	14	the	the	DET
easat-4026	106	15	differential	differential	NOUN
easat-4026	106	16	of	of	ADP
easat-4026	106	17	𝑅	𝑅	PROPN
easat-4026	106	18	,	,	PUNCT
easat-4026	106	19	so	so	ADV
easat-4026	106	20	the	the	DET
easat-4026	106	21	complex	complex	ADJ
easat-4026	106	22	(	(	PUNCT
easat-4026	106	23	𝑅	𝑅	PROPN
easat-4026	106	24	,	,	PUNCT
easat-4026	106	25	𝒹𝑅)and	𝒹𝑅)and	PROPN
easat-4026	106	26	the	the	DET
easat-4026	106	27	left	left	ADJ
easat-4026	106	28	multiplication	multiplication	NOUN
easat-4026	106	29	ℳ⊗𝑅	ℳ⊗𝑅	PROPN
easat-4026	106	30	→	→	SYM
easat-4026	106	31	𝑅	𝑅	PROPN
easat-4026	106	32	combined	combine	VERB
easat-4026	106	33	to	to	PART
easat-4026	106	34	create	create	VERB
easat-4026	106	35	the	the	DET
easat-4026	106	36	left	left	ADJ
easat-4026	106	37	differential	differential	NOUN
easat-4026	106	38	graded	grade	VERB
easat-4026	106	39	ℳ-module	ℳ-module	PROPN
easat-4026	107	1	such	such	ADJ
easat-4026	107	2	that	that	SCONJ
easat-4026	107	3	the	the	DET
easat-4026	107	4	leibniz	leibniz	NOUN
easat-4026	107	5	rule	rule	NOUN
easat-4026	107	6	satisfies	satisfie	NOUN
easat-4026	107	7	for	for	ADP
easat-4026	107	8	all	all	DET
easat-4026	107	9	𝓂	𝓂	PROPN
easat-4026	107	10	∈	∈	PROPN
easat-4026	107	11	ℳ	ℳ	PROPN
easat-4026	107	12	,	,	PUNCT
easat-4026	107	13	𝑟	𝑟	X
easat-4026	107	14	∈	∈	PROPN
easat-4026	107	15	𝑅	𝑅	PROPN
easat-4026	107	16	:	:	PUNCT
easat-4026	107	17	𝒹𝑅(𝓂𝑟	𝒹𝑅(𝓂𝑟	NUM
easat-4026	107	18	)	)	PUNCT
easat-4026	107	19	=	=	SYM
easat-4026	107	20	𝒹(𝓂)𝑟	𝒹(𝓂)𝑟	PROPN
easat-4026	107	21	+	+	CCONJ
easat-4026	107	22	(	(	PUNCT
easat-4026	107	23	−1)|𝓂|𝓂𝒹𝑅(𝑟	−1)|𝓂|𝓂𝒹𝑅(𝑟	NOUN
easat-4026	107	24	)	)	PUNCT
easat-4026	107	25	.	.	PUNCT
easat-4026	108	1	an	an	DET
easat-4026	108	2	ℳ-module	ℳ-module	PROPN
easat-4026	108	3	with	with	ADP
easat-4026	108	4	differential	differential	ADJ
easat-4026	108	5	grading	grading	NOUN
easat-4026	108	6	is	be	AUX
easat-4026	108	7	merely	merely	ADV
easat-4026	108	8	a	a	DET
easat-4026	108	9	complex	complex	NOUN
easat-4026	108	10	.	.	PUNCT
easat-4026	109	1	similar	similar	ADJ
easat-4026	109	2	terms	term	NOUN
easat-4026	109	3	are	be	AUX
easat-4026	109	4	used	use	VERB
easat-4026	109	5	to	to	PART
easat-4026	109	6	define	define	VERB
easat-4026	109	7	a	a	DET
easat-4026	109	8	right	right	ADJ
easat-4026	109	9	differential	differential	NOUN
easat-4026	109	10	graded	grade	VERB
easat-4026	109	11	ℳ-module	ℳ-module	PROPN
easat-4026	109	12	.	.	PUNCT
easat-4026	110	1	after	after	ADP
easat-4026	110	2	that	that	PRON
easat-4026	110	3	,	,	PUNCT
easat-4026	110	4	we	we	PRON
easat-4026	110	5	define	define	VERB
easat-4026	110	6	graded	grade	VERB
easat-4026	110	7	spaces	space	NOUN
easat-4026	110	8	using	use	VERB
easat-4026	110	9	morphisms	morphism	NOUN
easat-4026	110	10	and	and	CCONJ
easat-4026	110	11	methods	method	NOUN
easat-4026	110	12	to	to	PART
easat-4026	110	13	determine	determine	VERB
easat-4026	110	14	derivatives	derivative	NOUN
easat-4026	110	15	within	within	ADP
easat-4026	110	16	these	these	DET
easat-4026	110	17	graded	grade	VERB
easat-4026	110	18	spaces	space	NOUN
easat-4026	110	19	.	.	PUNCT
easat-4026	111	1	2.12	2.12	NUM
easat-4026	111	2	.	.	PUNCT
easat-4026	111	3	definition	definition	NOUN
easat-4026	111	4	[	[	X
easat-4026	111	5	16	16	NUM
easat-4026	111	6	]	]	PUNCT
easat-4026	111	7	let	let	VERB
easat-4026	111	8	𝐻𝑜𝑚ℳ(𝑅	𝐻𝑜𝑚ℳ(𝑅	NOUN
easat-4026	111	9	,	,	PUNCT
easat-4026	111	10	𝑆	𝑆	PROPN
easat-4026	111	11	)	)	PUNCT
easat-4026	111	12	be	be	VERB
easat-4026	111	13	a	a	DET
easat-4026	111	14	graded	grade	VERB
easat-4026	111	15	vector	vector	NOUN
easat-4026	111	16	space	space	NOUN
easat-4026	111	17	that	that	PRON
easat-4026	111	18	contains	contain	VERB
easat-4026	111	19	ℳ-homomorphisms	ℳ-homomorphisms	PROPN
easat-4026	111	20	as	as	ADP
easat-4026	111	21	of	of	ADP
easat-4026	111	22	𝑅	𝑅	NOUN
easat-4026	111	23	to	to	ADP
easat-4026	111	24	𝑆.	𝑆.	PROPN
easat-4026	111	25	graded	grade	VERB
easat-4026	111	26	modules	module	NOUN
easat-4026	111	27	:	:	PUNCT
easat-4026	111	28	𝐻𝑜𝑚ℳ(𝑅	𝐻𝑜𝑚ℳ(𝑅	PROPN
easat-4026	111	29	,	,	PUNCT
easat-4026	111	30	𝑆	𝑆	PROPN
easat-4026	111	31	)	)	PUNCT
easat-4026	111	32	∶=⊕	∶=⊕	NOUN
easat-4026	111	33	𝚤∈ℤ	𝚤∈ℤ	X
easat-4026	112	1	𝐻𝑜𝑚ℳ	𝐻𝑜𝑚ℳ	ADJ
easat-4026	112	2	𝐺𝑟(𝑅	𝐺𝑟(𝑅	NOUN
easat-4026	112	3	,	,	PUNCT
easat-4026	112	4	𝑆)𝚤	𝑆)𝚤	NUM
easat-4026	112	5	,	,	PUNCT
easat-4026	112	6	using	use	VERB
easat-4026	112	7	the	the	DET
easat-4026	112	8	differential	differential	NOUN
easat-4026	112	9	𝒹𝐻𝑜𝑚	𝒹𝐻𝑜𝑚	PROPN
easat-4026	112	10	determined	determine	VERB
easat-4026	112	11	to	to	PART
easat-4026	112	12	be	be	AUX
easat-4026	112	13	:	:	PUNCT
easat-4026	112	14	𝒹𝐻𝑜𝑚(𝒽	𝒹𝐻𝑜𝑚(𝒽	NOUN
easat-4026	112	15	)	)	PUNCT
easat-4026	112	16	=	=	PUNCT
easat-4026	112	17	𝒹𝑆	𝒹𝑆	VERB
easat-4026	112	18	∘	∘	NOUN
easat-4026	112	19	𝒽	𝒽	PRON
easat-4026	112	20	−	−	PROPN
easat-4026	112	21	(	(	PUNCT
easat-4026	112	22	−1)|𝒽|𝒽	−1)|𝒽|𝒽	VERB
easat-4026	112	23	∘	∘	X
easat-4026	112	24	𝒹𝑅	𝒹𝑅	PROPN
easat-4026	112	25	,	,	PUNCT
easat-4026	112	26	∀	∀	X
easat-4026	112	27	𝒽	𝒽	PRON
easat-4026	112	28	∈	∈	PROPN
easat-4026	112	29	𝐻𝑜𝑚ℳ(𝑅	𝐻𝑜𝑚ℳ(𝑅	PROPN
easat-4026	112	30	,	,	PUNCT
easat-4026	112	31	𝑆	𝑆	PROPN
easat-4026	112	32	)	)	PUNCT
easat-4026	112	33	.	.	PUNCT
easat-4026	113	1	in	in	ADP
easat-4026	113	2	particular	particular	ADJ
easat-4026	113	3	,	,	PUNCT
easat-4026	113	4	differential	differential	ADJ
easat-4026	113	5	graded	grade	VERB
easat-4026	113	6	algebras	algebra	NOUN
easat-4026	113	7	are	be	AUX
easat-4026	113	8	used	use	VERB
easat-4026	113	9	to	to	PART
easat-4026	113	10	create	create	VERB
easat-4026	113	11	the	the	DET
easat-4026	113	12	graded	grade	VERB
easat-4026	113	13	vector	vector	NOUN
easat-4026	113	14	space	space	NOUN
easat-4026	113	15	𝐻𝑜𝑚ℳ(𝑅	𝐻𝑜𝑚ℳ(𝑅	PROPN
easat-4026	113	16	,	,	PUNCT
easat-4026	113	17	𝑅	𝑅	PROPN
easat-4026	113	18	)	)	PUNCT
easat-4026	113	19	,	,	PUNCT
easat-4026	113	20	where	where	SCONJ
easat-4026	113	21	𝐻𝑜𝑚ℳ(𝐿	𝐻𝑜𝑚ℳ(𝐿	PROPN
easat-4026	113	22	,	,	PUNCT
easat-4026	113	23	𝑅	𝑅	NOUN
easat-4026	113	24	)	)	PUNCT
easat-4026	113	25	is	be	AUX
easat-4026	113	26	a	a	DET
easat-4026	113	27	differential	differential	ADV
easat-4026	113	28	-	-	PUNCT
easat-4026	113	29	graded	grade	VERB
easat-4026	113	30	module	module	NOUN
easat-4026	113	31	over	over	ADP
easat-4026	113	32	𝐻𝑜𝑚ℳ(𝑅	𝐻𝑜𝑚ℳ(𝑅	PROPN
easat-4026	113	33	,	,	PUNCT
easat-4026	113	34	𝑅	𝑅	NOUN
easat-4026	113	35	)	)	PUNCT
easat-4026	113	36	differential	differential	NOUN
easat-4026	113	37	-	-	PUNCT
easat-4026	113	38	graded	grade	VERB
easat-4026	113	39	algebras	algebra	NOUN
easat-4026	113	40	.	.	PUNCT
easat-4026	114	1	next	next	ADV
easat-4026	114	2	,	,	PUNCT
easat-4026	114	3	we	we	PRON
easat-4026	114	4	will	will	AUX
easat-4026	114	5	describe	describe	VERB
easat-4026	114	6	how	how	SCONJ
easat-4026	114	7	to	to	PART
easat-4026	114	8	handle	handle	VERB
easat-4026	114	9	the	the	DET
easat-4026	114	10	tensor	tensor	NOUN
easat-4026	114	11	product	product	NOUN
easat-4026	114	12	of	of	ADP
easat-4026	114	13	graded	grade	VERB
easat-4026	114	14	spaces	space	NOUN
easat-4026	114	15	and	and	CCONJ
easat-4026	114	16	determine	determine	VERB
easat-4026	114	17	the	the	DET
easat-4026	114	18	graded	grade	VERB
easat-4026	114	19	map	map	NOUN
easat-4026	114	20	's	's	PART
easat-4026	114	21	properties	property	NOUN
easat-4026	114	22	in	in	ADP
easat-4026	114	23	this	this	DET
easat-4026	114	24	context	context	NOUN
easat-4026	114	25	.	.	PUNCT
easat-4026	115	1	2.13	2.13	NUM
easat-4026	115	2	.	.	PUNCT
easat-4026	115	3	definition	definition	NOUN
easat-4026	115	4	[	[	X
easat-4026	115	5	17	17	NUM
easat-4026	115	6	]	]	PUNCT
easat-4026	115	7	by	by	ADP
easat-4026	115	8	the	the	DET
easat-4026	115	9	differential	differential	ADJ
easat-4026	115	10	𝒹⊗	𝒹⊗	NOUN
easat-4026	115	11	and	and	CCONJ
easat-4026	115	12	the	the	DET
easat-4026	115	13	complex	complex	ADJ
easat-4026	115	14	ℳ	ℳ	PROPN
easat-4026	115	15	,	,	PUNCT
easat-4026	115	16	we	we	PRON
easat-4026	115	17	can	can	AUX
easat-4026	115	18	define	define	VERB
easat-4026	115	19	the	the	DET
easat-4026	115	20	graded	grade	VERB
easat-4026	115	21	vector	vector	NOUN
easat-4026	115	22	space	space	NOUN
easat-4026	115	23	for	for	ADP
easat-4026	115	24	a	a	DET
easat-4026	115	25	tensor	tensor	NOUN
easat-4026	115	26	product	product	NOUN
easat-4026	115	27	𝑅	𝑅	PROPN
easat-4026	115	28	⊗ℳ	⊗ℳ	PUNCT
easat-4026	115	29	𝑆	𝑆	PROPN
easat-4026	115	30	over	over	ADP
easat-4026	115	31	ℳ	ℳ	PROPN
easat-4026	115	32	as	as	ADP
easat-4026	115	33	:	:	PUNCT
easat-4026	115	34	𝒹⊗(𝑟	𝒹⊗(𝑟	NOUN
easat-4026	115	35	⊗ℳ	⊗ℳ	PROPN
easat-4026	115	36	𝑠	𝑠	NOUN
easat-4026	115	37	)	)	PUNCT
easat-4026	115	38	=	=	SYM
easat-4026	115	39	𝒹𝑅(𝑟	𝒹𝑅(𝑟	PROPN
easat-4026	115	40	)	)	PUNCT
easat-4026	115	41	⊗ℳ	⊗ℳ	PROPN
easat-4026	115	42	𝑠	𝑠	PROPN
easat-4026	116	1	+	+	CCONJ
easat-4026	116	2	(	(	PUNCT
easat-4026	116	3	−1)|𝑟|𝑟	−1)|𝑟|𝑟	INTJ
easat-4026	116	4	⊗ℳ	⊗ℳ	INTJ
easat-4026	116	5	𝒹𝑆(𝑠	𝒹𝑆(𝑠	PROPN
easat-4026	116	6	)	)	PUNCT
easat-4026	116	7	.	.	PUNCT
easat-4026	117	1	there	there	PRON
easat-4026	117	2	is	be	VERB
easat-4026	117	3	an	an	DET
easat-4026	117	4	adjoint	adjoint	NOUN
easat-4026	117	5	property	property	NOUN
easat-4026	117	6	between	between	ADP
easat-4026	117	7	𝐻𝑜𝑚ℳ	𝐻𝑜𝑚ℳ	ADJ
easat-4026	117	8	and	and	CCONJ
easat-4026	117	9	⊗ℳ	⊗ℳ	ADJ
easat-4026	117	10	:	:	PUNCT
easat-4026	118	1	𝐻𝑜𝑚ℳ(𝐿	𝐻𝑜𝑚ℳ(𝐿	PROPN
easat-4026	118	2	⊗ℳ	⊗ℳ	PROPN
easat-4026	118	3	𝑅	𝑅	PROPN
easat-4026	118	4	,	,	PUNCT
easat-4026	118	5	𝑆	𝑆	PROPN
easat-4026	118	6	)	)	PUNCT
easat-4026	118	7	≅	≅	PROPN
easat-4026	118	8	𝐻𝑜𝑚ℳ(𝐿	𝐻𝑜𝑚ℳ(𝐿	PROPN
easat-4026	118	9	,	,	PUNCT
easat-4026	118	10	𝐻𝑜𝑚ℳ(𝑅	𝐻𝑜𝑚ℳ(𝑅	PROPN
easat-4026	118	11	,	,	PUNCT
easat-4026	118	12	𝑆	𝑆	PROPN
easat-4026	118	13	)	)	PUNCT
easat-4026	118	14	)	)	PUNCT
easat-4026	118	15	.	.	PUNCT
easat-4026	119	1	here	here	ADV
easat-4026	119	2	,	,	PUNCT
easat-4026	119	3	we	we	PRON
easat-4026	119	4	will	will	AUX
easat-4026	119	5	define	define	VERB
easat-4026	119	6	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	119	7	and	and	CCONJ
easat-4026	119	8	how	how	SCONJ
easat-4026	119	9	to	to	PART
easat-4026	119	10	handle	handle	VERB
easat-4026	119	11	their	their	PRON
easat-4026	119	12	fundamental	fundamental	ADJ
easat-4026	119	13	properties	property	NOUN
easat-4026	119	14	,	,	PUNCT
easat-4026	119	15	such	such	ADJ
easat-4026	119	16	as	as	ADP
easat-4026	119	17	the	the	DET
easat-4026	119	18	stasheff	stasheff	NOUN
easat-4026	119	19	identity	identity	NOUN
easat-4026	119	20	.	.	PUNCT
easat-4026	120	1	2.14	2.14	NUM
easat-4026	120	2	.	.	PUNCT
easat-4026	121	1	definition	definition	NOUN
easat-4026	121	2	[	[	X
easat-4026	121	3	18	18	NUM
easat-4026	121	4	]	]	PUNCT
easat-4026	121	5	a	a	DET
easat-4026	121	6	vector	vector	NOUN
easat-4026	121	7	space	space	NOUN
easat-4026	121	8	with	with	ADP
easat-4026	121	9	ℤ	ℤ	PROPN
easat-4026	121	10	grades	grade	NOUN
easat-4026	121	11	is	be	AUX
easat-4026	121	12	an	an	DET
easat-4026	121	13	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	121	14	on	on	ADP
easat-4026	121	15	a	a	DET
easat-4026	121	16	field	field	NOUN
easat-4026	121	17	𝑀	𝑀	NOUN
easat-4026	121	18	such	such	ADJ
easat-4026	121	19	that	that	SCONJ
easat-4026	121	20	:	:	PUNCT
easat-4026	121	21	𝒜	𝒜	NOUN
easat-4026	121	22	=	=	SYM
easat-4026	121	23	⊕	⊕	PROPN
easat-4026	121	24	𝓅∈ℤ	𝓅∈ℤ	ADJ
easat-4026	121	25	𝒜𝒫	𝒜𝒫	PROPN
easat-4026	121	26	,	,	PUNCT
easat-4026	121	27	supplied	supply	VERB
easat-4026	121	28	with	with	ADP
easat-4026	121	29	graded	grade	VERB
easat-4026	121	30	maps	map	NOUN
easat-4026	121	31	that	that	PRON
easat-4026	121	32	are	be	AUX
easat-4026	121	33	homogenous	homogenous	ADJ
easat-4026	121	34	𝑀-linear	𝑀-linear	PROPN
easat-4026	121	35	mappings	mapping	NOUN
easat-4026	121	36	;	;	PUNCT
easat-4026	121	37	𝑟𝑛:𝒜	𝑟𝑛:𝒜	PROPN
easat-4026	121	38	⊗𝑛	⊗𝑛	PROPN
easat-4026	121	39	→	→	SYM
easat-4026	121	40	𝒜	𝒜	PROPN
easat-4026	121	41	,	,	PUNCT
easat-4026	121	42	𝑛	𝑛	PRON
easat-4026	121	43	≥	≥	NUM
easat-4026	121	44	1	1	NUM
easat-4026	121	45	.	.	PUNCT
easat-4026	121	46	of	of	ADP
easat-4026	121	47	(	(	PUNCT
easat-4026	121	48	|𝑟𝑛|	|𝑟𝑛|	PROPN
easat-4026	121	49	=	=	SYM
easat-4026	121	50	2	2	NUM
easat-4026	121	51	−	−	NOUN
easat-4026	121	52	𝑛)-degree	𝑛)-degree	PUNCT
easat-4026	121	53	,	,	PUNCT
easat-4026	121	54	fulfilling	fulfil	VERB
easat-4026	121	55	the	the	DET
easat-4026	121	56	subsequent	subsequent	ADJ
easat-4026	121	57	requirements	requirement	NOUN
easat-4026	121	58	of	of	ADP
easat-4026	121	59	stasheff	stasheff	NOUN
easat-4026	121	60	identities	identity	NOUN
easat-4026	121	61	:	:	PUNCT
easat-4026	121	62	∀	∀	PUNCT
easat-4026	121	63	𝑛	𝑛	PRON
easat-4026	121	64	∈	∈	PROPN
easat-4026	121	65	ℕ	ℕ	PROPN
easat-4026	121	66	,	,	PUNCT
easat-4026	121	67	(	(	PUNCT
easat-4026	121	68	𝑆𝐿(𝑛	𝑆𝐿(𝑛	NOUN
easat-4026	121	69	)	)	PUNCT
easat-4026	121	70	)	)	PUNCT
easat-4026	122	1	∑(−1)𝑚+𝑠𝑡𝑟𝓆(𝑖𝑑	∑(−1)𝑚+𝑠𝑡𝑟𝓆(𝑖𝑑	PROPN
easat-4026	122	2	⊗𝑚	⊗𝑚	VERB
easat-4026	122	3	⊗	⊗	PROPN
easat-4026	122	4	𝑟𝑠⊗	𝑟𝑠⊗	ADP
easat-4026	122	5	𝑖𝑑⊗𝑡	𝑖𝑑⊗𝑡	NOUN
easat-4026	122	6	)	)	PUNCT
easat-4026	122	7	=	=	SYM
easat-4026	122	8	0	0	NUM
easat-4026	123	1	(	(	PUNCT
easat-4026	123	2	3	3	NUM
easat-4026	123	3	)	)	PUNCT
easat-4026	123	4	where	where	SCONJ
easat-4026	123	5	the	the	DET
easat-4026	123	6	total	total	NOUN
easat-4026	123	7	is	be	AUX
easat-4026	123	8	applied	apply	VERB
easat-4026	123	9	to	to	ADP
easat-4026	123	10	all	all	DET
easat-4026	123	11	decompositions	decomposition	NOUN
easat-4026	124	1	𝑛	𝑛	PRON
easat-4026	124	2	=	=	SYM
easat-4026	124	3	𝑚	𝑚	PROPN
easat-4026	124	4	+	+	NUM
easat-4026	124	5	𝑠	𝑠	PROPN
easat-4026	124	6	+	+	CCONJ
easat-4026	124	7	𝑡	𝑡	PROPN
easat-4026	124	8	,	,	PUNCT
easat-4026	124	9	𝑚	𝑚	PROPN
easat-4026	124	10	,	,	PUNCT
easat-4026	124	11	𝑡	𝑡	X
easat-4026	124	12	≥	≥	NOUN
easat-4026	124	13	0	0	NUM
easat-4026	124	14	and	and	CCONJ
easat-4026	124	15	𝑠	𝑠	PRON
easat-4026	124	16	≥	≥	NUM
easat-4026	124	17	1	1	NUM
easat-4026	124	18	,	,	PUNCT
easat-4026	124	19	and	and	CCONJ
easat-4026	124	20	𝓆	𝓆	X
easat-4026	124	21	=	=	SYM
easat-4026	124	22	𝑚	𝑚	PROPN
easat-4026	124	23	+	+	NOUN
easat-4026	124	24	1	1	NUM
easat-4026	124	25	+	+	NUM
easat-4026	124	26	𝑡.	𝑡.	NOUN
easat-4026	124	27	thus	thus	ADV
easat-4026	124	28	,	,	PUNCT
easat-4026	124	29	𝑖𝑑	𝑖𝑑	ADP
easat-4026	124	30	refers	refer	NOUN
easat-4026	124	31	to	to	ADP
easat-4026	124	32	the	the	DET
easat-4026	124	33	identification	identification	NOUN
easat-4026	124	34	map	map	NOUN
easat-4026	124	35	of	of	ADP
easat-4026	124	36	𝒜.	𝒜.	NOUN
easat-4026	124	37	because	because	SCONJ
easat-4026	124	38	of	of	ADP
easat-4026	124	39	the	the	DET
easat-4026	124	40	koszul	koszul	ADJ
easat-4026	124	41	sign	sign	NOUN
easat-4026	124	42	rule	rule	NOUN
easat-4026	124	43	,	,	PUNCT
easat-4026	124	44	additional	additional	ADJ
easat-4026	124	45	signs	sign	NOUN
easat-4026	124	46	occur	occur	VERB
easat-4026	124	47	when	when	SCONJ
easat-4026	124	48	these	these	DET
easat-4026	124	49	formulas	formula	NOUN
easat-4026	124	50	get	get	AUX
easat-4026	124	51	used	use	VERB
easat-4026	124	52	on	on	ADP
easat-4026	124	53	elements	element	NOUN
easat-4026	124	54	.	.	PUNCT
easat-4026	125	1	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	125	2	are	be	AUX
easat-4026	125	3	known	know	VERB
easat-4026	125	4	as	as	ADP
easat-4026	125	5	strongly	strongly	ADV
easat-4026	125	6	homotopy	homotopy	VERB
easat-4026	125	7	associative	associative	ADJ
easat-4026	125	8	algebras	algebra	NOUN
easat-4026	125	9	.	.	PUNCT
easat-4026	126	1	for	for	ADP
easat-4026	126	2	example	example	NOUN
easat-4026	126	3	:	:	PUNCT
easat-4026	126	4	1	1	X
easat-4026	126	5	)	)	PUNCT
easat-4026	126	6	𝑆𝐿(1	𝑆𝐿(1	PROPN
easat-4026	126	7	)	)	PUNCT
easat-4026	126	8	means	mean	VERB
easat-4026	126	9	that	that	SCONJ
easat-4026	126	10	:	:	PUNCT
easat-4026	126	11	𝑟1	𝑟1	NOUN
easat-4026	126	12	∘	∘	NOUN
easat-4026	126	13	𝑟1	𝑟1	PROPN
easat-4026	126	14	=	=	PUNCT
easat-4026	126	15	0	0	X
easat-4026	126	16	.	.	PUNCT
easat-4026	126	17	where	where	SCONJ
easat-4026	126	18	𝑟1	𝑟1	PROPN
easat-4026	126	19	has	have	VERB
easat-4026	126	20	a	a	DET
easat-4026	126	21	degree	degree	NOUN
easat-4026	126	22	of	of	ADP
easat-4026	126	23	1	1	NUM
easat-4026	126	24	,	,	PUNCT
easat-4026	126	25	which	which	PRON
easat-4026	126	26	means	mean	VERB
easat-4026	126	27	that	that	SCONJ
easat-4026	126	28	𝑟1	𝑟1	PROPN
easat-4026	126	29	is	be	AUX
easat-4026	126	30	a	a	DET
easat-4026	126	31	derivative	derivative	NOUN
easat-4026	126	32	of	of	ADP
easat-4026	126	33	𝒜.	𝒜.	PROPN
easat-4026	126	34	2	2	NUM
easat-4026	126	35	)	)	PUNCT
easat-4026	126	36	𝑆𝐿(2	𝑆𝐿(2	PROPN
easat-4026	126	37	)	)	PUNCT
easat-4026	126	38	says	say	VERB
easat-4026	126	39	that	that	SCONJ
easat-4026	126	40	𝑟1	𝑟1	PROPN
easat-4026	126	41	is	be	AUX
easat-4026	126	42	a	a	DET
easat-4026	126	43	derivation	derivation	NOUN
easat-4026	126	44	for	for	ADP
easat-4026	126	45	𝑟2	𝑟2	NOUN
easat-4026	126	46	,	,	PUNCT
easat-4026	126	47	such	such	ADJ
easat-4026	126	48	that	that	SCONJ
easat-4026	126	49	:	:	PUNCT
easat-4026	126	50	𝑟1	𝑟1	NOUN
easat-4026	126	51	∘	∘	NOUN
easat-4026	126	52	𝑟2	𝑟2	NOUN
easat-4026	126	53	=	=	SYM
easat-4026	126	54	𝑟2	𝑟2	NOUN
easat-4026	126	55	∘	∘	X
easat-4026	126	56	(	(	PUNCT
easat-4026	126	57	𝑟1⊗	𝑟1⊗	NOUN
easat-4026	126	58	𝑖𝑑	𝑖𝑑	ADP
easat-4026	126	59	+	+	CCONJ
easat-4026	126	60	𝑖𝑑	𝑖𝑑	ADP
easat-4026	126	61	⊗	⊗	PROPN
easat-4026	126	62	𝑟1	𝑟1	PROPN
easat-4026	126	63	)	)	PUNCT
easat-4026	126	64	the	the	DET
easat-4026	126	65	degree	degree	NOUN
easat-4026	126	66	of	of	ADP
easat-4026	126	67	𝑟2	𝑟2	NOUN
easat-4026	126	68	is	be	AUX
easat-4026	126	69	zero	zero	NUM
easat-4026	126	70	.	.	PUNCT
easat-4026	127	1	3	3	X
easat-4026	127	2	)	)	PUNCT
easat-4026	127	3	𝑆𝐿(3	𝑆𝐿(3	PROPN
easat-4026	127	4	)	)	PUNCT
easat-4026	127	5	indicates	indicate	VERB
easat-4026	127	6	that	that	SCONJ
easat-4026	127	7	𝑟2	𝑟2	NOUN
easat-4026	127	8	is	be	AUX
easat-4026	127	9	associative	associative	ADJ
easat-4026	127	10	till	till	SCONJ
easat-4026	127	11	the	the	DET
easat-4026	127	12	homotopy	homotopy	NOUN
easat-4026	127	13	𝑟3	𝑟3	NOUN
easat-4026	127	14	can	can	AUX
easat-4026	127	15	be	be	AUX
easat-4026	127	16	re	re	VERB
easat-4026	127	17	-	-	VERB
easat-4026	127	18	written	write	VERB
easat-4026	127	19	as	as	ADP
easat-4026	127	20	:	:	PUNCT
easat-4026	127	21	9477	9477	NUM
easat-4026	127	22	edelweiss	edelweiss	PROPN
easat-4026	127	23	applied	apply	VERB
easat-4026	127	24	science	science	NOUN
easat-4026	127	25	and	and	CCONJ
easat-4026	127	26	technology	technology	NOUN
easat-4026	127	27	issn	issn	PROPN
easat-4026	127	28	:	:	PUNCT
easat-4026	127	29	2576	2576	NUM
easat-4026	127	30	-	-	SYM
easat-4026	127	31	8484	8484	NUM
easat-4026	127	32	vol	vol	NOUN
easat-4026	127	33	.	.	PROPN
easat-4026	127	34	8	8	NUM
easat-4026	127	35	,	,	PUNCT
easat-4026	127	36	no	no	INTJ
easat-4026	127	37	.	.	NOUN
easat-4026	127	38	6	6	NUM
easat-4026	127	39	:	:	SYM
easat-4026	127	40	9472	9472	NUM
easat-4026	127	41	-	-	SYM
easat-4026	127	42	9486	9486	NUM
easat-4026	127	43	,	,	PUNCT
easat-4026	127	44	2024	2024	NUM
easat-4026	127	45	doi	doi	NOUN
easat-4026	127	46	:	:	PUNCT
easat-4026	127	47	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	127	48	©	©	PROPN
easat-4026	127	49	2024	2024	NUM
easat-4026	127	50	by	by	ADP
easat-4026	127	51	the	the	DET
easat-4026	127	52	authors	author	NOUN
easat-4026	127	53	;	;	PUNCT
easat-4026	127	54	licensee	licensee	PROPN
easat-4026	127	55	learning	learning	NOUN
easat-4026	127	56	gate	gate	NOUN
easat-4026	127	57	𝑟2	𝑟2	NOUN
easat-4026	127	58	∘	∘	X
easat-4026	127	59	(	(	PUNCT
easat-4026	127	60	𝑖𝑑	𝑖𝑑	ADP
easat-4026	127	61	⊗	⊗	PROPN
easat-4026	127	62	𝑟2	𝑟2	NOUN
easat-4026	127	63	−	−	NOUN
easat-4026	127	64	𝑟2	𝑟2	NOUN
easat-4026	127	65	⊗	⊗	PROPN
easat-4026	127	66	𝑖𝑑	𝑖𝑑	ADP
easat-4026	127	67	)	)	PUNCT
easat-4026	127	68	=	=	NOUN
easat-4026	127	69	𝜕(𝑟3	𝜕(𝑟3	NOUN
easat-4026	127	70	)	)	PUNCT
easat-4026	127	71	=	=	SYM
easat-4026	127	72	𝑟1	𝑟1	NOUN
easat-4026	127	73	∘	∘	NOUN
easat-4026	127	74	𝑟3	𝑟3	NOUN
easat-4026	127	75	+	+	CCONJ
easat-4026	127	76	𝑟3	𝑟3	NOUN
easat-4026	127	77	∘	∘	X
easat-4026	127	78	(	(	PUNCT
easat-4026	127	79	𝑟1⊗	𝑟1⊗	NOUN
easat-4026	127	80	𝑖𝑑	𝑖𝑑	ADP
easat-4026	127	81	⊗	⊗	PROPN
easat-4026	127	82	𝑖𝑑	𝑖𝑑	ADP
easat-4026	128	1	+	+	CCONJ
easat-4026	128	2	𝑖𝑑	𝑖𝑑	ADP
easat-4026	128	3	⊗	⊗	PROPN
easat-4026	128	4	𝑟1⊗	𝑟1⊗	PROPN
easat-4026	129	1	𝑖𝑑	𝑖𝑑	ADP
easat-4026	129	2	+	+	CCONJ
easat-4026	130	1	𝑖𝑑	𝑖𝑑	ADP
easat-4026	130	2	⊗	⊗	PROPN
easat-4026	130	3	𝑖𝑑	𝑖𝑑	ADP
easat-4026	130	4	⊗	⊗	PROPN
easat-4026	130	5	𝑟1	𝑟1	PROPN
easat-4026	130	6	)	)	PUNCT
easat-4026	130	7	where	where	SCONJ
easat-4026	130	8	𝜕	𝜕	NOUN
easat-4026	130	9	is	be	AUX
easat-4026	130	10	the	the	DET
easat-4026	130	11	differential	differential	NOUN
easat-4026	130	12	of	of	ADP
easat-4026	130	13	𝐻𝑜𝑚(𝒜⊗3,𝒜	𝐻𝑜𝑚(𝒜⊗3,𝒜	NOUN
easat-4026	130	14	)	)	PUNCT
easat-4026	130	15	induced	induce	VERB
easat-4026	130	16	by	by	ADP
easat-4026	130	17	𝑟1	𝑟1	NOUN
easat-4026	130	18	.	.	PUNCT
easat-4026	131	1	4	4	NUM
easat-4026	131	2	)	)	PUNCT
easat-4026	131	3	pentagonal	pentagonal	PROPN
easat-4026	131	4	homotopy	homotopy	NOUN
easat-4026	131	5	associative	associative	NOUN
easat-4026	131	6	algebra	algebra	PROPN
easat-4026	131	7	(	(	PUNCT
easat-4026	131	8	𝒜	𝒜	NOUN
easat-4026	131	9	,	,	PUNCT
easat-4026	131	10	𝑟1	𝑟1	NOUN
easat-4026	131	11	,	,	PUNCT
easat-4026	131	12	𝑟2	𝑟2	NOUN
easat-4026	131	13	,	,	PUNCT
easat-4026	131	14	𝑟3	𝑟3	NOUN
easat-4026	131	15	):	):	PUNCT
easat-4026	131	16	𝑆𝐿(4	𝑆𝐿(4	NOUN
easat-4026	131	17	):	):	PUNCT
easat-4026	131	18	𝑟2	𝑟2	NOUN
easat-4026	131	19	∘	∘	X
easat-4026	131	20	(	(	PUNCT
easat-4026	131	21	𝑖𝑑	𝑖𝑑	ADP
easat-4026	131	22	⊗	⊗	ADJ
easat-4026	131	23	𝑟3	𝑟3	NOUN
easat-4026	131	24	+	+	CCONJ
easat-4026	131	25	𝑟3	𝑟3	NOUN
easat-4026	131	26	⊗	⊗	NOUN
easat-4026	131	27	𝑖𝑑	𝑖𝑑	ADP
easat-4026	131	28	)	)	PUNCT
easat-4026	131	29	=	=	SYM
easat-4026	131	30	𝑟3	𝑟3	NOUN
easat-4026	131	31	∘	∘	NOUN
easat-4026	131	32	(	(	PUNCT
easat-4026	131	33	𝑟2⊗	𝑟2⊗	X
easat-4026	131	34	𝑖𝑑	𝑖𝑑	ADP
easat-4026	131	35	⊗	⊗	PROPN
easat-4026	131	36	𝑖𝑑	𝑖𝑑	ADP
easat-4026	131	37	−	−	PROPN
easat-4026	131	38	𝑖𝑑	𝑖𝑑	ADP
easat-4026	131	39	⊗	⊗	PROPN
easat-4026	131	40	𝑟2	𝑟2	PROPN
easat-4026	132	1	⊗	⊗	PROPN
easat-4026	132	2	𝑖𝑑	𝑖𝑑	ADP
easat-4026	133	1	+	+	CCONJ
easat-4026	134	1	𝑖𝑑	𝑖𝑑	ADP
easat-4026	134	2	⊗	⊗	PROPN
easat-4026	134	3	𝑖𝑑	𝑖𝑑	ADP
easat-4026	134	4	⊗	⊗	PROPN
easat-4026	134	5	𝑟2	𝑟2	NOUN
easat-4026	134	6	)	)	PUNCT
easat-4026	134	7	.	.	PUNCT
easat-4026	135	1	we	we	PRON
easat-4026	135	2	will	will	AUX
easat-4026	135	3	present	present	VERB
easat-4026	135	4	an	an	DET
easat-4026	135	5	example	example	NOUN
easat-4026	135	6	of	of	ADP
easat-4026	135	7	an	an	DET
easat-4026	135	8	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-4026	135	9	with	with	ADP
easat-4026	135	10	degree	degree	NOUN
easat-4026	135	11	0	0	PUNCT
easat-4026	135	12	and	and	CCONJ
easat-4026	135	13	analyze	analyze	VERB
easat-4026	135	14	its	its	PRON
easat-4026	135	15	properties	property	NOUN
easat-4026	135	16	through	through	ADP
easat-4026	135	17	specific	specific	ADJ
easat-4026	135	18	assignments	assignment	NOUN
easat-4026	135	19	.	.	PUNCT
easat-4026	136	1	2.15	2.15	NUM
easat-4026	136	2	.	.	PUNCT
easat-4026	136	3	example	example	NOUN
easat-4026	136	4	let	let	VERB
easat-4026	136	5	the	the	DET
easat-4026	136	6	associative	associative	ADJ
easat-4026	136	7	algebras	algebras	PROPN
easat-4026	136	8	𝒜	𝒜	NOUN
easat-4026	136	9	be	be	AUX
easat-4026	136	10	an	an	DET
easat-4026	136	11	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	136	12	focused	focus	VERB
easat-4026	136	13	at	at	ADP
easat-4026	136	14	degree	degree	NOUN
easat-4026	136	15	0	0	NUM
easat-4026	136	16	through	through	ADP
easat-4026	136	17	all	all	DET
easat-4026	136	18	multiplications	multiplication	NOUN
easat-4026	136	19	𝑟𝑛	𝑟𝑛	ADP
easat-4026	137	1	=	=	SYM
easat-4026	137	2	0	0	NUM
easat-4026	137	3	for	for	ADP
easat-4026	137	4	𝑛	𝑛	DET
easat-4026	137	5	≠	≠	PROPN
easat-4026	137	6	2	2	NUM
easat-4026	137	7	.	.	PUNCT
easat-4026	137	8	as	as	ADP
easat-4026	137	9	a	a	DET
easat-4026	137	10	result	result	NOUN
easat-4026	137	11	,	,	PUNCT
easat-4026	137	12	the	the	DET
easat-4026	137	13	associative	associative	NOUN
easat-4026	137	14	algebras	algebra	NOUN
easat-4026	137	15	make	make	VERB
easat-4026	137	16	a	a	DET
easat-4026	137	17	subclass	subclass	NOUN
easat-4026	137	18	of	of	ADP
easat-4026	137	19	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	137	20	with	with	ADP
easat-4026	137	21	the	the	DET
easat-4026	137	22	form	form	NOUN
easat-4026	137	23	(	(	PUNCT
easat-4026	137	24	𝒜	𝒜	NOUN
easat-4026	137	25	,	,	PUNCT
easat-4026	137	26	𝑟2	𝑟2	NOUN
easat-4026	137	27	)	)	PUNCT
easat-4026	137	28	.	.	PUNCT
easat-4026	138	1	we	we	PRON
easat-4026	138	2	will	will	AUX
easat-4026	138	3	define	define	VERB
easat-4026	138	4	morphisms	morphism	NOUN
easat-4026	138	5	between	between	ADP
easat-4026	138	6	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	138	7	and	and	CCONJ
easat-4026	138	8	describe	describe	VERB
easat-4026	138	9	how	how	SCONJ
easat-4026	138	10	to	to	PART
easat-4026	138	11	handle	handle	VERB
easat-4026	138	12	these	these	DET
easat-4026	138	13	morphisms	morphism	NOUN
easat-4026	138	14	'	'	PART
easat-4026	138	15	necessary	necessary	ADJ
easat-4026	138	16	conditions	condition	NOUN
easat-4026	138	17	and	and	CCONJ
easat-4026	138	18	properties	property	NOUN
easat-4026	138	19	.	.	PUNCT
easat-4026	139	1	2.16	2.16	NUM
easat-4026	139	2	.	.	PUNCT
easat-4026	140	1	definition	definition	NOUN
easat-4026	140	2	[	[	X
easat-4026	140	3	19	19	NUM
easat-4026	140	4	]	]	PUNCT
easat-4026	140	5	suppose	suppose	VERB
easat-4026	140	6	that	that	SCONJ
easat-4026	140	7	𝒜	𝒜	NOUN
easat-4026	140	8	and	and	CCONJ
easat-4026	140	9	ℬ	ℬ	NOUN
easat-4026	140	10	are	be	AUX
easat-4026	140	11	both	both	PRON
easat-4026	140	12	𝒜∞-algebras	𝒜∞-algebras	NUM
easat-4026	140	13	,	,	PUNCT
easat-4026	140	14	then	then	ADV
easat-4026	140	15	the	the	DET
easat-4026	140	16	family	family	NOUN
easat-4026	140	17	of	of	ADP
easat-4026	140	18	𝑀-linear	𝑀-linear	PROPN
easat-4026	140	19	graded	grade	VERB
easat-4026	140	20	maps	map	NOUN
easat-4026	140	21	is	be	AUX
easat-4026	140	22	the	the	DET
easat-4026	140	23	morphism	morphism	NOUN
easat-4026	140	24	𝒽:𝒜	𝒽:𝒜	PROPN
easat-4026	140	25	→	→	SYM
easat-4026	140	26	ℬ	ℬ	NOUN
easat-4026	140	27	of	of	ADP
easat-4026	140	28	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	140	29	such	such	ADJ
easat-4026	140	30	that	that	PRON
easat-4026	140	31	:	:	PUNCT
easat-4026	140	32	𝒽𝑛:𝒜	𝒽𝑛:𝒜	PROPN
easat-4026	140	33	⊗𝑛	⊗𝑛	PROPN
easat-4026	140	34	→	→	SYM
easat-4026	140	35	ℬ	ℬ	SYM
easat-4026	140	36	𝑛	𝑛	PRON
easat-4026	140	37	≥	≥	NUM
easat-4026	140	38	1	1	NUM
easat-4026	140	39	of	of	ADP
easat-4026	140	40	degree	degree	NOUN
easat-4026	140	41	(	(	PUNCT
easat-4026	140	42	1	1	NUM
easat-4026	140	43	−	−	NUM
easat-4026	140	44	𝑛	𝑛	NOUN
easat-4026	140	45	)	)	PUNCT
easat-4026	140	46	,	,	PUNCT
easat-4026	140	47	where	where	SCONJ
easat-4026	140	48	the	the	DET
easat-4026	140	49	subsequent	subsequent	ADJ
easat-4026	140	50	identities	identity	NOUN
easat-4026	140	51	apply	apply	VERB
easat-4026	140	52	for	for	ADP
easat-4026	140	53	each	each	DET
easat-4026	140	54	𝑛	𝑛	DET
easat-4026	140	55	≥	≥	NOUN
easat-4026	140	56	1	1	NUM
easat-4026	140	57	:	:	PUNCT
easat-4026	140	58	∑	∑	PUNCT
easat-4026	140	59	(	(	PUNCT
easat-4026	140	60	−1)𝑚+𝑠𝑡𝒽𝑚+1+𝑡	−1)𝑚+𝑠𝑡𝒽𝑚+1+𝑡	PROPN
easat-4026	140	61	∘	∘	X
easat-4026	140	62	(	(	PUNCT
easat-4026	140	63	𝑖𝑑	𝑖𝑑	INTJ
easat-4026	140	64	⊗𝑚	⊗𝑚	PROPN
easat-4026	140	65	⊗𝑟𝑠	⊗𝑟𝑠	PROPN
easat-4026	140	66	⊗	⊗	NUM
easat-4026	140	67	𝑖𝑑⊗𝑡	𝑖𝑑⊗𝑡	NOUN
easat-4026	140	68	)	)	PUNCT
easat-4026	140	69	𝑚+𝑠+𝑡=1	𝑚+𝑠+𝑡=1	PROPN
easat-4026	140	70	𝑚,𝑡≥0	𝑚,𝑡≥0	DET
easat-4026	140	71	𝑠≥1	𝑠≥1	PROPN
easat-4026	140	72	=	=	SYM
easat-4026	140	73	∑	∑	PROPN
easat-4026	140	74	∑	∑	PUNCT
easat-4026	140	75	(	(	PUNCT
easat-4026	140	76	−1)𝓆𝑟𝑗	−1)𝓆𝑟𝑗	PROPN
easat-4026	140	77	′(𝒽𝑖1⊗𝒽𝑖2	′(𝒽𝑖1⊗𝒽𝑖2	VERB
easat-4026	140	78	⊗	⊗	NOUN
easat-4026	140	79	…	…	SYM
easat-4026	140	80	⊗𝒽𝑖𝑗	⊗𝒽𝑖𝑗	NOUN
easat-4026	140	81	)	)	PUNCT
easat-4026	140	82	𝑖1+···+𝑖𝑗=𝑛	𝑖1+···+𝑖𝑗=𝑛	NOUN
easat-4026	141	1	𝑛	𝑛	PRON
easat-4026	141	2	𝑗=1	𝑗=1	PROPN
easat-4026	141	3	,	,	PUNCT
easat-4026	141	4	(	(	PUNCT
easat-4026	141	5	𝑀𝐿(𝑛	𝑀𝐿(𝑛	NOUN
easat-4026	141	6	)	)	PUNCT
easat-4026	141	7	)	)	PUNCT
easat-4026	142	1	where	where	SCONJ
easat-4026	142	2	:	:	PUNCT
easat-4026	142	3	𝓆	𝓆	X
easat-4026	142	4	=	=	SYM
easat-4026	142	5	(	(	PUNCT
easat-4026	142	6	𝑖𝑗−1	𝑖𝑗−1	NOUN
easat-4026	142	7	−	−	PROPN
easat-4026	142	8	1	1	NUM
easat-4026	142	9	)	)	PUNCT
easat-4026	142	10	+	+	CCONJ
easat-4026	142	11	2(𝑖𝑗−2	2(𝑖𝑗−2	NUM
easat-4026	142	12	−	−	NUM
easat-4026	142	13	1	1	NUM
easat-4026	142	14	)	)	PUNCT
easat-4026	142	15	+	+	NOUN
easat-4026	142	16	⋯+	⋯+	NOUN
easat-4026	142	17	(	(	PUNCT
easat-4026	142	18	𝑗	𝑗	X
easat-4026	142	19	−	−	PROPN
easat-4026	142	20	2)(𝑖2	2)(𝑖2	NUM
easat-4026	142	21	−	−	NOUN
easat-4026	142	22	1	1	NUM
easat-4026	142	23	)	)	PUNCT
easat-4026	142	24	+	+	CCONJ
easat-4026	142	25	(	(	PUNCT
easat-4026	142	26	𝑗	𝑗	INTJ
easat-4026	142	27	−	−	PROPN
easat-4026	142	28	1)(𝑖1	1)(𝑖1	NUM
easat-4026	142	29	−	−	PROPN
easat-4026	142	30	1	1	NUM
easat-4026	142	31	)	)	PUNCT
easat-4026	142	32	.	.	PUNCT
easat-4026	143	1	assuming	assume	VERB
easat-4026	143	2	𝒜	𝒜	NOUN
easat-4026	143	3	and	and	CCONJ
easat-4026	143	4	ℬ	ℬ	NOUN
easat-4026	143	5	are	be	AUX
easat-4026	143	6	both	both	ADV
easat-4026	143	7	unital	unital	ADJ
easat-4026	143	8	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	143	9	of	of	ADP
easat-4026	143	10	stringent	stringent	ADJ
easat-4026	143	11	units	unit	NOUN
easat-4026	143	12	1𝒜	1𝒜	NOUN
easat-4026	143	13	and	and	CCONJ
easat-4026	143	14	1ℬ	1ℬ	NOUN
easat-4026	143	15	respectively	respectively	ADV
easat-4026	143	16	,	,	PUNCT
easat-4026	143	17	so	so	CCONJ
easat-4026	143	18	𝒽	𝒽	DET
easat-4026	143	19	additionally	additionally	ADV
easat-4026	143	20	needs	need	VERB
easat-4026	143	21	to	to	PART
easat-4026	143	22	fulfill	fulfill	VERB
easat-4026	143	23	the	the	DET
easat-4026	143	24	following	follow	VERB
easat-4026	143	25	unital	unital	ADJ
easat-4026	143	26	morphism	morphism	NOUN
easat-4026	143	27	requirements	requirement	NOUN
easat-4026	143	28	:	:	PUNCT
easat-4026	143	29	1	1	X
easat-4026	143	30	)	)	PUNCT
easat-4026	143	31	as	as	SCONJ
easat-4026	143	32	required	require	VERB
easat-4026	143	33	for	for	ADP
easat-4026	143	34	ring	ring	NOUN
easat-4026	143	35	morphisms	morphism	NOUN
easat-4026	143	36	,	,	PUNCT
easat-4026	143	37	𝒽1(1𝒜	𝒽1(1𝒜	ADJ
easat-4026	143	38	)	)	PUNCT
easat-4026	143	39	=	=	SYM
easat-4026	144	1	1ℬ.	1ℬ.	NUM
easat-4026	144	2	2	2	X
easat-4026	144	3	)	)	PUNCT
easat-4026	144	4	if	if	SCONJ
easat-4026	144	5	𝒶𝑖	𝒶𝑖	ADV
easat-4026	144	6	=	=	PUNCT
easat-4026	144	7	1𝒜	1𝒜	PROPN
easat-4026	144	8	,	,	PUNCT
easat-4026	144	9	then	then	ADV
easat-4026	144	10	𝒽𝑛(𝒶1⊗	𝒽𝑛(𝒶1⊗	NUM
easat-4026	144	11	…	…	SYM
easat-4026	144	12	⊗𝒶𝑛	⊗𝒶𝑛	NUM
easat-4026	144	13	)	)	PUNCT
easat-4026	145	1	=	=	SYM
easat-4026	145	2	0	0	NUM
easat-4026	145	3	for	for	ADP
easat-4026	145	4	all	all	DET
easat-4026	145	5	𝑛	𝑛	PRON
easat-4026	145	6	≥	≥	NUM
easat-4026	145	7	2	2	NUM
easat-4026	145	8	,	,	PUNCT
easat-4026	145	9	𝑖	𝑖	SYM
easat-4026	145	10	∈	∈	PROPN
easat-4026	145	11	{	{	PUNCT
easat-4026	145	12	1	1	NUM
easat-4026	145	13	,	,	PUNCT
easat-4026	145	14	…	…	PUNCT
easat-4026	145	15	,	,	PUNCT
easat-4026	145	16	𝑛	𝑛	PROPN
easat-4026	145	17	}	}	PUNCT
easat-4026	145	18	.	.	PUNCT
easat-4026	146	1	3	3	X
easat-4026	146	2	)	)	PUNCT
easat-4026	146	3	a	a	DET
easat-4026	146	4	stringent	stringent	ADJ
easat-4026	146	5	morphism	morphism	NOUN
easat-4026	146	6	over	over	ADP
easat-4026	146	7	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	146	8	is	be	AUX
easat-4026	146	9	𝒽:𝒜	𝒽:𝒜	NOUN
easat-4026	146	10	→	→	SYM
easat-4026	146	11	ℬ	ℬ	NOUN
easat-4026	146	12	when	when	SCONJ
easat-4026	146	13	𝒽𝑛	𝒽𝑛	PROPN
easat-4026	146	14	=	=	SYM
easat-4026	146	15	0	0	PROPN
easat-4026	146	16	for	for	ADP
easat-4026	146	17	all	all	DET
easat-4026	146	18	𝑛	𝑛	PRON
easat-4026	146	19	≥	≥	NOUN
easat-4026	146	20	2	2	NUM
easat-4026	146	21	.	.	NOUN
easat-4026	146	22	4	4	NUM
easat-4026	146	23	)	)	PUNCT
easat-4026	146	24	when	when	SCONJ
easat-4026	146	25	𝒽:𝒜	𝒽:𝒜	PROPN
easat-4026	146	26	→	→	SYM
easat-4026	146	27	ℬ	ℬ	PROPN
easat-4026	146	28	is	be	AUX
easat-4026	146	29	a	a	DET
easat-4026	146	30	stringent	stringent	ADJ
easat-4026	146	31	morphism	morphism	NOUN
easat-4026	146	32	,	,	PUNCT
easat-4026	146	33	the	the	DET
easat-4026	146	34	morphism	morphism	NOUN
easat-4026	146	35	identification	identification	NOUN
easat-4026	146	36	𝑀𝐿(𝑛	𝑀𝐿(𝑛	NOUN
easat-4026	146	37	)	)	PUNCT
easat-4026	146	38	changes	change	VERB
easat-4026	146	39	to	to	ADP
easat-4026	146	40	𝒽1𝑟𝑛	𝒽1𝑟𝑛	NOUN
easat-4026	146	41	=	=	SYM
easat-4026	146	42	𝑟𝑛(𝒽1⊗	𝑟𝑛(𝒽1⊗	NOUN
easat-4026	146	43	…	…	SYM
easat-4026	146	44	⊗𝒽1	⊗𝒽1	PROPN
easat-4026	146	45	)	)	PUNCT
easat-4026	146	46	.	.	PUNCT
easat-4026	147	1	as	as	ADP
easat-4026	147	2	a	a	DET
easat-4026	147	3	result	result	NOUN
easat-4026	147	4	,	,	PUNCT
easat-4026	147	5	ring	ring	NOUN
easat-4026	147	6	homeomorphisms	homeomorphism	NOUN
easat-4026	147	7	and	and	CCONJ
easat-4026	147	8	stringent	stringent	ADJ
easat-4026	147	9	morphisms	morphism	NOUN
easat-4026	147	10	of	of	ADP
easat-4026	147	11	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	147	12	tend	tend	VERB
easat-4026	147	13	to	to	PART
easat-4026	147	14	be	be	AUX
easat-4026	147	15	identical	identical	ADJ
easat-4026	147	16	.	.	PUNCT
easat-4026	148	1	5	5	X
easat-4026	148	2	)	)	PUNCT
easat-4026	148	3	we	we	PRON
easat-4026	148	4	state	state	VERB
easat-4026	148	5	that	that	SCONJ
easat-4026	148	6	𝒽	𝒽	PRON
easat-4026	148	7	is	be	AUX
easat-4026	148	8	a	a	DET
easat-4026	148	9	stringent	stringent	ADJ
easat-4026	148	10	isomorphism	isomorphism	NOUN
easat-4026	148	11	if	if	SCONJ
easat-4026	148	12	𝒽1	𝒽1	PROPN
easat-4026	148	13	is	be	AUX
easat-4026	148	14	a	a	DET
easat-4026	148	15	vector	vector	NOUN
easat-4026	148	16	spaces	space	NOUN
easat-4026	148	17	'	'	PART
easat-4026	148	18	isomorphism	isomorphism	NOUN
easat-4026	148	19	and	and	CCONJ
easat-4026	148	20	𝒽	𝒽	NOUN
easat-4026	148	21	is	be	AUX
easat-4026	148	22	a	a	DET
easat-4026	148	23	stringent	stringent	ADJ
easat-4026	148	24	morphism	morphism	NOUN
easat-4026	148	25	.	.	PUNCT
easat-4026	149	1	it	it	PRON
easat-4026	149	2	proves	prove	VERB
easat-4026	149	3	to	to	PART
easat-4026	149	4	be	be	AUX
easat-4026	149	5	even	even	ADV
easat-4026	149	6	more	more	ADV
easat-4026	149	7	significant	significant	ADJ
easat-4026	149	8	to	to	PART
easat-4026	149	9	take	take	VERB
easat-4026	149	10	into	into	ADP
easat-4026	149	11	consideration	consideration	NOUN
easat-4026	149	12	the	the	DET
easat-4026	149	13	homology	homology	NOUN
easat-4026	149	14	of	of	ADP
easat-4026	149	15	the	the	DET
easat-4026	149	16	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	149	17	,	,	PUNCT
easat-4026	149	18	similar	similar	ADJ
easat-4026	149	19	to	to	ADP
easat-4026	149	20	how	how	SCONJ
easat-4026	149	21	it	it	PRON
easat-4026	149	22	is	be	AUX
easat-4026	149	23	conducted	conduct	VERB
easat-4026	149	24	via	via	ADP
easat-4026	149	25	chain	chain	NOUN
easat-4026	149	26	complexes	complex	NOUN
easat-4026	149	27	.	.	PUNCT
easat-4026	150	1	here	here	ADV
easat-4026	150	2	,	,	PUNCT
easat-4026	150	3	we	we	PRON
easat-4026	150	4	will	will	AUX
easat-4026	150	5	define	define	VERB
easat-4026	150	6	strict	strict	ADJ
easat-4026	150	7	morphisms	morphism	NOUN
easat-4026	150	8	and	and	CCONJ
easat-4026	150	9	how	how	SCONJ
easat-4026	150	10	to	to	PART
easat-4026	150	11	handle	handle	VERB
easat-4026	150	12	them	they	PRON
easat-4026	150	13	within	within	ADP
easat-4026	150	14	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	150	15	,	,	PUNCT
easat-4026	150	16	including	include	VERB
easat-4026	150	17	the	the	DET
easat-4026	150	18	use	use	NOUN
easat-4026	150	19	of	of	ADP
easat-4026	150	20	these	these	DET
easat-4026	150	21	definitions	definition	NOUN
easat-4026	150	22	in	in	ADP
easat-4026	150	23	analysis	analysis	NOUN
easat-4026	150	24	.	.	PUNCT
easat-4026	151	1	2.17	2.17	NUM
easat-4026	151	2	.	.	PUNCT
easat-4026	152	1	definition	definition	NOUN
easat-4026	152	2	[	[	X
easat-4026	152	3	19	19	NUM
easat-4026	152	4	]	]	PUNCT
easat-4026	152	5	the	the	DET
easat-4026	152	6	morphism	morphism	NOUN
easat-4026	152	7	𝒽	𝒽	NOUN
easat-4026	152	8	is	be	AUX
easat-4026	152	9	considered	consider	VERB
easat-4026	152	10	stringent	stringent	ADJ
easat-4026	152	11	if	if	SCONJ
easat-4026	152	12	𝒽𝑖	𝒽𝑖	ADP
easat-4026	152	13	=	=	SYM
easat-4026	152	14	0	0	NUM
easat-4026	152	15	for	for	ADP
easat-4026	152	16	any	any	DET
easat-4026	152	17	𝑖	𝑖	SYM
easat-4026	152	18	≠	≠	PROPN
easat-4026	152	19	1	1	NUM
easat-4026	152	20	.	.	PUNCT
easat-4026	153	1	the	the	DET
easat-4026	153	2	stringent	stringent	ADJ
easat-4026	153	3	morphism	morphism	NOUN
easat-4026	153	4	𝒽	𝒽	PROPN
easat-4026	153	5	that	that	PRON
easat-4026	153	6	makes	make	VERB
easat-4026	153	7	𝒽1	𝒽1	ADP
easat-4026	153	8	the	the	DET
easat-4026	153	9	identity	identity	NOUN
easat-4026	153	10	of	of	ADP
easat-4026	153	11	𝒜	𝒜	NOUN
easat-4026	153	12	is	be	AUX
easat-4026	153	13	the	the	DET
easat-4026	153	14	identity	identity	NOUN
easat-4026	153	15	morphism	morphism	NOUN
easat-4026	153	16	.	.	PUNCT
easat-4026	154	1	while	while	SCONJ
easat-4026	154	2	𝒽:𝒜	𝒽:𝒜	PROPN
easat-4026	154	3	→	→	SYM
easat-4026	154	4	ℬ	ℬ	PROPN
easat-4026	154	5	is	be	AUX
easat-4026	154	6	a	a	DET
easat-4026	154	7	stringent	stringent	ADJ
easat-4026	154	8	morphism	morphism	NOUN
easat-4026	154	9	,	,	PUNCT
easat-4026	154	10	the	the	DET
easat-4026	154	11	identity	identity	NOUN
easat-4026	154	12	𝑀𝐿(𝑛	𝑀𝐿(𝑛	NOUN
easat-4026	154	13	)	)	PUNCT
easat-4026	154	14	becomes	become	VERB
easat-4026	154	15	:	:	PUNCT
easat-4026	154	16	𝒽1𝑟𝑛	𝒽1𝑟𝑛	PUNCT
easat-4026	154	17	=	=	SYM
easat-4026	154	18	𝑟𝑛(𝒽1⊗⋯⊗𝒽1	𝑟𝑛(𝒽1⊗⋯⊗𝒽1	ADJ
easat-4026	154	19	)	)	PUNCT
easat-4026	154	20	.	.	PUNCT
easat-4026	155	1	9478	9478	NUM
easat-4026	155	2	edelweiss	edelweiss	PROPN
easat-4026	155	3	applied	apply	VERB
easat-4026	155	4	science	science	NOUN
easat-4026	155	5	and	and	CCONJ
easat-4026	155	6	technology	technology	NOUN
easat-4026	155	7	issn	issn	PROPN
easat-4026	155	8	:	:	PUNCT
easat-4026	155	9	2576	2576	NUM
easat-4026	155	10	-	-	SYM
easat-4026	155	11	8484	8484	NUM
easat-4026	155	12	vol	vol	NOUN
easat-4026	155	13	.	.	PROPN
easat-4026	155	14	8	8	NUM
easat-4026	155	15	,	,	PUNCT
easat-4026	155	16	no	no	INTJ
easat-4026	155	17	.	.	NOUN
easat-4026	156	1	6	6	NUM
easat-4026	156	2	:	:	SYM
easat-4026	156	3	9472	9472	NUM
easat-4026	156	4	-	-	SYM
easat-4026	156	5	9486	9486	NUM
easat-4026	156	6	,	,	PUNCT
easat-4026	156	7	2024	2024	NUM
easat-4026	156	8	doi	doi	NOUN
easat-4026	156	9	:	:	PUNCT
easat-4026	156	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	156	11	©	©	PROPN
easat-4026	156	12	2024	2024	NUM
easat-4026	156	13	by	by	ADP
easat-4026	156	14	the	the	DET
easat-4026	156	15	authors	author	NOUN
easat-4026	156	16	;	;	PUNCT
easat-4026	156	17	licensee	licensee	PROPN
easat-4026	156	18	learning	learning	NOUN
easat-4026	156	19	gate	gate	NOUN
easat-4026	156	20	in	in	ADP
easat-4026	156	21	classical	classical	ADJ
easat-4026	156	22	ring	ring	NOUN
easat-4026	156	23	theory	theory	NOUN
easat-4026	156	24	,	,	PUNCT
easat-4026	156	25	homeomorphisms	homeomorphisms	PROPN
easat-4026	156	26	are	be	AUX
easat-4026	156	27	similar	similar	ADJ
easat-4026	156	28	to	to	ADP
easat-4026	156	29	strict	strict	ADJ
easat-4026	156	30	morphisms	morphism	NOUN
easat-4026	156	31	.	.	PUNCT
easat-4026	157	1	morphism	morphism	NOUN
easat-4026	157	2	𝒽	𝒽	PRON
easat-4026	157	3	called	call	VERB
easat-4026	157	4	a	a	DET
easat-4026	157	5	strict	strict	ADJ
easat-4026	157	6	isomorphism	isomorphism	NOUN
easat-4026	157	7	if	if	SCONJ
easat-4026	157	8	it	it	PRON
easat-4026	157	9	is	be	AUX
easat-4026	157	10	𝒽	𝒽	PRON
easat-4026	157	11	strict	strict	ADJ
easat-4026	157	12	and	and	CCONJ
easat-4026	157	13	𝒽1is	𝒽1i	NOUN
easat-4026	157	14	an	an	DET
easat-4026	157	15	isomorphism	isomorphism	NOUN
easat-4026	157	16	of	of	ADP
easat-4026	157	17	vector	vector	NOUN
easat-4026	157	18	spaces	space	NOUN
easat-4026	157	19	.	.	PUNCT
easat-4026	158	1	in	in	ADP
easat-4026	158	2	this	this	DET
easat-4026	158	3	instance	instance	NOUN
easat-4026	158	4	,	,	PUNCT
easat-4026	158	5	the	the	DET
easat-4026	158	6	inverse	inverse	NOUN
easat-4026	158	7	morphism	morphism	NOUN
easat-4026	158	8	of	of	ADP
easat-4026	158	9	𝒽	𝒽	PROPN
easat-4026	158	10	is	be	AUX
easat-4026	158	11	denoted	denote	VERB
easat-4026	158	12	by	by	ADP
easat-4026	158	13	𝒽1	𝒽1	PROPN
easat-4026	158	14	−1	−1	NOUN
easat-4026	158	15	:	:	PUNCT
easat-4026	158	16	ℬ	ℬ	PROPN
easat-4026	158	17	→	→	SYM
easat-4026	158	18	𝒜.	𝒜.	NOUN
easat-4026	158	19	we	we	PRON
easat-4026	158	20	will	will	AUX
easat-4026	158	21	explain	explain	VERB
easat-4026	158	22	how	how	SCONJ
easat-4026	158	23	to	to	PART
easat-4026	158	24	classify	classify	VERB
easat-4026	158	25	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	158	26	through	through	ADP
easat-4026	158	27	equivalence	equivalence	NOUN
easat-4026	158	28	classes	class	NOUN
easat-4026	158	29	and	and	CCONJ
easat-4026	158	30	how	how	SCONJ
easat-4026	158	31	to	to	PART
easat-4026	158	32	handle	handle	VERB
easat-4026	158	33	different	different	ADJ
easat-4026	158	34	models	model	NOUN
easat-4026	158	35	of	of	ADP
easat-4026	158	36	algebras	algebra	NOUN
easat-4026	158	37	in	in	ADP
easat-4026	158	38	the	the	DET
easat-4026	158	39	following	following	NOUN
easat-4026	158	40	.	.	PUNCT
easat-4026	159	1	2.18	2.18	NUM
easat-4026	159	2	.	.	PUNCT
easat-4026	159	3	definition	definition	NOUN
easat-4026	159	4	[	[	X
easat-4026	159	5	20	20	NUM
easat-4026	159	6	]	]	PUNCT
easat-4026	159	7	assume	assume	VERB
easat-4026	159	8	that	that	SCONJ
easat-4026	159	9	𝒜	𝒜	NOUN
easat-4026	159	10	and	and	CCONJ
easat-4026	159	11	ℬ	ℬ	NOUN
easat-4026	159	12	are	be	AUX
easat-4026	159	13	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	159	14	and	and	CCONJ
easat-4026	159	15	that	that	SCONJ
easat-4026	159	16	𝒽:𝒜	𝒽:𝒜	NOUN
easat-4026	159	17	→	→	SYM
easat-4026	159	18	ℬ	ℬ	PROPN
easat-4026	159	19	is	be	AUX
easat-4026	159	20	a	a	DET
easat-4026	159	21	𝒜∞-morphism	𝒜∞-morphism	PROPN
easat-4026	159	22	.	.	PUNCT
easat-4026	160	1	if	if	SCONJ
easat-4026	160	2	𝒽1	𝒽1	NOUN
easat-4026	160	3	represents	represent	VERB
easat-4026	160	4	a	a	DET
easat-4026	160	5	quasi	quasi	NOUN
easat-4026	160	6	-	-	NOUN
easat-4026	160	7	isomorphism	isomorphism	NOUN
easat-4026	160	8	of	of	ADP
easat-4026	160	9	complexes	complex	NOUN
easat-4026	160	10	,	,	PUNCT
easat-4026	160	11	then	then	ADV
easat-4026	160	12	we	we	PRON
easat-4026	160	13	can	can	AUX
easat-4026	160	14	argue	argue	VERB
easat-4026	160	15	that	that	SCONJ
easat-4026	160	16	𝒽	𝒽	PRON
easat-4026	160	17	is	be	AUX
easat-4026	160	18	a	a	DET
easat-4026	160	19	quasi	quasi	NOUN
easat-4026	160	20	-	-	NOUN
easat-4026	160	21	isomorphism	isomorphism	NOUN
easat-4026	160	22	as	as	ADV
easat-4026	160	23	well	well	ADV
easat-4026	160	24	.	.	PUNCT
easat-4026	161	1	it	it	PRON
easat-4026	161	2	follows	follow	VERB
easat-4026	161	3	that	that	SCONJ
easat-4026	161	4	the	the	DET
easat-4026	161	5	induced	induced	ADJ
easat-4026	161	6	map	map	NOUN
easat-4026	161	7	ℋ(𝒽1):ℋ(𝒜•	ℋ(𝒽1):ℋ(𝒜•	NOUN
easat-4026	161	8	)	)	PUNCT
easat-4026	161	9	→	→	SYM
easat-4026	161	10	ℋ(ℬ•	ℋ(ℬ•	NUM
easat-4026	161	11	)	)	PUNCT
easat-4026	161	12	is	be	AUX
easat-4026	161	13	isomorphic	isomorphic	ADJ
easat-4026	161	14	.	.	PUNCT
easat-4026	162	1	here	here	ADV
easat-4026	162	2	,	,	PUNCT
easat-4026	162	3	we	we	PRON
easat-4026	162	4	use	use	VERB
easat-4026	162	5	the	the	DET
easat-4026	162	6	notation	notation	NOUN
easat-4026	162	7	𝒜	𝒜	NOUN
easat-4026	162	8	≃	≃	NOUN
easat-4026	162	9	ℬ.	ℬ.	NOUN
easat-4026	162	10	we	we	PRON
easat-4026	162	11	will	will	AUX
easat-4026	162	12	classify	classify	VERB
easat-4026	162	13	two	two	NUM
easat-4026	162	14	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	162	15	as	as	ADP
easat-4026	162	16	belonging	belong	VERB
easat-4026	162	17	to	to	ADP
easat-4026	162	18	the	the	DET
easat-4026	162	19	same	same	ADJ
easat-4026	162	20	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	162	21	if	if	SCONJ
easat-4026	162	22	they	they	PRON
easat-4026	162	23	are	be	AUX
easat-4026	162	24	quasi	quasi	ADJ
easat-4026	162	25	-	-	ADJ
easat-4026	162	26	isomorphic	isomorphic	ADJ
easat-4026	162	27	.	.	PUNCT
easat-4026	163	1	in	in	ADP
easat-4026	163	2	other	other	ADJ
easat-4026	163	3	words	word	NOUN
easat-4026	163	4	,	,	PUNCT
easat-4026	163	5	quasi	quasi	ADJ
easat-4026	163	6	-	-	ADJ
easat-4026	163	7	isomorphism	isomorphism	ADJ
easat-4026	163	8	classes	class	NOUN
easat-4026	163	9	of	of	ADP
easat-4026	163	10	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	163	11	will	will	AUX
easat-4026	163	12	be	be	AUX
easat-4026	163	13	taken	take	VERB
easat-4026	163	14	into	into	ADP
easat-4026	163	15	consideration	consideration	NOUN
easat-4026	163	16	.	.	PUNCT
easat-4026	164	1	a	a	DET
easat-4026	164	2	model	model	NOUN
easat-4026	164	3	of	of	ADP
easat-4026	164	4	𝒜	𝒜	NOUN
easat-4026	164	5	is	be	AUX
easat-4026	164	6	a	a	DET
easat-4026	164	7	representation	representation	NOUN
easat-4026	164	8	of	of	ADP
easat-4026	164	9	a	a	DET
easat-4026	164	10	class	class	NOUN
easat-4026	164	11	of	of	ADP
easat-4026	164	12	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	164	13	that	that	PRON
easat-4026	164	14	are	be	AUX
easat-4026	164	15	quasi	quasi	ADJ
easat-4026	164	16	-	-	ADJ
easat-4026	164	17	isomorphic	isomorphic	ADJ
easat-4026	164	18	to	to	ADP
easat-4026	164	19	𝒜	𝒜	NOUN
easat-4026	164	20	and	and	CCONJ
easat-4026	164	21	satisfy	satisfy	VERB
easat-4026	164	22	some	some	DET
easat-4026	164	23	helpful	helpful	ADJ
easat-4026	164	24	properties	property	NOUN
easat-4026	164	25	.	.	PUNCT
easat-4026	165	1	as	as	ADP
easat-4026	165	2	an	an	DET
easat-4026	165	3	illustration	illustration	NOUN
easat-4026	165	4	,	,	PUNCT
easat-4026	165	5	the	the	DET
easat-4026	165	6	term	term	NOUN
easat-4026	165	7	"	"	PUNCT
easat-4026	165	8	minimal	minimal	ADJ
easat-4026	165	9	model	model	NOUN
easat-4026	165	10	of	of	ADP
easat-4026	165	11	𝒜	𝒜	NOUN
easat-4026	165	12	"	"	PUNCT
easat-4026	165	13	refers	refer	VERB
easat-4026	165	14	to	to	ADP
easat-4026	165	15	a	a	DET
easat-4026	165	16	representative	representative	NOUN
easat-4026	165	17	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	165	18	containing	contain	VERB
easat-4026	165	19	a	a	DET
easat-4026	165	20	zero	zero	NUM
easat-4026	165	21	𝑟1	𝑟1	NOUN
easat-4026	165	22	.	.	PUNCT
easat-4026	166	1	as	as	ADP
easat-4026	166	2	a	a	DET
easat-4026	166	3	result	result	NOUN
easat-4026	166	4	,	,	PUNCT
easat-4026	166	5	we	we	PRON
easat-4026	166	6	can	can	AUX
easat-4026	166	7	examine	examine	VERB
easat-4026	166	8	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	166	9	by	by	ADP
easat-4026	166	10	taking	take	VERB
easat-4026	166	11	into	into	ADP
easat-4026	166	12	consideration	consideration	NOUN
easat-4026	166	13	alternative	alternative	ADJ
easat-4026	166	14	models	model	NOUN
easat-4026	166	15	of	of	ADP
easat-4026	166	16	𝒜.	𝒜.	NOUN
easat-4026	166	17	now	now	ADV
easat-4026	166	18	,	,	PUNCT
easat-4026	166	19	we	we	PRON
easat-4026	166	20	will	will	AUX
easat-4026	166	21	show	show	VERB
easat-4026	166	22	that	that	SCONJ
easat-4026	166	23	𝐸𝑥𝑡𝒜	𝐸𝑥𝑡𝒜	PROPN
easat-4026	166	24	∗	∗	NOUN
easat-4026	166	25	(	(	PUNCT
easat-4026	166	26	𝑀,𝑀	𝑀,𝑀	NOUN
easat-4026	166	27	)	)	PUNCT
easat-4026	166	28	is	be	AUX
easat-4026	166	29	an	an	DET
easat-4026	166	30	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-4026	166	31	and	and	CCONJ
easat-4026	166	32	how	how	SCONJ
easat-4026	166	33	to	to	PART
easat-4026	166	34	use	use	VERB
easat-4026	166	35	different	different	ADJ
easat-4026	166	36	solutions	solution	NOUN
easat-4026	166	37	in	in	ADP
easat-4026	166	38	this	this	DET
easat-4026	166	39	context	context	NOUN
easat-4026	166	40	.	.	PUNCT
easat-4026	167	1	2.19	2.19	NUM
easat-4026	167	2	.	.	PUNCT
easat-4026	167	3	theorem	theorem	VERB
easat-4026	168	1	[	[	X
easat-4026	168	2	21	21	NUM
easat-4026	168	3	]	]	PUNCT
easat-4026	168	4	assuming	assume	VERB
easat-4026	168	5	𝒜	𝒜	NOUN
easat-4026	168	6	is	be	AUX
easat-4026	168	7	the	the	DET
easat-4026	168	8	algebra	algebra	NOUN
easat-4026	168	9	over	over	ADP
easat-4026	168	10	𝑀	𝑀	PROPN
easat-4026	168	11	,	,	PUNCT
easat-4026	168	12	then	then	ADV
easat-4026	168	13	𝐸𝑥𝑡𝒜	𝐸𝑥𝑡𝒜	PROPN
easat-4026	168	14	∗	∗	NOUN
easat-4026	168	15	(	(	PUNCT
easat-4026	168	16	𝑀,𝑀	𝑀,𝑀	NOUN
easat-4026	168	17	)	)	PUNCT
easat-4026	168	18	is	be	AUX
easat-4026	168	19	an	an	DET
easat-4026	168	20	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	168	21	.	.	PUNCT
easat-4026	169	1	to	to	PART
easat-4026	169	2	demonstrate	demonstrate	VERB
easat-4026	169	3	this	this	PRON
easat-4026	169	4	,	,	PUNCT
easat-4026	169	5	we	we	PRON
easat-4026	169	6	take	take	VERB
easat-4026	169	7	the	the	DET
easat-4026	169	8	projective	projective	ADJ
easat-4026	169	9	resolution	resolution	NOUN
easat-4026	169	10	𝒫	𝒫	PROPN
easat-4026	169	11	of	of	ADP
easat-4026	169	12	𝑀𝒜	𝑀𝒜	PROPN
easat-4026	169	13	.	.	PUNCT
easat-4026	170	1	h	h	NOUN
easat-4026	170	2	consequently	consequently	ADV
easat-4026	170	3	,	,	PUNCT
easat-4026	170	4	the	the	DET
easat-4026	170	5	differential	differential	NOUN
easat-4026	170	6	graded	grade	VERB
easat-4026	170	7	algebra	algebra	NOUN
easat-4026	170	8	with	with	ADP
easat-4026	170	9	homology	homology	NOUN
easat-4026	170	10	determined	determine	VERB
easat-4026	170	11	by	by	ADP
easat-4026	170	12	the	the	DET
easat-4026	170	13	yoneda	yoneda	PROPN
easat-4026	170	14	algebra	algebra	PROPN
easat-4026	170	15	𝐸𝑥𝑡𝒜	𝐸𝑥𝑡𝒜	PROPN
easat-4026	170	16	∗	∗	NOUN
easat-4026	170	17	(	(	PUNCT
easat-4026	170	18	𝑀,𝑀	𝑀,𝑀	NOUN
easat-4026	170	19	)	)	PUNCT
easat-4026	170	20	is	be	AUX
easat-4026	170	21	the	the	DET
easat-4026	170	22	morphism	morphism	NOUN
easat-4026	170	23	complex	complex	NOUN
easat-4026	170	24	ℬ	ℬ	NOUN
easat-4026	170	25	=	=	SYM
easat-4026	170	26	𝐻𝑜𝑚𝒜(𝒫,𝒫	𝐻𝑜𝑚𝒜(𝒫,𝒫	PROPN
easat-4026	170	27	)	)	PUNCT
easat-4026	170	28	.	.	PUNCT
easat-4026	171	1	we	we	PRON
easat-4026	171	2	will	will	AUX
easat-4026	171	3	provide	provide	VERB
easat-4026	171	4	an	an	DET
easat-4026	171	5	illustrative	illustrative	ADJ
easat-4026	171	6	example	example	NOUN
easat-4026	171	7	of	of	ADP
easat-4026	171	8	applications	application	NOUN
easat-4026	171	9	of	of	ADP
easat-4026	171	10	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	171	11	and	and	CCONJ
easat-4026	171	12	how	how	SCONJ
easat-4026	171	13	to	to	PART
easat-4026	171	14	use	use	VERB
easat-4026	171	15	them	they	PRON
easat-4026	171	16	in	in	ADP
easat-4026	171	17	specific	specific	ADJ
easat-4026	171	18	cases	case	NOUN
easat-4026	171	19	of	of	ADP
easat-4026	171	20	tensor	tensor	NOUN
easat-4026	171	21	products	product	NOUN
easat-4026	171	22	and	and	CCONJ
easat-4026	171	23	maps	map	NOUN
easat-4026	171	24	.	.	PUNCT
easat-4026	172	1	2.20	2.20	NUM
easat-4026	172	2	.	.	PUNCT
easat-4026	172	3	example	example	VERB
easat-4026	172	4	the	the	DET
easat-4026	172	5	map	map	NOUN
easat-4026	172	6	𝑟2	𝑟2	NOUN
easat-4026	172	7	is	be	AUX
easat-4026	172	8	produced	produce	VERB
easat-4026	172	9	by	by	ADP
easat-4026	172	10	multiplying	multiply	VERB
easat-4026	172	11	𝑅	𝑅	PROPN
easat-4026	172	12	and	and	CCONJ
easat-4026	172	13	the	the	DET
easat-4026	172	14	mappings	mapping	NOUN
easat-4026	172	15	𝑟𝑛	𝑟𝑛	ADP
easat-4026	173	1	=	=	NOUN
easat-4026	173	2	0	0	NUM
easat-4026	173	3	for	for	SCONJ
easat-4026	173	4	all	all	DET
easat-4026	173	5	𝑛	𝑛	PRON
easat-4026	173	6	≠	≠	PROPN
easat-4026	173	7	2	2	NUM
easat-4026	173	8	,	,	PUNCT
easat-4026	173	9	the	the	DET
easat-4026	173	10	graded	grade	VERB
easat-4026	173	11	space	space	NOUN
easat-4026	173	12	𝒜	𝒜	NOUN
easat-4026	173	13	=	=	NOUN
easat-4026	173	14	𝑅[𝔷]/(ℰ2	𝑅[𝔷]/(ℰ2	NOUN
easat-4026	173	15	)	)	PUNCT
easat-4026	173	16	has	have	VERB
easat-4026	173	17	a	a	DET
easat-4026	173	18	trivial	trivial	ADJ
easat-4026	173	19	𝒜∞-structures	𝒜∞-structure	NOUN
easat-4026	173	20	when	when	SCONJ
easat-4026	173	21	𝑅	𝑅	PROPN
easat-4026	173	22	is	be	AUX
easat-4026	173	23	an	an	DET
easat-4026	173	24	extraordinary	extraordinary	ADJ
easat-4026	173	25	algebra	algebra	NOUN
easat-4026	173	26	for	for	ADP
easat-4026	173	27	𝑁	𝑁	PROPN
easat-4026	173	28	≥	≥	NOUN
easat-4026	173	29	1	1	NUM
easat-4026	173	30	and	and	CCONJ
easat-4026	173	31	𝔷	𝔷	PROPN
easat-4026	173	32	is	be	AUX
easat-4026	173	33	an	an	DET
easat-4026	173	34	undetermined	undetermined	ADJ
easat-4026	173	35	of	of	ADP
easat-4026	173	36	degree	degree	NOUN
easat-4026	173	37	2	2	NUM
easat-4026	173	38	−	−	NOUN
easat-4026	173	39	𝑁.	𝑁.	PROPN
easat-4026	174	1	we	we	PRON
easat-4026	174	2	define	define	VERB
easat-4026	174	3	the	the	DET
easat-4026	174	4	distorted	distorted	ADJ
easat-4026	174	5	multiplication	multiplication	NOUN
easat-4026	174	6	and	and	CCONJ
easat-4026	174	7	the	the	DET
easat-4026	174	8	linear	linear	ADJ
easat-4026	174	9	map	map	NOUN
easat-4026	174	10	𝒽	𝒽	NOUN
easat-4026	174	11	:	:	PUNCT
easat-4026	174	12	𝑅⊗𝑁	𝑅⊗𝑁	PROPN
easat-4026	174	13	→	→	SYM
easat-4026	174	14	𝑅	𝑅	PROPN
easat-4026	174	15	as	as	SCONJ
easat-4026	174	16	follows	follow	VERB
easat-4026	174	17	:	:	PUNCT
easat-4026	174	18	𝑟𝑛	𝑟𝑛	ADP
easat-4026	174	19	′	′	NOUN
easat-4026	175	1	=	=	PUNCT
easat-4026	176	1	{	{	PUNCT
easat-4026	176	2	𝑟𝑛	𝑟𝑛	ADP
easat-4026	176	3	𝑛	𝑛	DET
easat-4026	176	4	≠	≠	PROPN
easat-4026	176	5	𝑁	𝑁	PROPN
easat-4026	176	6	𝑟𝑁	𝑟𝑁	PROPN
easat-4026	176	7	+	+	PUNCT
easat-4026	176	8	ℰ𝒽	ℰ𝒽	NOUN
easat-4026	176	9	𝑛	𝑛	NOUN
easat-4026	176	10	=	=	SYM
easat-4026	176	11	𝑁	𝑁	PROPN
easat-4026	176	12	.	.	PUNCT
easat-4026	177	1	the	the	DET
easat-4026	177	2	given	give	VERB
easat-4026	177	3	𝑟𝑛	𝑟𝑛	ADP
easat-4026	177	4	′	′	NUM
easat-4026	177	5	thus	thus	ADV
easat-4026	177	6	becomes	become	VERB
easat-4026	177	7	an	an	DET
easat-4026	177	8	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	177	9	if	if	SCONJ
easat-4026	177	10	and	and	CCONJ
easat-4026	177	11	only	only	ADV
easat-4026	177	12	if	if	SCONJ
easat-4026	177	13	𝒽	𝒽	PRON
easat-4026	177	14	is	be	AUX
easat-4026	177	15	a	a	DET
easat-4026	177	16	simplicial	simplicial	ADJ
easat-4026	177	17	co	co	NOUN
easat-4026	177	18	-	-	NOUN
easat-4026	177	19	cycle	cycle	NOUN
easat-4026	177	20	over	over	ADP
easat-4026	177	21	𝑅.	𝑅.	NOUN
easat-4026	177	22	next	next	ADV
easat-4026	177	23	,	,	PUNCT
easat-4026	177	24	we	we	PRON
easat-4026	177	25	will	will	AUX
easat-4026	177	26	define	define	VERB
easat-4026	177	27	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	177	28	using	use	VERB
easat-4026	177	29	left	left	ADJ
easat-4026	177	30	models	model	NOUN
easat-4026	177	31	and	and	CCONJ
easat-4026	177	32	how	how	SCONJ
easat-4026	177	33	to	to	PART
easat-4026	177	34	handle	handle	VERB
easat-4026	177	35	fundamental	fundamental	ADJ
easat-4026	177	36	properties	property	NOUN
easat-4026	177	37	of	of	ADP
easat-4026	177	38	these	these	DET
easat-4026	177	39	models	model	NOUN
easat-4026	177	40	.	.	PUNCT
easat-4026	178	1	2.21	2.21	NUM
easat-4026	178	2	.	.	PUNCT
easat-4026	178	3	definition	definition	NOUN
easat-4026	178	4	[	[	X
easat-4026	178	5	11	11	NUM
easat-4026	178	6	]	]	PUNCT
easat-4026	178	7	consider	consider	VERB
easat-4026	178	8	𝒜	𝒜	NOUN
easat-4026	178	9	as	as	ADP
easat-4026	178	10	an	an	DET
easat-4026	178	11	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	178	12	.	.	PUNCT
easat-4026	179	1	the	the	DET
easat-4026	179	2	ℤ-graded	ℤ-graded	ADJ
easat-4026	179	3	vector	vector	NOUN
easat-4026	179	4	space	space	NOUN
easat-4026	179	5	𝑅	𝑅	PROPN
easat-4026	179	6	is	be	AUX
easat-4026	179	7	defined	define	VERB
easat-4026	179	8	as	as	ADP
easat-4026	179	9	the	the	DET
easat-4026	179	10	left	left	ADJ
easat-4026	179	11	𝒜∞-module	𝒜∞-module	PROPN
easat-4026	179	12	of	of	ADP
easat-4026	179	13	𝒜	𝒜	NOUN
easat-4026	179	14	,	,	PUNCT
easat-4026	179	15	equipped	equip	VERB
easat-4026	179	16	with	with	ADP
easat-4026	179	17	maps	map	NOUN
easat-4026	179	18	:	:	PUNCT
easat-4026	179	19	𝑟𝑛	𝑟𝑛	ADP
easat-4026	179	20	𝑅:𝒜⊗𝑛−1⊗𝑅	𝑅:𝒜⊗𝑛−1⊗𝑅	ADP
easat-4026	179	21	→	→	SYM
easat-4026	179	22	𝑅	𝑅	PROPN
easat-4026	179	23	,	,	PUNCT
easat-4026	179	24	𝑛	𝑛	DET
easat-4026	179	25	≥	≥	NOUN
easat-4026	179	26	1	1	NUM
easat-4026	179	27	of	of	ADP
easat-4026	179	28	degree	degree	NOUN
easat-4026	179	29	(	(	PUNCT
easat-4026	179	30	2	2	NUM
easat-4026	179	31	−	−	NUM
easat-4026	179	32	𝑛	𝑛	NOUN
easat-4026	179	33	)	)	PUNCT
easat-4026	179	34	,	,	PUNCT
easat-4026	179	35	fulfilling	fulfil	VERB
easat-4026	179	36	the	the	DET
easat-4026	179	37	identical	identical	ADJ
easat-4026	179	38	stasheff	stasheff	NOUN
easat-4026	179	39	identities	identity	NOUN
easat-4026	179	40	𝑆𝐿(𝑛	𝑆𝐿(𝑛	ADJ
easat-4026	179	41	):	):	PUNCT
easat-4026	179	42	∑(−1)𝑚+𝑠𝑡𝑟𝑚+1+𝑡(𝑖𝑑	∑(−1)𝑚+𝑠𝑡𝑟𝑚+1+𝑡(𝑖𝑑	PROPN
easat-4026	179	43	⊗𝑚	⊗𝑚	VERB
easat-4026	179	44	⊗	⊗	PROPN
easat-4026	179	45	𝑟𝑠	𝑟𝑠	PROPN
easat-4026	180	1	⊗	⊗	NUM
easat-4026	180	2	𝑖𝑑⊗𝑡	𝑖𝑑⊗𝑡	NOUN
easat-4026	180	3	)	)	PUNCT
easat-4026	180	4	=	=	SYM
easat-4026	180	5	0	0	NUM
easat-4026	180	6	,	,	PUNCT
easat-4026	180	7	as	as	ADP
easat-4026	180	8	in	in	ADP
easat-4026	180	9	the	the	DET
easat-4026	180	10	definition	definition	NOUN
easat-4026	180	11	of	of	ADP
easat-4026	180	12	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	180	13	.	.	PUNCT
easat-4026	181	1	we	we	PRON
easat-4026	181	2	will	will	AUX
easat-4026	181	3	describe	describe	VERB
easat-4026	181	4	how	how	SCONJ
easat-4026	181	5	to	to	PART
easat-4026	181	6	define	define	VERB
easat-4026	181	7	morphisms	morphism	NOUN
easat-4026	181	8	between	between	ADP
easat-4026	181	9	left	left	ADJ
easat-4026	181	10	models	model	NOUN
easat-4026	181	11	of	of	ADP
easat-4026	181	12	algebras	algebra	NOUN
easat-4026	181	13	and	and	CCONJ
easat-4026	181	14	how	how	SCONJ
easat-4026	181	15	to	to	PART
easat-4026	181	16	construct	construct	VERB
easat-4026	181	17	these	these	DET
easat-4026	181	18	morphisms	morphism	NOUN
easat-4026	181	19	to	to	PART
easat-4026	181	20	meet	meet	VERB
easat-4026	181	21	specified	specified	ADJ
easat-4026	181	22	conditions	condition	NOUN
easat-4026	181	23	.	.	PUNCT
easat-4026	182	1	9479	9479	NUM
easat-4026	182	2	edelweiss	edelweiss	PROPN
easat-4026	182	3	applied	apply	VERB
easat-4026	182	4	science	science	NOUN
easat-4026	182	5	and	and	CCONJ
easat-4026	182	6	technology	technology	NOUN
easat-4026	182	7	issn	issn	PROPN
easat-4026	182	8	:	:	PUNCT
easat-4026	182	9	2576	2576	NUM
easat-4026	182	10	-	-	SYM
easat-4026	182	11	8484	8484	NUM
easat-4026	182	12	vol	vol	NOUN
easat-4026	182	13	.	.	PROPN
easat-4026	182	14	8	8	NUM
easat-4026	182	15	,	,	PUNCT
easat-4026	182	16	no	no	INTJ
easat-4026	182	17	.	.	NOUN
easat-4026	183	1	6	6	NUM
easat-4026	183	2	:	:	SYM
easat-4026	183	3	9472	9472	NUM
easat-4026	183	4	-	-	SYM
easat-4026	183	5	9486	9486	NUM
easat-4026	183	6	,	,	PUNCT
easat-4026	183	7	2024	2024	NUM
easat-4026	183	8	doi	doi	NOUN
easat-4026	183	9	:	:	PUNCT
easat-4026	183	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	183	11	©	©	PROPN
easat-4026	183	12	2024	2024	NUM
easat-4026	183	13	by	by	ADP
easat-4026	183	14	the	the	DET
easat-4026	183	15	authors	author	NOUN
easat-4026	183	16	;	;	PUNCT
easat-4026	183	17	licensee	licensee	PROPN
easat-4026	183	18	learning	learning	NOUN
easat-4026	183	19	gate	gate	VERB
easat-4026	183	20	2.22	2.22	NUM
easat-4026	183	21	.	.	PUNCT
easat-4026	184	1	definition	definition	NOUN
easat-4026	184	2	[	[	X
easat-4026	184	3	8	8	NUM
easat-4026	184	4	]	]	PUNCT
easat-4026	184	5	assume	assume	VERB
easat-4026	184	6	the	the	DET
easat-4026	184	7	morphism	morphism	NOUN
easat-4026	184	8	of	of	ADP
easat-4026	184	9	the	the	DET
easat-4026	184	10	left	left	ADJ
easat-4026	184	11	𝒜∞-modules	𝒜∞-modules	PROPN
easat-4026	184	12	𝒽:𝑅	𝒽:𝑅	PROPN
easat-4026	184	13	→	→	SYM
easat-4026	184	14	𝑆	𝑆	PROPN
easat-4026	184	15	is	be	AUX
easat-4026	184	16	defined	define	VERB
easat-4026	184	17	as	as	ADP
easat-4026	184	18	the	the	DET
easat-4026	184	19	family	family	NOUN
easat-4026	184	20	of	of	ADP
easat-4026	184	21	the	the	DET
easat-4026	184	22	graded	grade	VERB
easat-4026	184	23	maps	map	NOUN
easat-4026	184	24	:	:	PUNCT
easat-4026	185	1	𝒽𝑛:𝒜	𝒽𝑛:𝒜	PROPN
easat-4026	185	2	⊗𝑛−1⊗𝑅	⊗𝑛−1⊗𝑅	PROPN
easat-4026	186	1	→	→	PUNCT
easat-4026	186	2	𝑆	𝑆	PROPN
easat-4026	186	3	of	of	ADP
easat-4026	186	4	degree	degree	NOUN
easat-4026	186	5	(	(	PUNCT
easat-4026	186	6	1	1	NUM
easat-4026	186	7	−	−	NUM
easat-4026	186	8	𝑛	𝑛	NOUN
easat-4026	186	9	)	)	PUNCT
easat-4026	186	10	for	for	ADP
easat-4026	186	11	all	all	DET
easat-4026	186	12	𝑛	𝑛	DET
easat-4026	186	13	≥	≥	NOUN
easat-4026	186	14	1	1	NUM
easat-4026	186	15	.	.	PUNCT
easat-4026	187	1	when	when	SCONJ
easat-4026	187	2	𝒽1	𝒽1	PROPN
easat-4026	187	3	=	=	SYM
easat-4026	187	4	𝑖𝑑𝑅	𝑖𝑑𝑅	PROPN
easat-4026	187	5	and	and	CCONJ
easat-4026	187	6	𝒽𝑖	𝒽𝑖	ADP
easat-4026	187	7	=	=	SYM
easat-4026	187	8	0	0	NUM
easat-4026	187	9	for	for	ADP
easat-4026	187	10	all	all	DET
easat-4026	187	11	𝑖	𝑖	PRON
easat-4026	187	12	≥	≥	NOUN
easat-4026	187	13	2	2	NUM
easat-4026	187	14	,	,	PUNCT
easat-4026	187	15	the	the	DET
easat-4026	187	16	identity	identity	NOUN
easat-4026	187	17	morphism	morphism	NOUN
easat-4026	187	18	𝒽	𝒽	PROPN
easat-4026	187	19	:	:	PUNCT
easat-4026	187	20	𝑅	𝑅	PROPN
easat-4026	187	21	→	→	SYM
easat-4026	187	22	𝑅	𝑅	PROPN
easat-4026	187	23	is	be	AUX
easat-4026	187	24	produced	produce	VERB
easat-4026	187	25	.	.	PUNCT
easat-4026	188	1	the	the	DET
easat-4026	188	2	composition	composition	NOUN
easat-4026	188	3	of	of	ADP
easat-4026	188	4	two	two	NUM
easat-4026	188	5	morphisms	morphism	NOUN
easat-4026	188	6	𝒽	𝒽	NOUN
easat-4026	188	7	:	:	PUNCT
easat-4026	188	8	𝑅	𝑅	PROPN
easat-4026	188	9	→	→	SYM
easat-4026	188	10	𝑆	𝑆	PROPN
easat-4026	188	11	and	and	CCONJ
easat-4026	188	12	ℊ	ℊ	PROPN
easat-4026	188	13	:	:	PUNCT
easat-4026	188	14	𝐿	𝐿	PROPN
easat-4026	188	15	→	→	SYM
easat-4026	188	16	𝑅	𝑅	PROPN
easat-4026	188	17	is	be	AUX
easat-4026	188	18	given	give	VERB
easat-4026	188	19	by	by	ADP
easat-4026	188	20	:	:	PUNCT
easat-4026	188	21	(	(	PUNCT
easat-4026	188	22	𝒽	𝒽	DET
easat-4026	188	23	∘	∘	NUM
easat-4026	188	24	ℊ)𝑛	ℊ)𝑛	NOUN
easat-4026	189	1	=	=	PUNCT
easat-4026	190	1	∑𝒽1+𝓌	∑𝒽1+𝓌	PUNCT
easat-4026	190	2	∘	∘	X
easat-4026	190	3	(	(	PUNCT
easat-4026	190	4	𝑖𝑑⊗𝓌	𝑖𝑑⊗𝓌	INTJ
easat-4026	190	5	⊗ℊ𝓋	⊗ℊ𝓋	NOUN
easat-4026	190	6	)	)	PUNCT
easat-4026	190	7	,	,	PUNCT
easat-4026	190	8	(	(	PUNCT
easat-4026	190	9	4	4	X
easat-4026	190	10	)	)	PUNCT
easat-4026	190	11	where	where	SCONJ
easat-4026	190	12	the	the	DET
easat-4026	190	13	sum	sum	NOUN
easat-4026	190	14	is	be	AUX
easat-4026	190	15	taken	take	VERB
easat-4026	190	16	over	over	ADP
easat-4026	190	17	all	all	DET
easat-4026	190	18	decompositions	decomposition	NOUN
easat-4026	190	19	𝑛	𝑛	PRON
easat-4026	190	20	=	=	SYM
easat-4026	190	21	𝓋	𝓋	PROPN
easat-4026	191	1	+	+	NOUN
easat-4026	191	2	𝓌.	𝓌.	NOUN
easat-4026	191	3	we	we	PRON
easat-4026	191	4	now	now	ADV
easat-4026	191	5	classify	classify	VERB
easat-4026	191	6	nested	nested	ADJ
easat-4026	191	7	categories	category	NOUN
easat-4026	191	8	and	and	CCONJ
easat-4026	191	9	apply	apply	VERB
easat-4026	191	10	functorial	functorial	NOUN
easat-4026	191	11	transformations	transformation	NOUN
easat-4026	191	12	between	between	ADP
easat-4026	191	13	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	191	14	.	.	PUNCT
easat-4026	192	1	2.23	2.23	NUM
easat-4026	192	2	.	.	PUNCT
easat-4026	193	1	theorem	theorem	VERB
easat-4026	193	2	[	[	X
easat-4026	193	3	8	8	NUM
easat-4026	193	4	]	]	PUNCT
easat-4026	193	5	assume	assume	VERB
easat-4026	193	6	that	that	SCONJ
easat-4026	193	7	𝒽:𝒜	𝒽:𝒜	NOUN
easat-4026	193	8	→	→	SYM
easat-4026	193	9	ℬ	ℬ	PROPN
easat-4026	193	10	is	be	AUX
easat-4026	193	11	a	a	DET
easat-4026	193	12	quasi	quasi	NOUN
easat-4026	193	13	-	-	NOUN
easat-4026	193	14	isomorphism	isomorphism	NOUN
easat-4026	193	15	of	of	ADP
easat-4026	193	16	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	193	17	.	.	PUNCT
easat-4026	194	1	then	then	ADV
easat-4026	194	2	,	,	PUNCT
easat-4026	194	3	the	the	DET
easat-4026	194	4	equivalent	equivalent	ADJ
easat-4026	194	5	functor	functor	NOUN
easat-4026	194	6	for	for	ADP
easat-4026	194	7	triangulated	triangulate	VERB
easat-4026	194	8	categories	category	NOUN
easat-4026	194	9	that	that	PRON
easat-4026	194	10	maps	map	VERB
easat-4026	194	11	ℬ	ℬ	NOUN
easat-4026	194	12	to	to	ADP
easat-4026	194	13	𝒜	𝒜	NOUN
easat-4026	194	14	is	be	AUX
easat-4026	194	15	:	:	PUNCT
easat-4026	194	16	𝒽∗:𝒜∞(ℬ	𝒽∗:𝒜∞(ℬ	NUM
easat-4026	194	17	)	)	PUNCT
easat-4026	194	18	→	→	SYM
easat-4026	194	19	𝒜∞(𝒜	𝒜∞(𝒜	PROPN
easat-4026	194	20	)	)	PUNCT
easat-4026	194	21	.	.	PUNCT
easat-4026	195	1	this	this	DET
easat-4026	195	2	theorem	theorem	NOUN
easat-4026	195	3	shows	show	VERB
easat-4026	195	4	how	how	SCONJ
easat-4026	195	5	to	to	PART
easat-4026	195	6	establish	establish	VERB
easat-4026	195	7	correspondences	correspondence	NOUN
easat-4026	195	8	between	between	ADP
easat-4026	195	9	nested	nested	ADJ
easat-4026	195	10	categories	category	NOUN
easat-4026	195	11	of	of	ADP
easat-4026	195	12	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	195	13	and	and	CCONJ
easat-4026	195	14	how	how	SCONJ
easat-4026	195	15	to	to	PART
easat-4026	195	16	use	use	VERB
easat-4026	195	17	these	these	DET
easat-4026	195	18	correspondences	correspondence	NOUN
easat-4026	195	19	in	in	ADP
easat-4026	195	20	practical	practical	ADJ
easat-4026	195	21	applications	application	NOUN
easat-4026	195	22	.	.	PUNCT
easat-4026	196	1	2.24	2.24	NUM
easat-4026	196	2	.	.	PUNCT
easat-4026	197	1	theorem	theorem	VERB
easat-4026	197	2	[	[	X
easat-4026	197	3	22	22	NUM
easat-4026	197	4	]	]	PUNCT
easat-4026	197	5	let	let	VERB
easat-4026	197	6	(	(	PUNCT
easat-4026	197	7	𝒜	𝒜	NOUN
easat-4026	197	8	,	,	PUNCT
easat-4026	197	9	𝒹𝒜	𝒹𝒜	NOUN
easat-4026	197	10	)	)	PUNCT
easat-4026	197	11	be	be	VERB
easat-4026	197	12	a	a	DET
easat-4026	197	13	differential	differential	ADJ
easat-4026	197	14	graded	grade	VERB
easat-4026	197	15	algebra	algebra	NOUN
easat-4026	197	16	,	,	PUNCT
easat-4026	197	17	and	and	CCONJ
easat-4026	197	18	let	let	VERB
easat-4026	197	19	ℭ𝒹ℊ(𝒜	ℭ𝒹ℊ(𝒜	NOUN
easat-4026	197	20	)	)	PUNCT
easat-4026	197	21	e	e	X
easat-4026	197	22	the	the	DET
easat-4026	197	23	category	category	NOUN
easat-4026	197	24	of	of	ADP
easat-4026	197	25	differential	differential	ADJ
easat-4026	197	26	graded	grade	VERB
easat-4026	197	27	modules	module	NOUN
easat-4026	197	28	with	with	ADP
easat-4026	197	29	morphisms	morphism	NOUN
easat-4026	197	30	between	between	ADP
easat-4026	197	31	these	these	DET
easat-4026	197	32	modules	module	NOUN
easat-4026	197	33	,	,	PUNCT
easat-4026	197	34	such	such	ADJ
easat-4026	197	35	that	that	PRON
easat-4026	197	36	𝒜𝒹ℊ(𝒜	𝒜𝒹ℊ(𝒜	NUM
easat-4026	197	37	)	)	PUNCT
easat-4026	197	38	is	be	AUX
easat-4026	197	39	the	the	DET
easat-4026	197	40	category	category	NOUN
easat-4026	197	41	.	.	PUNCT
easat-4026	198	1	then	then	ADV
easat-4026	198	2	,	,	PUNCT
easat-4026	198	3	the	the	DET
easat-4026	198	4	equivalence	equivalence	NOUN
easat-4026	198	5	for	for	ADP
easat-4026	198	6	the	the	DET
easat-4026	198	7	triangulated	triangulate	VERB
easat-4026	198	8	categories	category	NOUN
easat-4026	198	9	:	:	PUNCT
easat-4026	198	10	𝒜𝒹ℊ(𝒜	𝒜𝒹ℊ(𝒜	PUNCT
easat-4026	198	11	)	)	PUNCT
easat-4026	198	12	→	→	SYM
easat-4026	198	13	𝒜∞(𝒜	𝒜∞(𝒜	PROPN
easat-4026	198	14	)	)	PUNCT
easat-4026	198	15	is	be	AUX
easat-4026	198	16	produced	produce	VERB
easat-4026	198	17	by	by	ADP
easat-4026	198	18	the	the	DET
easat-4026	198	19	inclusion	inclusion	NOUN
easat-4026	198	20	functor	functor	PROPN
easat-4026	198	21	ℭ𝒹ℊ(𝒜	ℭ𝒹ℊ(𝒜	NOUN
easat-4026	198	22	)	)	PUNCT
easat-4026	198	23	→	→	SYM
easat-4026	198	24	ℭ∞(𝒜	ℭ∞(𝒜	PROPN
easat-4026	198	25	)	)	PUNCT
easat-4026	198	26	.	.	PUNCT
easat-4026	199	1	next	next	ADV
easat-4026	199	2	we	we	PRON
easat-4026	199	3	define	define	VERB
easat-4026	199	4	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	199	5	using	use	VERB
easat-4026	199	6	graded	grade	VERB
easat-4026	199	7	algebra	algebra	NOUN
easat-4026	199	8	properties	property	NOUN
easat-4026	199	9	and	and	CCONJ
easat-4026	199	10	examine	examine	VERB
easat-4026	199	11	how	how	SCONJ
easat-4026	199	12	to	to	PART
easat-4026	199	13	utilize	utilize	VERB
easat-4026	199	14	these	these	DET
easat-4026	199	15	definitions	definition	NOUN
easat-4026	199	16	in	in	ADP
easat-4026	199	17	analysis	analysis	NOUN
easat-4026	199	18	.	.	PUNCT
easat-4026	200	1	2.25	2.25	NUM
easat-4026	200	2	.	.	PUNCT
easat-4026	201	1	definition	definition	NOUN
easat-4026	201	2	[	[	X
easat-4026	201	3	14	14	NUM
easat-4026	201	4	]	]	PUNCT
easat-4026	201	5	assuming	assume	VERB
easat-4026	201	6	ℛ	ℛ	PROPN
easat-4026	201	7	is	be	AUX
easat-4026	201	8	a	a	DET
easat-4026	201	9	commutative	commutative	ADJ
easat-4026	201	10	ring	ring	NOUN
easat-4026	201	11	and	and	CCONJ
easat-4026	201	12	𝒜	𝒜	NOUN
easat-4026	201	13	=	=	PUNCT
easat-4026	201	14	∑	∑	PUNCT
easat-4026	201	15	𝒜𝑛	𝒜𝑛	PROPN
easat-4026	201	16	𝑛∈ℤ	𝑛∈ℤ	PROPN
easat-4026	201	17	is	be	AUX
easat-4026	201	18	a	a	DET
easat-4026	201	19	graded	grade	VERB
easat-4026	201	20	ℛ-module	ℛ-module	PROPN
easat-4026	201	21	.	.	PUNCT
easat-4026	201	22	therefore	therefore	ADV
easat-4026	201	23	,	,	PUNCT
easat-4026	201	24	the	the	DET
easat-4026	201	25	collection	collection	NOUN
easat-4026	201	26	of	of	ADP
easat-4026	201	27	multiplication	multiplication	NOUN
easat-4026	201	28	maps	map	NOUN
easat-4026	201	29	𝑟𝑛:𝒜	𝑟𝑛:𝒜	PROPN
easat-4026	201	30	⨂𝑛	⨂𝑛	ADJ
easat-4026	201	31	→	→	SYM
easat-4026	201	32	𝒜	𝒜	NOUN
easat-4026	201	33	of	of	ADP
easat-4026	201	34	degree	degree	NOUN
easat-4026	201	35	(	(	PUNCT
easat-4026	201	36	2	2	NUM
easat-4026	201	37	−	−	NUM
easat-4026	201	38	𝑛	𝑛	NOUN
easat-4026	201	39	)	)	PUNCT
easat-4026	201	40	is	be	AUX
easat-4026	201	41	an	an	DET
easat-4026	201	42	𝒜∞-structures	𝒜∞-structure	NOUN
easat-4026	201	43	on	on	ADP
easat-4026	201	44	𝒜	𝒜	NOUN
easat-4026	201	45	,	,	PUNCT
easat-4026	201	46	such	such	ADJ
easat-4026	201	47	that	that	SCONJ
easat-4026	201	48	:	:	PUNCT
easat-4026	201	49	∑	∑	PUNCT
easat-4026	201	50	(	(	PUNCT
easat-4026	201	51	−1)𝑚𝑠+𝑡𝑟𝑚+1+𝑡	−1)𝑚𝑠+𝑡𝑟𝑚+1+𝑡	X
easat-4026	201	52	(	(	PUNCT
easat-4026	201	53	1⨂𝑚	1⨂𝑚	NUM
easat-4026	201	54	⊗	⊗	PROPN
easat-4026	201	55	𝑟𝑠	𝑟𝑠	PROPN
easat-4026	201	56	⊗1⨂𝑡	⊗1⨂𝑡	PROPN
easat-4026	201	57	)	)	PUNCT
easat-4026	201	58	𝑚+𝑠+𝑡=𝑛	𝑚+𝑠+𝑡=𝑛	NUM
easat-4026	201	59	=	=	SYM
easat-4026	201	60	0	0	NUM
easat-4026	201	61	,	,	PUNCT
easat-4026	201	62	for	for	ADP
easat-4026	201	63	all	all	DET
easat-4026	201	64	𝑛.	𝑛.	NOUN
easat-4026	201	65	now	now	ADV
easat-4026	201	66	,	,	PUNCT
easat-4026	201	67	we	we	PRON
easat-4026	201	68	discuss	discuss	VERB
easat-4026	201	69	constructing	construct	VERB
easat-4026	201	70	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	201	71	using	use	VERB
easat-4026	201	72	homology	homology	NOUN
easat-4026	201	73	and	and	CCONJ
easat-4026	201	74	applying	apply	VERB
easat-4026	201	75	this	this	DET
easat-4026	201	76	structure	structure	NOUN
easat-4026	201	77	in	in	ADP
easat-4026	201	78	various	various	ADJ
easat-4026	201	79	contexts	contexts	NOUN
easat-4026	201	80	.	.	PUNCT
easat-4026	202	1	2.26	2.26	NUM
easat-4026	202	2	.	.	PUNCT
easat-4026	202	3	theorem	theorem	VERB
easat-4026	202	4	[	[	X
easat-4026	202	5	23	23	NUM
easat-4026	202	6	]	]	PUNCT
easat-4026	202	7	assume	assume	VERB
easat-4026	202	8	(	(	PUNCT
easat-4026	202	9	𝒞	𝒞	PROPN
easat-4026	202	10	,	,	PUNCT
easat-4026	202	11	ω	ω	PROPN
easat-4026	202	12	,	,	PUNCT
easat-4026	202	13	𝜂	𝜂	NOUN
easat-4026	202	14	)	)	PUNCT
easat-4026	202	15	is	be	AUX
easat-4026	202	16	a	a	DET
easat-4026	202	17	differential	differential	ADJ
easat-4026	202	18	graded	grade	VERB
easat-4026	202	19	algebra	algebra	NOUN
easat-4026	202	20	,	,	PUNCT
easat-4026	202	21	and	and	CCONJ
easat-4026	203	1	ℋ∗(𝒞	ℋ∗(𝒞	X
easat-4026	203	2	)	)	PUNCT
easat-4026	203	3	is	be	AUX
easat-4026	203	4	projective	projective	ADJ
easat-4026	203	5	over	over	ADP
easat-4026	203	6	𝑀	𝑀	PROPN
easat-4026	203	7	in	in	ADP
easat-4026	203	8	each	each	DET
easat-4026	203	9	degree	degree	NOUN
easat-4026	203	10	.	.	PUNCT
easat-4026	204	1	then	then	ADV
easat-4026	204	2	there	there	PRON
easat-4026	204	3	exists	exist	VERB
easat-4026	204	4	an	an	DET
easat-4026	204	5	𝒜∞-structure	𝒜∞-structure	PROPN
easat-4026	204	6	𝑟	𝑟	NOUN
easat-4026	204	7	on	on	ADP
easat-4026	204	8	ℋ∗(𝒞	ℋ∗(𝒞	NOUN
easat-4026	204	9	)	)	PUNCT
easat-4026	204	10	such	such	ADJ
easat-4026	204	11	that	that	SCONJ
easat-4026	204	12	:	:	PUNCT
easat-4026	204	13	(	(	PUNCT
easat-4026	204	14	i	i	NOUN
easat-4026	204	15	)	)	PUNCT
easat-4026	204	16	𝑟1	𝑟1	NOUN
easat-4026	204	17	=	=	SYM
easat-4026	204	18	0	0	PUNCT
easat-4026	204	19	(	(	PUNCT
easat-4026	204	20	minimal	minimal	ADJ
easat-4026	204	21	case	case	NOUN
easat-4026	204	22	)	)	PUNCT
easat-4026	204	23	,	,	PUNCT
easat-4026	204	24	(	(	PUNCT
easat-4026	204	25	ii	ii	NOUN
easat-4026	204	26	)	)	PUNCT
easat-4026	204	27	𝑟2	𝑟2	NOUN
easat-4026	204	28	is	be	AUX
easat-4026	204	29	induced	induce	VERB
easat-4026	204	30	by	by	ADP
easat-4026	204	31	𝜂	𝜂	PROPN
easat-4026	204	32	,	,	PUNCT
easat-4026	204	33	(	(	PUNCT
easat-4026	204	34	iii	iii	NOUN
easat-4026	204	35	)	)	PUNCT
easat-4026	204	36	(	(	PUNCT
easat-4026	204	37	𝐻∗(𝒞	𝐻∗(𝒞	X
easat-4026	204	38	)	)	PUNCT
easat-4026	204	39	,	,	PUNCT
easat-4026	204	40	𝑟	𝑟	X
easat-4026	204	41	)	)	PUNCT
easat-4026	204	42	is	be	AUX
easat-4026	204	43	quasi	quasi	ADJ
easat-4026	204	44	-	-	ADJ
easat-4026	204	45	isomorphic	isomorphic	ADJ
easat-4026	204	46	to	to	ADP
easat-4026	204	47	(	(	PUNCT
easat-4026	204	48	𝒞	𝒞	PROPN
easat-4026	204	49	,	,	PUNCT
easat-4026	204	50	𝛺	𝛺	PROPN
easat-4026	204	51	,	,	PUNCT
easat-4026	204	52	𝜂	𝜂	NOUN
easat-4026	204	53	)	)	PUNCT
easat-4026	204	54	.	.	PUNCT
easat-4026	205	1	next	next	ADV
easat-4026	205	2	,	,	PUNCT
easat-4026	205	3	we	we	PRON
easat-4026	205	4	introduce	introduce	VERB
easat-4026	205	5	the	the	DET
easat-4026	205	6	maurer	maurer	NOUN
easat-4026	205	7	-	-	PUNCT
easat-4026	205	8	cartan	cartan	PROPN
easat-4026	205	9	equation	equation	NOUN
easat-4026	205	10	,	,	PUNCT
easat-4026	205	11	which	which	PRON
easat-4026	205	12	allows	allow	VERB
easat-4026	205	13	us	we	PRON
easat-4026	205	14	to	to	PART
easat-4026	205	15	extend	extend	VERB
easat-4026	205	16	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	205	17	and	and	CCONJ
easat-4026	205	18	analyze	analyze	VERB
easat-4026	205	19	different	different	ADJ
easat-4026	205	20	applications	application	NOUN
easat-4026	205	21	.	.	PUNCT
easat-4026	206	1	2.27	2.27	NUM
easat-4026	206	2	.	.	PUNCT
easat-4026	206	3	definition	definition	NOUN
easat-4026	206	4	[	[	X
easat-4026	206	5	24	24	NUM
easat-4026	206	6	]	]	PUNCT
easat-4026	206	7	let	let	VERB
easat-4026	206	8	ℋ	ℋ	PRON
easat-4026	206	9	be	be	AUX
easat-4026	206	10	an	an	DET
easat-4026	206	11	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-4026	206	12	,	,	PUNCT
easat-4026	206	13	and	and	CCONJ
easat-4026	206	14	let	let	VERB
easat-4026	206	15	𝒶	𝒶	PROPN
easat-4026	206	16	∈	∈	PROPN
easat-4026	206	17	ℋ1	ℋ1	NOUN
easat-4026	206	18	,	,	PUNCT
easat-4026	206	19	we	we	PRON
easat-4026	206	20	say	say	VERB
easat-4026	206	21	𝒶	𝒶	PROPN
easat-4026	206	22	satisfies	satisfy	VERB
easat-4026	206	23	the	the	DET
easat-4026	206	24	maurer	maurer	PROPN
easat-4026	206	25	-	-	PUNCT
easat-4026	206	26	cartan	cartan	PROPN
easat-4026	206	27	equation	equation	NOUN
easat-4026	206	28	if	if	SCONJ
easat-4026	206	29	:	:	PUNCT
easat-4026	206	30	∑±𝑟𝑛(𝒶⨂𝑛	∑±𝑟𝑛(𝒶⨂𝑛	NOUN
easat-4026	206	31	)	)	PUNCT
easat-4026	206	32	𝑛∈ℕ	𝑛∈ℕ	X
easat-4026	206	33	=	=	SYM
easat-4026	207	1	0	0	X
easat-4026	207	2	.	.	PUNCT
easat-4026	208	1	the	the	DET
easat-4026	208	2	twisted	twisted	ADJ
easat-4026	208	3	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	208	4	ℋ𝒶	ℋ𝒶	PROPN
easat-4026	208	5	is	be	AUX
easat-4026	208	6	then	then	ADV
easat-4026	208	7	defined	define	VERB
easat-4026	208	8	by	by	ADP
easat-4026	208	9	:	:	PUNCT
easat-4026	208	10	9480	9480	NUM
easat-4026	208	11	edelweiss	edelweiss	PROPN
easat-4026	208	12	applied	apply	VERB
easat-4026	208	13	science	science	NOUN
easat-4026	208	14	and	and	CCONJ
easat-4026	208	15	technology	technology	NOUN
easat-4026	208	16	issn	issn	PROPN
easat-4026	208	17	:	:	PUNCT
easat-4026	208	18	2576	2576	NUM
easat-4026	208	19	-	-	SYM
easat-4026	208	20	8484	8484	NUM
easat-4026	208	21	vol	vol	NOUN
easat-4026	208	22	.	.	PROPN
easat-4026	209	1	8	8	NUM
easat-4026	209	2	,	,	PUNCT
easat-4026	209	3	no	no	INTJ
easat-4026	209	4	.	.	NOUN
easat-4026	210	1	6	6	NUM
easat-4026	210	2	:	:	SYM
easat-4026	210	3	9472	9472	NUM
easat-4026	210	4	-	-	SYM
easat-4026	210	5	9486	9486	NUM
easat-4026	210	6	,	,	PUNCT
easat-4026	210	7	2024	2024	NUM
easat-4026	210	8	doi	doi	NOUN
easat-4026	210	9	:	:	PUNCT
easat-4026	210	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	210	11	©	©	PROPN
easat-4026	210	12	2024	2024	NUM
easat-4026	210	13	by	by	ADP
easat-4026	210	14	the	the	DET
easat-4026	210	15	authors	author	NOUN
easat-4026	210	16	;	;	PUNCT
easat-4026	210	17	licensee	licensee	PROPN
easat-4026	210	18	learning	learning	NOUN
easat-4026	210	19	gate	gate	NOUN
easat-4026	210	20	𝑟𝑛	𝑟𝑛	ADP
easat-4026	210	21	𝒶(𝒾1,⋯	𝒶(𝒾1,⋯	PROPN
easat-4026	210	22	,	,	PUNCT
easat-4026	210	23	𝒾𝑛	𝒾𝑛	PROPN
easat-4026	210	24	)	)	PUNCT
easat-4026	210	25	=	=	SYM
easat-4026	210	26	∑	∑	PUNCT
easat-4026	210	27	±𝑟𝑛+ℓ(𝒶⨂ℓ1	±𝑟𝑛+ℓ(𝒶⨂ℓ1	PROPN
easat-4026	210	28	,	,	PUNCT
easat-4026	210	29	𝒾1	𝒾1	PROPN
easat-4026	210	30	,	,	PUNCT
easat-4026	210	31	𝒶⨂ℓ2	𝒶⨂ℓ2	VERB
easat-4026	210	32	,	,	PUNCT
easat-4026	210	33	𝒾2,⋯	𝒾2,⋯	NOUN
easat-4026	210	34	,	,	PUNCT
easat-4026	210	35	𝒾𝑛	𝒾𝑛	NOUN
easat-4026	210	36	,	,	PUNCT
easat-4026	210	37	𝒶⨂ℓ𝑛+1	𝒶⨂ℓ𝑛+1	NOUN
easat-4026	210	38	)	)	PUNCT
easat-4026	210	39	ℓ1+⋯+ℓ𝑛+1∈ℕ0	ℓ1+⋯+ℓ𝑛+1∈ℕ0	NOUN
easat-4026	210	40	,	,	PUNCT
easat-4026	210	41	where	where	SCONJ
easat-4026	210	42	ℓ	ℓ	NOUN
easat-4026	210	43	=	=	SYM
easat-4026	210	44	ℓ1	ℓ1	PROPN
easat-4026	210	45	+	+	NOUN
easat-4026	210	46	⋯+	⋯+	NOUN
easat-4026	210	47	ℓ𝑛+1	ℓ𝑛+1	NOUN
easat-4026	210	48	.	.	PUNCT
easat-4026	211	1	these	these	DET
easat-4026	211	2	formulas	formula	NOUN
easat-4026	211	3	also	also	ADV
easat-4026	211	4	apply	apply	VERB
easat-4026	211	5	to	to	ADP
easat-4026	211	6	𝒜∞-bimodules	𝒜∞-bimodules	PROPN
easat-4026	211	7	.	.	PUNCT
easat-4026	212	1	hence	hence	ADV
easat-4026	212	2	,	,	PUNCT
easat-4026	212	3	if	if	SCONJ
easat-4026	212	4	𝒩	𝒩	PROPN
easat-4026	212	5	is	be	AUX
easat-4026	212	6	an	an	DET
easat-4026	212	7	𝒜∞-bimodules	𝒜∞-bimodule	NOUN
easat-4026	212	8	over	over	ADP
easat-4026	212	9	ℋ	ℋ	PROPN
easat-4026	212	10	then	then	ADV
easat-4026	212	11	,	,	PUNCT
easat-4026	212	12	𝒩𝒶	𝒩𝒶	PROPN
easat-4026	212	13	is	be	AUX
easat-4026	212	14	an	an	DET
easat-4026	212	15	𝒜∞-bimodules	𝒜∞-bimodules	PROPN
easat-4026	212	16	over	over	ADP
easat-4026	212	17	ℋ𝒶.	ℋ𝒶.	PROPN
easat-4026	212	18	now	now	ADV
easat-4026	212	19	,	,	PUNCT
easat-4026	212	20	we	we	PRON
easat-4026	212	21	define	define	VERB
easat-4026	212	22	simple	simple	ADJ
easat-4026	212	23	homology	homology	NOUN
easat-4026	212	24	for	for	ADP
easat-4026	212	25	𝒜∞-algebras	𝒜∞-algebras	PRON
easat-4026	212	26	using	use	VERB
easat-4026	212	27	complexes	complex	NOUN
easat-4026	212	28	and	and	CCONJ
easat-4026	212	29	their	their	PRON
easat-4026	212	30	sequences	sequence	NOUN
easat-4026	212	31	.	.	PUNCT
easat-4026	213	1	2.28	2.28	NUM
easat-4026	213	2	.	.	PUNCT
easat-4026	214	1	definition	definition	NOUN
easat-4026	214	2	[	[	X
easat-4026	214	3	25	25	NUM
easat-4026	214	4	]	]	PUNCT
easat-4026	214	5	let	let	VERB
easat-4026	214	6	a	a	DET
easat-4026	214	7	space	space	NOUN
easat-4026	214	8	𝒳	𝒳	PRON
easat-4026	214	9	be	be	VERB
easat-4026	214	10	an	an	DET
easat-4026	214	11	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-4026	214	12	,	,	PUNCT
easat-4026	214	13	and	and	CCONJ
easat-4026	214	14	let	let	VERB
easat-4026	214	15	(	(	PUNCT
easat-4026	214	16	𝒳∗	𝒳∗	INTJ
easat-4026	214	17	,	,	PUNCT
easat-4026	214	18	𝒹∗	𝒹∗	PROPN
easat-4026	214	19	)	)	PUNCT
easat-4026	214	20	=	=	PRON
easat-4026	215	1	{	{	PUNCT
easat-4026	215	2	𝒳𝑛	𝒳𝑛	PROPN
easat-4026	215	3	,	,	PUNCT
easat-4026	215	4	𝒹𝑛	𝒹𝑛	AUX
easat-4026	215	5	}	}	PUNCT
easat-4026	215	6	represent	represent	VERB
easat-4026	215	7	a	a	DET
easat-4026	215	8	chain	chain	NOUN
easat-4026	215	9	complex	complex	NOUN
easat-4026	215	10	such	such	ADJ
easat-4026	215	11	that	that	SCONJ
easat-4026	215	12	:	:	PUNCT
easat-4026	215	13	…	…	PUNCT
easat-4026	215	14	→	→	SYM
easat-4026	215	15	𝒳𝑛+1	𝒳𝑛+1	X
easat-4026	215	16	𝒹𝑛+1	𝒹𝑛+1	X
easat-4026	215	17	→	→	PUNCT
easat-4026	215	18	𝒳𝑛	𝒳𝑛	NOUN
easat-4026	215	19	𝒹𝑛	𝒹𝑛	PRON
easat-4026	215	20	→	→	PUNCT
easat-4026	215	21	𝒳𝑛−1	𝒳𝑛−1	ADJ
easat-4026	215	22	𝒹𝑛−1	𝒹𝑛−1	PROPN
easat-4026	215	23	→	→	SYM
easat-4026	215	24	…	…	PUNCT
easat-4026	215	25	𝒹1	𝒹1	NOUN
easat-4026	215	26	→	→	SYM
easat-4026	215	27	𝒳0	𝒳0	PROPN
easat-4026	215	28	𝒹0	𝒹0	PROPN
easat-4026	215	29	→	→	SYM
easat-4026	215	30	0	0	NUM
easat-4026	215	31	then	then	ADV
easat-4026	215	32	,	,	PUNCT
easat-4026	215	33	the	the	DET
easat-4026	215	34	𝑛𝑡ℎ	𝑛𝑡ℎ	PROPN
easat-4026	215	35	homology	homology	NOUN
easat-4026	215	36	of	of	ADP
easat-4026	215	37	an	an	DET
easat-4026	215	38	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-4026	215	39	𝒳	𝒳	PROPN
easat-4026	215	40	is	be	AUX
easat-4026	215	41	defined	define	VERB
easat-4026	215	42	as	as	ADP
easat-4026	215	43	:	:	PUNCT
easat-4026	215	44	ℋ𝑛(𝒳	ℋ𝑛(𝒳	NUM
easat-4026	215	45	)	)	PUNCT
easat-4026	215	46	=	=	SYM
easat-4026	215	47	𝑘𝑒𝑟(𝒹𝑛	𝑘𝑒𝑟(𝒹𝑛	NOUN
easat-4026	215	48	)	)	PUNCT
easat-4026	215	49	𝐼𝑚(𝒹𝑛+1	𝐼𝑚(𝒹𝑛+1	NOUN
easat-4026	215	50	)	)	PUNCT
easat-4026	215	51	.	.	PUNCT
easat-4026	216	1	(	(	PUNCT
easat-4026	216	2	5	5	NUM
easat-4026	216	3	)	)	PUNCT
easat-4026	216	4	since	since	SCONJ
easat-4026	216	5	𝐼𝑚	𝐼𝑚	PROPN
easat-4026	216	6	𝒹𝑛+1	𝒹𝑛+1	NUM
easat-4026	216	7	⊂	⊂	ADJ
easat-4026	216	8	𝑘𝑒𝑟	𝑘𝑒𝑟	NOUN
easat-4026	216	9	𝒹𝑛	𝒹𝑛	NOUN
easat-4026	216	10	,	,	PUNCT
easat-4026	216	11	and	and	CCONJ
easat-4026	216	12	ℋ𝑛(𝒳	ℋ𝑛(𝒳	NUM
easat-4026	216	13	)	)	PUNCT
easat-4026	216	14	=	=	SYM
easat-4026	216	15	𝑛−cycles	𝑛−cycle	NOUN
easat-4026	216	16	𝑛−boundaries	𝑛−boundarie	NOUN
easat-4026	216	17	.	.	PUNCT
easat-4026	217	1	finally	finally	ADV
easat-4026	217	2	,	,	PUNCT
easat-4026	217	3	we	we	PRON
easat-4026	217	4	define	define	VERB
easat-4026	217	5	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	217	6	using	use	VERB
easat-4026	217	7	corrections	correction	NOUN
easat-4026	217	8	for	for	ADP
easat-4026	217	9	algebra	algebra	NOUN
easat-4026	217	10	models	model	NOUN
easat-4026	217	11	and	and	CCONJ
easat-4026	217	12	explain	explain	VERB
easat-4026	217	13	how	how	SCONJ
easat-4026	217	14	to	to	PART
easat-4026	217	15	handle	handle	VERB
easat-4026	217	16	derived	derived	ADJ
easat-4026	217	17	structures	structure	NOUN
easat-4026	217	18	.	.	PUNCT
easat-4026	218	1	2.29	2.29	NUM
easat-4026	218	2	.	.	PUNCT
easat-4026	218	3	definition	definition	NOUN
easat-4026	218	4	[	[	X
easat-4026	218	5	22	22	NUM
easat-4026	218	6	]	]	PUNCT
easat-4026	218	7	let	let	VERB
easat-4026	218	8	ℬ	ℬ	PRON
easat-4026	218	9	be	be	AUX
easat-4026	218	10	a	a	DET
easat-4026	218	11	unital	unital	ADJ
easat-4026	218	12	associative	associative	ADJ
easat-4026	218	13	algebra	algebra	NOUN
easat-4026	218	14	,	,	PUNCT
easat-4026	218	15	𝑅	𝑅	PROPN
easat-4026	218	16	a	a	DET
easat-4026	218	17	right	right	ADJ
easat-4026	218	18	ℬ-module	ℬ-module	PROPN
easat-4026	218	19	,	,	PUNCT
easat-4026	218	20	and	and	CCONJ
easat-4026	218	21	𝒫	𝒫	PROPN
easat-4026	218	22	→	→	SYM
easat-4026	218	23	𝑅	𝑅	PROPN
easat-4026	218	24	the	the	DET
easat-4026	218	25	projective	projective	ADJ
easat-4026	218	26	resolution	resolution	NOUN
easat-4026	218	27	.	.	PUNCT
easat-4026	219	1	consider	consider	VERB
easat-4026	219	2	𝒜	𝒜	NOUN
easat-4026	219	3	=	=	SYM
easat-4026	219	4	𝐻𝑜𝑚ℬ(𝒫,𝒫	𝐻𝑜𝑚ℬ(𝒫,𝒫	PROPN
easat-4026	219	5	)	)	PUNCT
easat-4026	219	6	represent	represent	VERB
easat-4026	219	7	the	the	DET
easat-4026	219	8	differential	differential	NOUN
easat-4026	219	9	graded	grade	VERB
easat-4026	219	10	endomorphism	endomorphism	PROPN
easat-4026	219	11	algebras	algebra	NOUN
easat-4026	219	12	of	of	ADP
easat-4026	219	13	𝒫	𝒫	PROPN
easat-4026	219	14	,	,	PUNCT
easat-4026	219	15	with	with	ADP
easat-4026	219	16	its	its	PRON
easat-4026	219	17	nth	nth	NOUN
easat-4026	219	18	component	component	NOUN
easat-4026	219	19	consisting	consist	VERB
easat-4026	219	20	of	of	ADP
easat-4026	219	21	the	the	DET
easat-4026	219	22	morphisms	morphism	NOUN
easat-4026	219	23	of	of	ADP
easat-4026	219	24	graded	grade	VERB
easat-4026	219	25	items	item	NOUN
easat-4026	219	26	of	of	ADP
easat-4026	219	27	degree	degree	NOUN
easat-4026	219	28	𝑛	𝑛	PROPN
easat-4026	219	29	and	and	CCONJ
easat-4026	219	30	its	its	PRON
easat-4026	219	31	differential	differential	NOUN
easat-4026	219	32	being	be	AUX
easat-4026	219	33	the	the	DET
easat-4026	219	34	super	super	ADJ
easat-4026	219	35	commutator	commutator	NOUN
easat-4026	219	36	corresponding	correspond	VERB
easat-4026	219	37	to	to	ADP
easat-4026	219	38	the	the	DET
easat-4026	219	39	differential	differential	NOUN
easat-4026	219	40	of	of	ADP
easat-4026	219	41	𝒫.	𝒫.	PROPN
easat-4026	219	42	since	since	SCONJ
easat-4026	219	43	𝒜	𝒜	NOUN
easat-4026	219	44	is	be	AUX
easat-4026	219	45	specifically	specifically	ADV
easat-4026	219	46	an	an	DET
easat-4026	219	47	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-4026	219	48	,	,	PUNCT
easat-4026	219	49	it	it	PRON
easat-4026	219	50	includes	include	VERB
easat-4026	219	51	a	a	DET
easat-4026	219	52	minimal	minimal	ADJ
easat-4026	219	53	model	model	NOUN
easat-4026	219	54	.	.	PUNCT
easat-4026	220	1	the	the	DET
easat-4026	220	2	homology	homology	NOUN
easat-4026	220	3	ℋ∗𝒜	ℋ∗𝒜	NOUN
easat-4026	220	4	,	,	PUNCT
easat-4026	220	5	as	as	ADP
easat-4026	220	6	an	an	DET
easat-4026	220	7	algebra	algebra	NOUN
easat-4026	220	8	of	of	ADP
easat-4026	220	9	𝑟2	𝑟2	NOUN
easat-4026	220	10	,	,	PUNCT
easat-4026	220	11	is	be	AUX
easat-4026	220	12	now	now	ADV
easat-4026	220	13	isomorphic	isomorphic	ADJ
easat-4026	220	14	to	to	ADP
easat-4026	220	15	the	the	DET
easat-4026	220	16	yoneda	yoneda	PROPN
easat-4026	220	17	algebra	algebra	PROPN
easat-4026	220	18	𝐸𝑥𝑡ℬ	𝐸𝑥𝑡ℬ	PROPN
easat-4026	220	19	∗	∗	NOUN
easat-4026	220	20	(	(	PUNCT
easat-4026	220	21	𝑅	𝑅	PROPN
easat-4026	220	22	,	,	PUNCT
easat-4026	220	23	𝑅	𝑅	PROPN
easat-4026	220	24	)	)	PUNCT
easat-4026	220	25	.	.	PUNCT
easat-4026	221	1	next	next	ADV
easat-4026	221	2	,	,	PUNCT
easat-4026	221	3	we	we	PRON
easat-4026	221	4	explain	explain	VERB
easat-4026	221	5	simple	simple	ADJ
easat-4026	221	6	homology	homology	NOUN
easat-4026	221	7	for	for	ADP
easat-4026	221	8	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	221	9	and	and	CCONJ
easat-4026	221	10	how	how	SCONJ
easat-4026	221	11	to	to	PART
easat-4026	221	12	use	use	VERB
easat-4026	221	13	these	these	DET
easat-4026	221	14	definitions	definition	NOUN
easat-4026	221	15	in	in	ADP
easat-4026	221	16	various	various	ADJ
easat-4026	221	17	applications	application	NOUN
easat-4026	221	18	.	.	PUNCT
easat-4026	222	1	2.30	2.30	NUM
easat-4026	222	2	.	.	PUNCT
easat-4026	223	1	definition	definition	NOUN
easat-4026	223	2	[	[	X
easat-4026	223	3	26	26	NUM
easat-4026	223	4	]	]	PUNCT
easat-4026	223	5	assume	assume	VERB
easat-4026	223	6	that	that	SCONJ
easat-4026	223	7	(	(	PUNCT
easat-4026	223	8	𝒜	𝒜	NOUN
easat-4026	223	9	=	=	NOUN
easat-4026	223	10	⊕𝑗∈ℤ	⊕𝑗∈ℤ	ADP
easat-4026	223	11	𝒜𝑗	𝒜𝑗	PROPN
easat-4026	223	12	,	,	PUNCT
easat-4026	223	13	(	(	PUNCT
easat-4026	223	14	𝜂𝑛)𝑛∈ℕ	𝜂𝑛)𝑛∈ℕ	PROPN
easat-4026	223	15	)	)	PUNCT
easat-4026	223	16	is	be	AUX
easat-4026	223	17	an	an	DET
easat-4026	223	18	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	223	19	over	over	ADP
easat-4026	223	20	𝑀	𝑀	PROPN
easat-4026	223	21	and	and	CCONJ
easat-4026	223	22	let	let	VERB
easat-4026	223	23	(	(	PUNCT
easat-4026	223	24	𝒮	𝒮	NOUN
easat-4026	223	25	=	=	PROPN
easat-4026	223	26	⊕𝑗∈ℤ	⊕𝑗∈ℤ	NOUN
easat-4026	223	27	𝒮𝑗	𝒮𝑗	ADJ
easat-4026	223	28	,	,	PUNCT
easat-4026	223	29	(	(	PUNCT
easat-4026	223	30	𝜇𝑚,𝑠	𝜇𝑚,𝑠	PUNCT
easat-4026	223	31	𝒮	𝒮	PROPN
easat-4026	223	32	)	)	PUNCT
easat-4026	223	33	𝑚,𝑠∈ℕ0	𝑚,𝑠∈ℕ0	NUM
easat-4026	223	34	)	)	PUNCT
easat-4026	223	35	be	be	AUX
easat-4026	223	36	an	an	DET
easat-4026	223	37	𝒜∞-bimodules	𝒜∞-bimodules	PROPN
easat-4026	223	38	over	over	ADP
easat-4026	223	39	𝒜.	𝒜.	NOUN
easat-4026	223	40	we	we	PRON
easat-4026	223	41	further	far	ADV
easat-4026	223	42	denote	denote	VERB
easat-4026	223	43	the	the	DET
easat-4026	223	44	index	index	NOUN
easat-4026	223	45	of	of	ADP
easat-4026	223	46	𝑟	𝑟	DET
easat-4026	223	47	∈	∈	PROPN
easat-4026	223	48	𝒮	𝒮	NOUN
easat-4026	223	49	by	by	ADP
easat-4026	223	50	𝜇(𝑟	𝜇(𝑟	PROPN
easat-4026	223	51	)	)	PUNCT
easat-4026	223	52	∶=	∶=	NUM
easat-4026	223	53	𝜇𝒮(𝑟	𝜇𝒮(𝑟	NUM
easat-4026	223	54	)	)	PUNCT
easat-4026	223	55	.	.	PUNCT
easat-4026	224	1	consider	consider	VERB
easat-4026	224	2	the	the	DET
easat-4026	224	3	graded	grade	VERB
easat-4026	224	4	𝑀-bimodule	𝑀-bimodule	PROPN
easat-4026	224	5	:	:	PUNCT
easat-4026	224	6	ℋℋ∗(𝒜	ℋℋ∗(𝒜	PROPN
easat-4026	224	7	,	,	PUNCT
easat-4026	224	8	𝒮):=	𝒮):=	X
easat-4026	224	9	∞	∞	NUM
easat-4026	224	10	⊕	⊕	NOUN
easat-4026	224	11	𝑛	𝑛	NOUN
easat-4026	224	12	=	=	SYM
easat-4026	224	13	0	0	NUM
easat-4026	224	14	𝒮	𝒮	PROPN
easat-4026	224	15	⊗𝒜⊗𝑛	⊗𝒜⊗𝑛	NOUN
easat-4026	224	16	,	,	PUNCT
easat-4026	224	17	with	with	ADP
easat-4026	224	18	grading	grade	VERB
easat-4026	224	19	on	on	ADP
easat-4026	224	20	ℋℋ∗(𝒜	ℋℋ∗(𝒜	PROPN
easat-4026	224	21	,	,	PUNCT
easat-4026	224	22	𝒮	𝒮	PROPN
easat-4026	224	23	)	)	PUNCT
easat-4026	224	24	is	be	AUX
easat-4026	224	25	given	give	VERB
easat-4026	224	26	by	by	ADP
easat-4026	224	27	:	:	PUNCT
easat-4026	224	28	ℋℋ𝑗(𝒜	ℋℋ𝑗(𝒜	NUM
easat-4026	224	29	,	,	PUNCT
easat-4026	224	30	𝒮	𝒮	NOUN
easat-4026	224	31	)	)	PUNCT
easat-4026	225	1	=	=	VERB
easat-4026	225	2	⊕𝑛∈ℕ0	⊕𝑛∈ℕ0	VERB
easat-4026	225	3	⊕𝑗=𝑛−𝑗0−𝑗1−⋯−𝑗𝑛	⊕𝑗=𝑛−𝑗0−𝑗1−⋯−𝑗𝑛	NOUN
easat-4026	225	4	𝒮𝑗0	𝒮𝑗0	NOUN
easat-4026	225	5	⊗𝒜𝑗1	⊗𝒜𝑗1	PROPN
easat-4026	225	6	⊗	⊗	PROPN
easat-4026	225	7	…	…	PUNCT
easat-4026	225	8	⊗𝒜𝑗𝑛	⊗𝒜𝑗𝑛	PROPN
easat-4026	225	9	.	.	PUNCT
easat-4026	226	1	(	(	PUNCT
easat-4026	226	2	6	6	NUM
easat-4026	226	3	)	)	PUNCT
easat-4026	226	4	which	which	PRON
easat-4026	226	5	corresponds	correspond	VERB
easat-4026	226	6	to	to	ADP
easat-4026	226	7	the	the	DET
easat-4026	226	8	simplicial	simplicial	ADJ
easat-4026	226	9	homology	homology	NOUN
easat-4026	226	10	of	of	ADP
easat-4026	226	11	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	226	12	.	.	PUNCT
easat-4026	227	1	after	after	ADP
easat-4026	227	2	that	that	PRON
easat-4026	227	3	,	,	PUNCT
easat-4026	227	4	we	we	PRON
easat-4026	227	5	discuss	discuss	VERB
easat-4026	227	6	how	how	SCONJ
easat-4026	227	7	to	to	PART
easat-4026	227	8	handle	handle	VERB
easat-4026	227	9	homology	homology	NOUN
easat-4026	227	10	degrees	degree	NOUN
easat-4026	227	11	in	in	ADP
easat-4026	227	12	graded	grade	VERB
easat-4026	227	13	spaces	space	NOUN
easat-4026	227	14	and	and	CCONJ
easat-4026	227	15	explain	explain	VERB
easat-4026	227	16	how	how	SCONJ
easat-4026	227	17	to	to	PART
easat-4026	227	18	determine	determine	VERB
easat-4026	227	19	these	these	DET
easat-4026	227	20	degrees	degree	NOUN
easat-4026	227	21	using	use	VERB
easat-4026	227	22	advanced	advanced	ADJ
easat-4026	227	23	definitions	definition	NOUN
easat-4026	227	24	.	.	PUNCT
easat-4026	228	1	2.31	2.31	NUM
easat-4026	228	2	.	.	PUNCT
easat-4026	228	3	definition	definition	NOUN
easat-4026	228	4	[	[	X
easat-4026	228	5	27	27	NUM
easat-4026	228	6	]	]	PUNCT
easat-4026	228	7	for	for	ADP
easat-4026	228	8	𝒶	𝒶	PROPN
easat-4026	228	9	∈	∈	PROPN
easat-4026	228	10	ℋℋ∗(𝒜	ℋℋ∗(𝒜	PROPN
easat-4026	228	11	,	,	PUNCT
easat-4026	228	12	𝒮	𝒮	PROPN
easat-4026	228	13	)	)	PUNCT
easat-4026	228	14	we	we	PRON
easat-4026	228	15	write	write	VERB
easat-4026	228	16	𝑑𝑒𝑔(𝒶	𝑑𝑒𝑔(𝒶	PROPN
easat-4026	228	17	)	)	PUNCT
easat-4026	229	1	=	=	PUNCT
easat-4026	229	2	𝑗	𝑗	INTJ
easat-4026	229	3	if	if	SCONJ
easat-4026	230	1	and	and	CCONJ
easat-4026	230	2	only	only	ADV
easat-4026	230	3	if	if	SCONJ
easat-4026	230	4	𝒶	𝒶	X
easat-4026	230	5	∈	∈	PROPN
easat-4026	230	6	ℋℋ𝑗(𝒜	ℋℋ𝑗(𝒜	NOUN
easat-4026	230	7	,	,	PUNCT
easat-4026	230	8	𝒮	𝒮	NOUN
easat-4026	230	9	)	)	PUNCT
easat-4026	231	1	and	and	CCONJ
easat-4026	231	2	call	call	VERB
easat-4026	231	3	it	it	PRON
easat-4026	231	4	the	the	DET
easat-4026	231	5	degree	degree	NOUN
easat-4026	231	6	of	of	ADP
easat-4026	231	7	𝒶.	𝒶.	NOUN
easat-4026	231	8	note	note	VERB
easat-4026	231	9	that	that	SCONJ
easat-4026	231	10	for	for	ADP
easat-4026	231	11	all	all	DET
easat-4026	231	12	𝓈	𝓈	PROPN
easat-4026	231	13	∈	∈	PROPN
easat-4026	231	14	𝒮	𝒮	PROPN
easat-4026	231	15	,	,	PUNCT
easat-4026	231	16	𝑛	𝑛	DET
easat-4026	231	17	∈	∈	PROPN
easat-4026	231	18	ℕ0	ℕ0	NOUN
easat-4026	231	19	and	and	CCONJ
easat-4026	231	20	𝒶1	𝒶1	NOUN
easat-4026	231	21	,	,	PUNCT
easat-4026	231	22	𝒶2	𝒶2	PROPN
easat-4026	231	23	,	,	PUNCT
easat-4026	231	24	…	…	PUNCT
easat-4026	231	25	,	,	PUNCT
easat-4026	231	26	𝒶𝑛	𝒶𝑛	X
easat-4026	231	27	∈	∈	PROPN
easat-4026	231	28	𝒜	𝒜	VERB
easat-4026	231	29	the	the	DET
easat-4026	231	30	degree	degree	NOUN
easat-4026	231	31	of	of	ADP
easat-4026	231	32	𝓈	𝓈	PROPN
easat-4026	231	33	⊗𝒶1	⊗𝒶1	PROPN
easat-4026	231	34	⊗	⊗	NOUN
easat-4026	231	35	…	…	NUM
easat-4026	231	36	⊗𝒶𝑛	⊗𝒶𝑛	NUM
easat-4026	231	37	is	be	AUX
easat-4026	231	38	explicitly	explicitly	ADV
easat-4026	231	39	given	give	VERB
easat-4026	231	40	by	by	ADP
easat-4026	231	41	:	:	PUNCT
easat-4026	231	42	deg(𝓈	deg(𝓈	PROPN
easat-4026	231	43	⊗	⊗	PROPN
easat-4026	231	44	𝒶1	𝒶1	PROPN
easat-4026	231	45	⊗	⊗	NUM
easat-4026	231	46	…	…	PUNCT
easat-4026	231	47	⊗𝒶𝑛	⊗𝒶𝑛	NUM
easat-4026	231	48	)	)	PUNCT
easat-4026	232	1	=	=	SYM
easat-4026	232	2	𝑛	𝑛	PRON
easat-4026	232	3	−	−	PROPN
easat-4026	232	4	𝜂(𝓈	𝜂(𝓈	X
easat-4026	232	5	)	)	PUNCT
easat-4026	232	6	−∑𝜂(𝒶𝑗	−∑𝜂(𝒶𝑗	PROPN
easat-4026	232	7	)	)	PUNCT
easat-4026	232	8	𝑛	𝑛	PRON
easat-4026	232	9	𝑗=1	𝑗=1	PROPN
easat-4026	232	10	=	=	SYM
easat-4026	232	11	−𝜂(𝓈	−𝜂(𝓈	PROPN
easat-4026	232	12	)	)	PUNCT
easat-4026	232	13	−∑‖𝒶𝑗‖	−∑‖𝒶𝑗‖	NOUN
easat-4026	233	1	𝑛	𝑛	PROPN
easat-4026	233	2	𝑗=1	𝑗=1	PROPN
easat-4026	233	3	.	.	PUNCT
easat-4026	234	1	9481	9481	NUM
easat-4026	234	2	edelweiss	edelweiss	PROPN
easat-4026	234	3	applied	apply	VERB
easat-4026	234	4	science	science	NOUN
easat-4026	234	5	and	and	CCONJ
easat-4026	234	6	technology	technology	NOUN
easat-4026	234	7	issn	issn	PROPN
easat-4026	234	8	:	:	PUNCT
easat-4026	234	9	2576	2576	NUM
easat-4026	234	10	-	-	SYM
easat-4026	234	11	8484	8484	NUM
easat-4026	234	12	vol	vol	NOUN
easat-4026	234	13	.	.	PROPN
easat-4026	234	14	8	8	NUM
easat-4026	234	15	,	,	PUNCT
easat-4026	234	16	no	no	INTJ
easat-4026	234	17	.	.	NOUN
easat-4026	235	1	6	6	NUM
easat-4026	235	2	:	:	SYM
easat-4026	235	3	9472	9472	NUM
easat-4026	235	4	-	-	SYM
easat-4026	235	5	9486	9486	NUM
easat-4026	235	6	,	,	PUNCT
easat-4026	235	7	2024	2024	NUM
easat-4026	235	8	doi	doi	NOUN
easat-4026	235	9	:	:	PUNCT
easat-4026	235	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	235	11	©	©	PROPN
easat-4026	235	12	2024	2024	NUM
easat-4026	235	13	by	by	ADP
easat-4026	235	14	the	the	DET
easat-4026	235	15	authors	author	NOUN
easat-4026	235	16	;	;	PUNCT
easat-4026	235	17	licensee	licensee	PROPN
easat-4026	235	18	learning	learn	VERB
easat-4026	235	19	gate	gate	PROPN
easat-4026	235	20	remark	remark	PROPN
easat-4026	235	21	:	:	PUNCT
easat-4026	235	22	the	the	DET
easat-4026	235	23	degree	degree	NOUN
easat-4026	235	24	on	on	ADP
easat-4026	235	25	ℋℋ∗(𝒜	ℋℋ∗(𝒜	PROPN
easat-4026	235	26	,	,	PUNCT
easat-4026	235	27	𝒮	𝒮	PROPN
easat-4026	235	28	)	)	PUNCT
easat-4026	235	29	is	be	AUX
easat-4026	235	30	best	well	ADV
easat-4026	235	31	understood	understand	VERB
easat-4026	235	32	in	in	ADP
easat-4026	235	33	terms	term	NOUN
easat-4026	235	34	of	of	ADP
easat-4026	235	35	the	the	DET
easat-4026	235	36	shifted	shift	VERB
easat-4026	235	37	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	235	38	𝒜	𝒜	NOUN
easat-4026	235	39	[	[	NOUN
easat-4026	235	40	1	1	NUM
easat-4026	235	41	]	]	PUNCT
easat-4026	235	42	.	.	PUNCT
easat-4026	236	1	we	we	PRON
easat-4026	236	2	may	may	AUX
easat-4026	236	3	identify	identify	VERB
easat-4026	236	4	ℋℋ∗(𝒜	ℋℋ∗(𝒜	PROPN
easat-4026	236	5	,	,	PUNCT
easat-4026	236	6	𝒮	𝒮	PROPN
easat-4026	236	7	)	)	PUNCT
easat-4026	236	8	as	as	ADP
easat-4026	236	9	a	a	DET
easat-4026	236	10	group	group	NOUN
easat-4026	236	11	,	,	PUNCT
easat-4026	236	12	with	with	ADP
easat-4026	236	13	∞	∞	PROPN
easat-4026	236	14	⊕	⊕	PROPN
easat-4026	236	15	𝑛	𝑛	NOUN
easat-4026	236	16	=	=	SYM
easat-4026	236	17	0	0	NUM
easat-4026	236	18	𝒮	𝒮	NOUN
easat-4026	236	19	⊗𝒜⊗𝑛.	⊗𝒜⊗𝑛.	NOUN
easat-4026	236	20	the	the	DET
easat-4026	236	21	degree	degree	NOUN
easat-4026	236	22	on	on	ADP
easat-4026	236	23	ℋℋ∗(𝒜	ℋℋ∗(𝒜	PROPN
easat-4026	236	24	,	,	PUNCT
easat-4026	236	25	𝒮	𝒮	PROPN
easat-4026	236	26	)	)	PUNCT
easat-4026	236	27	then	then	ADV
easat-4026	236	28	coincides	coincide	VERB
easat-4026	236	29	with	with	ADP
easat-4026	236	30	the	the	DET
easat-4026	236	31	usual	usual	ADJ
easat-4026	236	32	product	product	NOUN
easat-4026	236	33	degree	degree	NOUN
easat-4026	236	34	of	of	ADP
easat-4026	236	35	this	this	DET
easat-4026	236	36	tensor	tensor	NOUN
easat-4026	236	37	algebras	algebra	NOUN
easat-4026	236	38	.	.	PUNCT
easat-4026	237	1	next	next	ADV
easat-4026	237	2	,	,	PUNCT
easat-4026	237	3	we	we	PRON
easat-4026	237	4	describe	describe	VERB
easat-4026	237	5	how	how	SCONJ
easat-4026	237	6	to	to	PART
easat-4026	237	7	deal	deal	VERB
easat-4026	237	8	with	with	ADP
easat-4026	237	9	the	the	DET
easat-4026	237	10	relative	relative	ADJ
easat-4026	237	11	homology	homology	NOUN
easat-4026	237	12	of	of	ADP
easat-4026	237	13	algebras	algebras	PROPN
easat-4026	237	14	and	and	CCONJ
easat-4026	237	15	how	how	SCONJ
easat-4026	237	16	to	to	PART
easat-4026	237	17	use	use	VERB
easat-4026	237	18	suitable	suitable	ADJ
easat-4026	237	19	models	model	NOUN
easat-4026	237	20	for	for	ADP
easat-4026	237	21	analysis	analysis	NOUN
easat-4026	237	22	in	in	ADP
easat-4026	237	23	this	this	DET
easat-4026	237	24	context	context	NOUN
easat-4026	237	25	.	.	PUNCT
easat-4026	238	1	2.32	2.32	NUM
easat-4026	238	2	.	.	PUNCT
easat-4026	238	3	definition	definition	NOUN
easat-4026	238	4	[	[	X
easat-4026	238	5	28	28	NUM
easat-4026	238	6	]	]	PUNCT
easat-4026	238	7	there	there	PRON
easat-4026	238	8	is	be	VERB
easat-4026	238	9	a	a	DET
easat-4026	238	10	map	map	NOUN
easat-4026	238	11	for	for	ADP
easat-4026	238	12	the	the	DET
easat-4026	238	13	relative	relative	ADJ
easat-4026	238	14	homology	homology	NOUN
easat-4026	238	15	of	of	ADP
easat-4026	238	16	𝒜	𝒜	NOUN
easat-4026	238	17	modulo	modulo	VERB
easat-4026	238	18	ℐ	ℐ	PRON
easat-4026	238	19	if	if	SCONJ
easat-4026	238	20	𝒜	𝒜	NOUN
easat-4026	238	21	is	be	AUX
easat-4026	238	22	an	an	DET
easat-4026	238	23	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	238	24	and	and	CCONJ
easat-4026	238	25	ℐ	ℐ	PRON
easat-4026	238	26	is	be	AUX
easat-4026	238	27	an	an	DET
easat-4026	238	28	ideal	ideal	NOUN
easat-4026	238	29	,	,	PUNCT
easat-4026	238	30	where	where	SCONJ
easat-4026	238	31	𝒜	𝒜	NOUN
easat-4026	238	32	→	→	SYM
easat-4026	238	33	𝒜/ℐ	𝒜/ℐ	NOUN
easat-4026	238	34	is	be	AUX
easat-4026	238	35	𝒜∞-split	𝒜∞-split	VERB
easat-4026	238	36	,	,	PUNCT
easat-4026	238	37	such	such	ADJ
easat-4026	238	38	that	that	SCONJ
easat-4026	238	39	:	:	PUNCT
easat-4026	238	40	𝔷:ℋℋ𝑛(ℐ	𝔷:ℋℋ𝑛(ℐ	NOUN
easat-4026	238	41	)	)	PUNCT
easat-4026	238	42	→	→	SYM
easat-4026	238	43	ℋℋ𝑛(𝒜/ℐ	ℋℋ𝑛(𝒜/ℐ	NOUN
easat-4026	238	44	)	)	PUNCT
easat-4026	238	45	,	,	PUNCT
easat-4026	238	46	if	if	SCONJ
easat-4026	238	47	this	this	DET
easat-4026	238	48	map	map	NOUN
easat-4026	238	49	is	be	AUX
easat-4026	238	50	an	an	DET
easat-4026	238	51	isomorphism	isomorphism	NOUN
easat-4026	238	52	,	,	PUNCT
easat-4026	238	53	it	it	PRON
easat-4026	238	54	is	be	AUX
easat-4026	238	55	claimed	claim	VERB
easat-4026	238	56	that	that	SCONJ
easat-4026	238	57	the	the	DET
easat-4026	238	58	ideal	ideal	NOUN
easat-4026	238	59	ℐ	ℐ	PRON
easat-4026	238	60	is	be	AUX
easat-4026	238	61	the	the	DET
easat-4026	238	62	excision	excision	NOUN
easat-4026	238	63	of	of	ADP
easat-4026	238	64	simplicial	simplicial	ADJ
easat-4026	238	65	homology	homology	NOUN
easat-4026	238	66	.	.	PUNCT
easat-4026	239	1	this	this	PRON
easat-4026	239	2	leads	lead	VERB
easat-4026	239	3	to	to	ADP
easat-4026	239	4	the	the	DET
easat-4026	239	5	following	follow	VERB
easat-4026	239	6	exact	exact	ADJ
easat-4026	239	7	sequence	sequence	NOUN
easat-4026	239	8	:	:	PUNCT
easat-4026	239	9	…	…	PUNCT
easat-4026	239	10	→	→	SYM
easat-4026	239	11	ℋℋ𝑛(ℐ	ℋℋ𝑛(ℐ	NOUN
easat-4026	239	12	)	)	PUNCT
easat-4026	239	13	→	→	SYM
easat-4026	239	14	ℋℋ𝑛(𝒜	ℋℋ𝑛(𝒜	NOUN
easat-4026	239	15	)	)	PUNCT
easat-4026	239	16	→	→	SYM
easat-4026	239	17	ℋℋ𝑛(𝒜/ℐ	ℋℋ𝑛(𝒜/ℐ	NOUN
easat-4026	239	18	)	)	PUNCT
easat-4026	239	19	→	→	SYM
easat-4026	239	20	ℋℋ𝑛−1(ℐ	ℋℋ𝑛−1(ℐ	NOUN
easat-4026	239	21	)	)	PUNCT
easat-4026	239	22	→	→	SYM
easat-4026	239	23	ℋℋ𝑛−1(𝒜	ℋℋ𝑛−1(𝒜	NUM
easat-4026	239	24	)	)	PUNCT
easat-4026	239	25	→	→	SYM
easat-4026	239	26	…	…	PUNCT
easat-4026	239	27	finally	finally	ADV
easat-4026	239	28	,	,	PUNCT
easat-4026	239	29	the	the	DET
easat-4026	239	30	exact	exact	ADJ
easat-4026	239	31	sequence	sequence	NOUN
easat-4026	239	32	for	for	ADP
easat-4026	239	33	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	239	34	with	with	ADP
easat-4026	239	35	ideal	ideal	NOUN
easat-4026	239	36	ℐ	ℐ	PRON
easat-4026	239	37	includes	include	VERB
easat-4026	239	38	a	a	DET
easat-4026	239	39	boundary	boundary	ADJ
easat-4026	239	40	map	map	NOUN
easat-4026	239	41	𝛿	𝛿	NOUN
easat-4026	239	42	,	,	PUNCT
easat-4026	239	43	which	which	PRON
easat-4026	239	44	is	be	AUX
easat-4026	239	45	clearly	clearly	ADV
easat-4026	239	46	defined	define	VERB
easat-4026	239	47	in	in	ADP
easat-4026	239	48	terms	term	NOUN
easat-4026	239	49	of	of	ADP
easat-4026	239	50	relative	relative	ADJ
easat-4026	239	51	cycles	cycle	NOUN
easat-4026	239	52	.	.	PUNCT
easat-4026	240	1	2.33	2.33	NUM
easat-4026	240	2	.	.	PUNCT
easat-4026	240	3	proposition	proposition	NOUN
easat-4026	240	4	[	[	X
easat-4026	240	5	29	29	NUM
easat-4026	240	6	]	]	PUNCT
easat-4026	240	7	given	give	VERB
easat-4026	240	8	𝒜	𝒜	NOUN
easat-4026	240	9	be	be	AUX
easat-4026	240	10	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	240	11	and	and	CCONJ
easat-4026	240	12	ℐ-ideal	ℐ-ideal	PROPN
easat-4026	240	13	since	since	SCONJ
easat-4026	240	14	ℐ	ℐ	PROPN
easat-4026	240	15	⊂	⊂	PROPN
easat-4026	240	16	𝒜	𝒜	PROPN
easat-4026	240	17	,	,	PUNCT
easat-4026	240	18	we	we	PRON
easat-4026	240	19	have	have	VERB
easat-4026	240	20	the	the	DET
easat-4026	240	21	following	follow	VERB
easat-4026	240	22	exact	exact	ADJ
easat-4026	240	23	sequence	sequence	NOUN
easat-4026	240	24	:	:	PUNCT
easat-4026	240	25	ℋℋ𝑛(ℐ	ℋℋ𝑛(ℐ	NOUN
easat-4026	240	26	)	)	PUNCT
easat-4026	240	27	𝑖∗	𝑖∗	PROPN
easat-4026	240	28	→	→	SYM
easat-4026	240	29	ℋℋ𝑛(𝒜	ℋℋ𝑛(𝒜	NOUN
easat-4026	240	30	)	)	PUNCT
easat-4026	240	31	𝛿(−1	𝛿(−1	PROPN
easat-4026	240	32	)	)	PUNCT
easat-4026	241	1	𝑗∗	𝑗∗	PROPN
easat-4026	241	2	ℋℋ𝑛(𝒜/ℐ	ℋℋ𝑛(𝒜/ℐ	PROPN
easat-4026	241	3	)	)	PUNCT
easat-4026	241	4	actually	actually	ADV
easat-4026	241	5	,	,	PUNCT
easat-4026	241	6	in	in	ADP
easat-4026	241	7	this	this	DET
easat-4026	241	8	application	application	NOUN
easat-4026	241	9	of	of	ADP
easat-4026	241	10	relative	relative	ADJ
easat-4026	241	11	homology	homology	NOUN
easat-4026	241	12	,	,	PUNCT
easat-4026	241	13	the	the	DET
easat-4026	241	14	boundary	boundary	ADJ
easat-4026	241	15	map	map	NOUN
easat-4026	241	16	𝛿	𝛿	NOUN
easat-4026	241	17	contains	contain	VERB
easat-4026	241	18	an	an	DET
easat-4026	241	19	obvious	obvious	ADJ
easat-4026	241	20	description	description	NOUN
easat-4026	241	21	:	:	PUNCT
easat-4026	241	22	the	the	DET
easat-4026	241	23	(	(	PUNCT
easat-4026	241	24	𝑛	𝑛	PROPN
easat-4026	241	25	−	−	PROPN
easat-4026	241	26	1)-homology	1)-homology	PROPN
easat-4026	241	27	class	class	NOUN
easat-4026	241	28	provided	provide	VERB
easat-4026	241	29	by	by	ADP
easat-4026	241	30	[	[	X
easat-4026	241	31	𝛿𝜑	𝛿𝜑	X
easat-4026	241	32	]	]	X
easat-4026	241	33	∈	∈	PROPN
easat-4026	241	34	ℋ𝑛−1(𝒜	ℋ𝑛−1(𝒜	PROPN
easat-4026	241	35	)	)	PUNCT
easat-4026	241	36	is	be	AUX
easat-4026	241	37	𝛿[𝜑	𝛿[𝜑	NOUN
easat-4026	241	38	]	]	PUNCT
easat-4026	241	39	if	if	SCONJ
easat-4026	241	40	𝜑	𝜑	PROPN
easat-4026	241	41	∈	∈	PROPN
easat-4026	241	42	𝒞𝑛(𝒴,𝒜	𝒞𝑛(𝒴,𝒜	NOUN
easat-4026	241	43	)	)	PUNCT
easat-4026	241	44	denotes	denote	VERB
easat-4026	241	45	a	a	DET
easat-4026	241	46	relative	relative	ADJ
easat-4026	241	47	cycle	cycle	NOUN
easat-4026	241	48	.	.	PUNCT
easat-4026	242	1	3	3	X
easat-4026	242	2	.	.	X
easat-4026	242	3	main	main	ADJ
easat-4026	242	4	result	result	NOUN
easat-4026	242	5	this	this	DET
easat-4026	242	6	text	text	NOUN
easat-4026	242	7	explores	explore	VERB
easat-4026	242	8	essential	essential	ADJ
easat-4026	242	9	topics	topic	NOUN
easat-4026	242	10	in	in	ADP
easat-4026	242	11	algebraic	algebraic	ADJ
easat-4026	242	12	topology	topology	NOUN
easat-4026	242	13	and	and	CCONJ
easat-4026	242	14	homology	homology	NOUN
easat-4026	242	15	related	relate	VERB
easat-4026	242	16	to	to	ADP
easat-4026	242	17	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	242	18	.	.	PUNCT
easat-4026	243	1	it	it	PRON
easat-4026	243	2	covers	cover	VERB
easat-4026	243	3	how	how	SCONJ
easat-4026	243	4	isomorphisms	isomorphism	NOUN
easat-4026	243	5	between	between	ADP
easat-4026	243	6	homology	homology	NOUN
easat-4026	243	7	groups	group	NOUN
easat-4026	243	8	are	be	AUX
easat-4026	243	9	preserved	preserve	VERB
easat-4026	243	10	under	under	ADP
easat-4026	243	11	specific	specific	ADJ
easat-4026	243	12	conditions	condition	NOUN
easat-4026	243	13	,	,	PUNCT
easat-4026	243	14	defines	define	VERB
easat-4026	243	15	simplicial	simplicial	NOUN
easat-4026	243	16	and	and	CCONJ
easat-4026	243	17	bar	bar	NOUN
easat-4026	243	18	homology	homology	NOUN
easat-4026	243	19	with	with	ADP
easat-4026	243	20	module	module	NOUN
easat-4026	243	21	coefficients	coefficient	NOUN
easat-4026	243	22	,	,	PUNCT
easat-4026	243	23	and	and	CCONJ
easat-4026	243	24	discusses	discuss	VERB
easat-4026	243	25	ℋ-unitarity	ℋ-unitarity	PROPN
easat-4026	243	26	.	.	PUNCT
easat-4026	244	1	the	the	DET
easat-4026	244	2	discussion	discussion	NOUN
easat-4026	244	3	also	also	ADV
easat-4026	244	4	includes	include	VERB
easat-4026	244	5	the	the	DET
easat-4026	244	6	relationships	relationship	NOUN
easat-4026	244	7	between	between	ADP
easat-4026	244	8	different	different	ADJ
easat-4026	244	9	homological	homological	ADJ
easat-4026	244	10	constructs	construct	NOUN
easat-4026	244	11	and	and	CCONJ
easat-4026	244	12	the	the	DET
easat-4026	244	13	conditions	condition	NOUN
easat-4026	244	14	required	require	VERB
easat-4026	244	15	for	for	ADP
easat-4026	244	16	quasi	quasi	NOUN
easat-4026	244	17	-	-	NOUN
easat-4026	244	18	isomorphisms	isomorphisms	X
easat-4026	244	19	,	,	PUNCT
easat-4026	244	20	providing	provide	VERB
easat-4026	244	21	a	a	DET
easat-4026	244	22	comprehensive	comprehensive	ADJ
easat-4026	244	23	view	view	NOUN
easat-4026	244	24	of	of	ADP
easat-4026	244	25	these	these	DET
easat-4026	244	26	concepts	concept	NOUN
easat-4026	244	27	and	and	CCONJ
easat-4026	244	28	their	their	PRON
easat-4026	244	29	interconnections	interconnection	NOUN
easat-4026	244	30	.	.	PUNCT
easat-4026	245	1	the	the	DET
easat-4026	245	2	frist	frist	PROPN
easat-4026	245	3	theorem	theorem	PROPN
easat-4026	245	4	discusses	discuss	VERB
easat-4026	245	5	the	the	DET
easat-4026	245	6	excision	excision	NOUN
easat-4026	245	7	theorem	theorem	VERB
easat-4026	245	8	for	for	ADP
easat-4026	245	9	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	245	10	,	,	PUNCT
easat-4026	245	11	demonstrating	demonstrate	VERB
easat-4026	245	12	how	how	SCONJ
easat-4026	245	13	isomorphisms	isomorphism	NOUN
easat-4026	245	14	between	between	ADP
easat-4026	245	15	homology	homology	NOUN
easat-4026	245	16	groups	group	NOUN
easat-4026	245	17	persist	persist	VERB
easat-4026	245	18	under	under	ADP
easat-4026	245	19	specific	specific	ADJ
easat-4026	245	20	conditions	condition	NOUN
easat-4026	245	21	and	and	CCONJ
easat-4026	245	22	the	the	DET
easat-4026	245	23	inclusion	inclusion	NOUN
easat-4026	245	24	maps	map	NOUN
easat-4026	245	25	.	.	PUNCT
easat-4026	246	1	it	it	PRON
easat-4026	246	2	includes	include	VERB
easat-4026	246	3	proof	proof	ADJ
easat-4026	246	4	strategies	strategy	NOUN
easat-4026	246	5	involving	involve	VERB
easat-4026	246	6	chain	chain	NOUN
easat-4026	246	7	complexes	complex	NOUN
easat-4026	246	8	and	and	CCONJ
easat-4026	246	9	homotopy	homotopy	NOUN
easat-4026	246	10	equivalence	equivalence	NOUN
easat-4026	246	11	.	.	PUNCT
easat-4026	247	1	3.1	3.1	NUM
easat-4026	247	2	.	.	PUNCT
easat-4026	247	3	theorem	theorem	PROPN
easat-4026	247	4	(	(	PUNCT
easat-4026	247	5	excision	excision	NOUN
easat-4026	247	6	theory	theory	NOUN
easat-4026	247	7	)	)	PUNCT
easat-4026	247	8	suppose	suppose	VERB
easat-4026	247	9	that	that	SCONJ
easat-4026	247	10	ℰ	ℰ	PROPN
easat-4026	247	11	is	be	AUX
easat-4026	247	12	a	a	DET
easat-4026	247	13	subset	subset	NOUN
easat-4026	247	14	of	of	ADP
easat-4026	247	15	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	247	16	such	such	ADJ
easat-4026	247	17	that	that	SCONJ
easat-4026	248	1	ℰ	ℰ	PROPN
easat-4026	248	2	⊂	⊂	PROPN
easat-4026	248	3	𝒜	𝒜	NOUN
easat-4026	248	4	⊂	⊂	PROPN
easat-4026	248	5	𝒳.	𝒳.	PROPN
easat-4026	248	6	then	then	ADV
easat-4026	248	7	,	,	PUNCT
easat-4026	248	8	for	for	SCONJ
easat-4026	248	9	all	all	DET
easat-4026	248	10	𝑛	𝑛	PRON
easat-4026	248	11	the	the	DET
easat-4026	248	12	isomorphisms	isomorphisms	PROPN
easat-4026	248	13	ℋ𝑛(𝒳\ℰ,𝒜\ℰ	ℋ𝑛(𝒳\ℰ,𝒜\ℰ	VERB
easat-4026	248	14	)	)	PUNCT
easat-4026	249	1	→	→	SYM
easat-4026	249	2	ℋ𝑛(𝒳,𝒜	ℋ𝑛(𝒳,𝒜	X
easat-4026	249	3	)	)	PUNCT
easat-4026	249	4	that	that	SCONJ
easat-4026	249	5	given	give	VERB
easat-4026	249	6	by	by	ADP
easat-4026	249	7	the	the	DET
easat-4026	249	8	inclusion	inclusion	NOUN
easat-4026	249	9	(	(	PUNCT
easat-4026	249	10	𝒳\ℰ,𝒜\ℰ	𝒳\ℰ,𝒜\ℰ	NOUN
easat-4026	249	11	)	)	PUNCT
easat-4026	249	12	↪	↪	PROPN
easat-4026	249	13	(	(	PUNCT
easat-4026	249	14	𝒳,𝒜	𝒳,𝒜	NOUN
easat-4026	249	15	)	)	PUNCT
easat-4026	249	16	.	.	PUNCT
easat-4026	250	1	if	if	SCONJ
easat-4026	250	2	𝒳	𝒳	PROPN
easat-4026	250	3	is	be	AUX
easat-4026	250	4	covered	cover	VERB
easat-4026	250	5	by	by	ADP
easat-4026	250	6	the	the	DET
easat-4026	250	7	interiors	interior	NOUN
easat-4026	250	8	of	of	ADP
easat-4026	250	9	the	the	DET
easat-4026	250	10	spaces	space	NOUN
easat-4026	250	11	𝒜	𝒜	NOUN
easat-4026	250	12	,	,	PUNCT
easat-4026	250	13	ℬ	ℬ	NOUN
easat-4026	250	14	such	such	ADJ
easat-4026	250	15	that	that	DET
easat-4026	250	16	𝒜	𝒜	NOUN
easat-4026	250	17	,	,	PUNCT
easat-4026	250	18	ℬ	ℬ	PROPN
easat-4026	250	19	⊂	⊂	PROPN
easat-4026	250	20	𝒳	𝒳	PROPN
easat-4026	250	21	,	,	PUNCT
easat-4026	250	22	then	then	ADV
easat-4026	250	23	the	the	DET
easat-4026	250	24	inclusion	inclusion	NOUN
easat-4026	250	25	(	(	PUNCT
easat-4026	250	26	ℬ,𝒜	ℬ,𝒜	NOUN
easat-4026	250	27	∩	∩	ADJ
easat-4026	250	28	ℬ	ℬ	NOUN
easat-4026	250	29	)	)	PUNCT
easat-4026	250	30	↪	↪	PROPN
easat-4026	250	31	(	(	PUNCT
easat-4026	250	32	𝒳,𝒜	𝒳,𝒜	NOUN
easat-4026	250	33	)	)	PUNCT
easat-4026	250	34	is	be	AUX
easat-4026	250	35	the	the	DET
easat-4026	250	36	equivalent	equivalent	ADJ
easat-4026	250	37	statement	statement	NOUN
easat-4026	250	38	that	that	PRON
easat-4026	250	39	persuades	persuade	VERB
easat-4026	250	40	the	the	DET
easat-4026	250	41	isomorphisms	isomorphisms	PROPN
easat-4026	250	42	ℋ𝑛(ℬ,𝒜	ℋ𝑛(ℬ,𝒜	PROPN
easat-4026	250	43	∩	∩	ADJ
easat-4026	250	44	ℬ	ℬ	NOUN
easat-4026	250	45	)	)	PUNCT
easat-4026	250	46	→	→	SYM
easat-4026	250	47	ℋ𝑛(𝒳,𝒜	ℋ𝑛(𝒳,𝒜	NOUN
easat-4026	250	48	)	)	PUNCT
easat-4026	250	49	for	for	ADP
easat-4026	250	50	all	all	DET
easat-4026	250	51	𝑛	𝑛	NOUN
easat-4026	250	52	,	,	PUNCT
easat-4026	250	53	where	where	SCONJ
easat-4026	250	54	the	the	DET
easat-4026	250	55	space	space	NOUN
easat-4026	250	56	ℬ	ℬ	NOUN
easat-4026	250	57	is	be	AUX
easat-4026	250	58	given	give	VERB
easat-4026	250	59	by	by	ADP
easat-4026	250	60	ℬ	ℬ	PROPN
easat-4026	250	61	=	=	SYM
easat-4026	250	62	𝒳\ℰ.	𝒳\ℰ.	PROPN
easat-4026	250	63	proof	proof	NOUN
easat-4026	250	64	:	:	PUNCT
easat-4026	250	65	by	by	ADP
easat-4026	250	66	using	use	VERB
easat-4026	250	67	[	[	X
easat-4026	250	68	30	30	NUM
easat-4026	250	69	]	]	PUNCT
easat-4026	250	70	,	,	PUNCT
easat-4026	250	71	[	[	X
easat-4026	250	72	31	31	NUM
easat-4026	250	73	]	]	PUNCT
easat-4026	250	74	and	and	CCONJ
easat-4026	250	75	consider	consider	VERB
easat-4026	250	76	𝒳	𝒳	PROPN
easat-4026	250	77	as	as	ADP
easat-4026	250	78	the	the	DET
easat-4026	250	79	combination	combination	NOUN
easat-4026	250	80	of	of	ADP
easat-4026	250	81	𝒜	𝒜	NOUN
easat-4026	250	82	and	and	CCONJ
easat-4026	250	83	ℬ	ℬ	NOUN
easat-4026	250	84	through	through	ADP
easat-4026	250	85	interiors	interior	NOUN
easat-4026	250	86	covering	cover	VERB
easat-4026	250	87	𝒳.	𝒳.	PROPN
easat-4026	250	88	next	next	ADV
easat-4026	250	89	,	,	PUNCT
easat-4026	250	90	there	there	PRON
easat-4026	250	91	are	be	VERB
easat-4026	250	92	maps	map	NOUN
easat-4026	250	93	of	of	ADP
easat-4026	250	94	natural	natural	ADJ
easat-4026	250	95	inclusion	inclusion	NOUN
easat-4026	250	96	.	.	PUNCT
easat-4026	251	1	9482	9482	NUM
easat-4026	251	2	edelweiss	edelweiss	PROPN
easat-4026	251	3	applied	apply	VERB
easat-4026	251	4	science	science	NOUN
easat-4026	251	5	and	and	CCONJ
easat-4026	251	6	technology	technology	NOUN
easat-4026	251	7	issn	issn	PROPN
easat-4026	251	8	:	:	PUNCT
easat-4026	251	9	2576	2576	NUM
easat-4026	251	10	-	-	SYM
easat-4026	251	11	8484	8484	NUM
easat-4026	251	12	vol	vol	NOUN
easat-4026	251	13	.	.	PROPN
easat-4026	251	14	8	8	NUM
easat-4026	251	15	,	,	PUNCT
easat-4026	251	16	no	no	INTJ
easat-4026	251	17	.	.	NOUN
easat-4026	252	1	6	6	NUM
easat-4026	252	2	:	:	SYM
easat-4026	252	3	9472	9472	NUM
easat-4026	252	4	-	-	SYM
easat-4026	252	5	9486	9486	NUM
easat-4026	252	6	,	,	PUNCT
easat-4026	252	7	2024	2024	NUM
easat-4026	252	8	doi	doi	NOUN
easat-4026	252	9	:	:	PUNCT
easat-4026	252	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	252	11	©	©	PROPN
easat-4026	252	12	2024	2024	NUM
easat-4026	252	13	by	by	ADP
easat-4026	252	14	the	the	DET
easat-4026	252	15	authors	author	NOUN
easat-4026	252	16	;	;	PUNCT
easat-4026	252	17	licensee	licensee	PROPN
easat-4026	252	18	learning	learn	VERB
easat-4026	252	19	gate	gate	PROPN
easat-4026	252	20	𝒞•(𝒳	𝒞•(𝒳	PROPN
easat-4026	252	21	)	)	PUNCT
easat-4026	252	22	𝚤	𝚤	PROPN
easat-4026	252	23	↑	↑	PROPN
easat-4026	252	24	𝒞•(𝒜	𝒞•(𝒜	PROPN
easat-4026	252	25	)	)	PUNCT
easat-4026	252	26	+	+	SYM
easat-4026	252	27	𝒞•(ℬ	𝒞•(ℬ	ADJ
easat-4026	252	28	)	)	PUNCT
easat-4026	252	29	𝒞•(𝒜	𝒞•(𝒜	NOUN
easat-4026	252	30	)	)	PUNCT
easat-4026	252	31	𝒞•(ℬ	𝒞•(ℬ	NOUN
easat-4026	252	32	)	)	PUNCT
easat-4026	252	33	𝒞•(𝒜	𝒞•(𝒜	NOUN
easat-4026	252	34	∩	∩	ADJ
easat-4026	252	35	ℬ	ℬ	NOUN
easat-4026	252	36	)	)	PUNCT
easat-4026	252	37	we	we	PRON
easat-4026	252	38	would	would	AUX
easat-4026	252	39	obtain	obtain	VERB
easat-4026	252	40	:	:	PUNCT
easat-4026	252	41	𝒞•(𝒳)/𝒞•(𝒜	𝒞•(𝒳)/𝒞•(𝒜	X
easat-4026	252	42	)	)	PUNCT
easat-4026	252	43	=	=	SYM
easat-4026	252	44	𝒞•(ℬ)/𝒞•(𝒜	𝒞•(ℬ)/𝒞•(𝒜	NOUN
easat-4026	252	45	∩	∩	ADJ
easat-4026	252	46	ℬ	ℬ	NOUN
easat-4026	252	47	)	)	PUNCT
easat-4026	252	48	,	,	PUNCT
easat-4026	252	49	where	where	SCONJ
easat-4026	252	50	the	the	DET
easat-4026	252	51	map	map	NOUN
easat-4026	252	52	𝜄	𝜄	PROPN
easat-4026	252	53	remains	remain	VERB
easat-4026	252	54	an	an	DET
easat-4026	252	55	isomorphism	isomorphism	NOUN
easat-4026	252	56	,	,	PUNCT
easat-4026	252	57	which	which	PRON
easat-4026	252	58	leads	lead	VERB
easat-4026	252	59	to	to	ADP
easat-4026	252	60	the	the	DET
easat-4026	252	61	desired	desire	VERB
easat-4026	252	62	result	result	NOUN
easat-4026	252	63	.	.	PUNCT
easat-4026	253	1	however	however	ADV
easat-4026	253	2	,	,	PUNCT
easat-4026	253	3	there	there	PRON
easat-4026	253	4	are	be	VERB
easat-4026	253	5	terrible	terrible	ADJ
easat-4026	253	6	simplices	simplice	NOUN
easat-4026	253	7	that	that	PRON
easat-4026	253	8	can	can	AUX
easat-4026	253	9	have	have	VERB
easat-4026	253	10	non	non	ADJ
easat-4026	253	11	-	-	ADJ
easat-4026	253	12	empty	empty	ADJ
easat-4026	253	13	intersections	intersection	NOUN
easat-4026	253	14	with	with	ADP
easat-4026	253	15	(	(	PUNCT
easat-4026	253	16	𝒜	𝒜	NOUN
easat-4026	253	17	−𝒜	−𝒜	NUM
easat-4026	253	18	∩	∩	ADJ
easat-4026	253	19	ℬ	ℬ	NOUN
easat-4026	253	20	)	)	PUNCT
easat-4026	253	21	and	and	CCONJ
easat-4026	253	22	(	(	PUNCT
easat-4026	253	23	ℬ	ℬ	PROPN
easat-4026	253	24	−𝒜	−𝒜	PROPN
easat-4026	253	25	∩	∩	ADJ
easat-4026	253	26	ℬ	ℬ	NOUN
easat-4026	253	27	)	)	PUNCT
easat-4026	253	28	;	;	PUNCT
easat-4026	253	29	therefore	therefore	ADV
easat-4026	253	30	,	,	PUNCT
easat-4026	253	31	the	the	DET
easat-4026	253	32	fact	fact	NOUN
easat-4026	253	33	that	that	SCONJ
easat-4026	253	34	the	the	DET
easat-4026	253	35	map	map	NOUN
easat-4026	253	36	𝜄	𝜄	NOUN
easat-4026	253	37	is	be	AUX
easat-4026	253	38	not	not	PART
easat-4026	253	39	an	an	DET
easat-4026	253	40	isomorphism	isomorphism	NOUN
easat-4026	253	41	is	be	AUX
easat-4026	253	42	an	an	DET
easat-4026	253	43	issue	issue	NOUN
easat-4026	253	44	.	.	PUNCT
easat-4026	254	1	assisted	assist	VERB
easat-4026	254	2	by	by	ADP
easat-4026	254	3	the	the	DET
easat-4026	254	4	chain	chain	NOUN
easat-4026	254	5	map	map	NOUN
easat-4026	254	6	𝜉	𝜉	NOUN
easat-4026	254	7	:	:	PUNCT
easat-4026	254	8	𝒞•(𝒳	𝒞•(𝒳	X
easat-4026	254	9	)	)	PUNCT
easat-4026	254	10	→	→	SYM
easat-4026	254	11	𝒞•(𝒜	𝒞•(𝒜	NOUN
easat-4026	254	12	)	)	PUNCT
easat-4026	254	13	+	+	SYM
easat-4026	254	14	𝒞•(ℬ	𝒞•(ℬ	NOUN
easat-4026	254	15	)	)	PUNCT
easat-4026	254	16	,	,	PUNCT
easat-4026	254	17	we	we	PRON
easat-4026	254	18	would	would	AUX
easat-4026	254	19	desire	desire	VERB
easat-4026	254	20	to	to	PART
easat-4026	254	21	demonstrate	demonstrate	VERB
easat-4026	254	22	how	how	SCONJ
easat-4026	254	23	to	to	PART
easat-4026	254	24	split	split	VERB
easat-4026	254	25	up	up	ADP
easat-4026	254	26	terrible	terrible	ADJ
easat-4026	254	27	simplices	simplice	NOUN
easat-4026	254	28	into	into	ADP
easat-4026	254	29	little	little	ADJ
easat-4026	254	30	good	good	ADJ
easat-4026	254	31	ones	one	NOUN
easat-4026	254	32	without	without	ADP
easat-4026	254	33	changing	change	VERB
easat-4026	254	34	the	the	DET
easat-4026	254	35	homology	homology	NOUN
easat-4026	254	36	.	.	PUNCT
easat-4026	255	1	we	we	PRON
easat-4026	255	2	demonstrate	demonstrate	VERB
easat-4026	255	3	that	that	SCONJ
easat-4026	255	4	𝒞•(𝒜	𝒞•(𝒜	NOUN
easat-4026	255	5	)	)	PUNCT
easat-4026	255	6	+	+	SYM
easat-4026	255	7	𝒞•(ℬ	𝒞•(ℬ	NOUN
easat-4026	255	8	)	)	PUNCT
easat-4026	255	9	be	be	VERB
easat-4026	255	10	the	the	DET
easat-4026	255	11	distorted	distorted	ADJ
easat-4026	255	12	retracting	retracting	NOUN
easat-4026	255	13	of	of	ADP
easat-4026	255	14	𝒞•(𝒳	𝒞•(𝒳	NOUN
easat-4026	255	15	)	)	PUNCT
easat-4026	255	16	,	,	PUNCT
easat-4026	255	17	indicating	indicate	VERB
easat-4026	255	18	that	that	SCONJ
easat-4026	255	19	𝜉	𝜉	ADP
easat-4026	255	20	∘	∘	NOUN
easat-4026	255	21	𝚤	𝚤	NOUN
easat-4026	255	22	=	=	SYM
easat-4026	255	23	𝐼𝑑	𝐼𝑑	PROPN
easat-4026	255	24	and	and	CCONJ
easat-4026	255	25	𝚤	𝚤	ADP
easat-4026	255	26	∘	∘	NOUN
easat-4026	255	27	𝜉	𝜉	X
easat-4026	255	28	=	=	PUNCT
easat-4026	255	29	𝒹𝔇+𝔇𝒹	𝒹𝔇+𝔇𝒹	PROPN
easat-4026	255	30	for	for	ADP
easat-4026	255	31	particular	particular	ADJ
easat-4026	255	32	chain	chain	NOUN
easat-4026	255	33	homotopy	homotopy	NOUN
easat-4026	255	34	𝔇.	𝔇.	PROPN
easat-4026	255	35	in	in	ADP
easat-4026	255	36	order	order	NOUN
easat-4026	255	37	to	to	PART
easat-4026	255	38	retain	retain	VERB
easat-4026	255	39	the	the	DET
easat-4026	255	40	sub	sub	ADJ
easat-4026	255	41	-	-	ADJ
easat-4026	255	42	complexes	complex	NOUN
easat-4026	255	43	𝒞•(𝒜	𝒞•(𝒜	NOUN
easat-4026	255	44	)	)	PUNCT
easat-4026	255	45	and	and	CCONJ
easat-4026	255	46	𝒞•(ℬ	𝒞•(ℬ	PROPN
easat-4026	255	47	)	)	PUNCT
easat-4026	255	48	,	,	PUNCT
easat-4026	255	49	we	we	PRON
easat-4026	255	50	choose	choose	VERB
easat-4026	255	51	𝔇	𝔇	PROPN
easat-4026	255	52	,	,	PUNCT
easat-4026	255	53	indicating	indicate	VERB
easat-4026	255	54	that	that	SCONJ
easat-4026	255	55	we	we	PRON
easat-4026	255	56	achieve	achieve	VERB
easat-4026	255	57	the	the	DET
easat-4026	255	58	equivalence	equivalence	NOUN
easat-4026	255	59	:	:	PUNCT
easat-4026	255	60	𝒞•(𝒳)/𝒞•(𝒜	𝒞•(𝒳)/𝒞•(𝒜	NOUN
easat-4026	255	61	)	)	PUNCT
easat-4026	255	62	→	→	SYM
easat-4026	255	63	𝒞•(ℬ)/𝒞•(𝒜	𝒞•(ℬ)/𝒞•(𝒜	NOUN
easat-4026	255	64	∩	∩	ADJ
easat-4026	255	65	ℬ	ℬ	NOUN
easat-4026	255	66	)	)	PUNCT
easat-4026	255	67	as	as	ADP
easat-4026	255	68	chain	chain	NOUN
easat-4026	255	69	homotopy	homotopy	NOUN
easat-4026	255	70	equivalence	equivalence	NOUN
easat-4026	255	71	.	.	PUNCT
easat-4026	256	1	next	next	ADV
easat-4026	256	2	,	,	PUNCT
easat-4026	256	3	we	we	PRON
easat-4026	256	4	define	define	VERB
easat-4026	256	5	simplicial	simplicial	ADJ
easat-4026	256	6	homology	homology	NOUN
easat-4026	256	7	isomorphisms	isomorphism	NOUN
easat-4026	256	8	in	in	ADP
easat-4026	256	9	the	the	DET
easat-4026	256	10	context	context	NOUN
easat-4026	256	11	of	of	ADP
easat-4026	256	12	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	256	13	,	,	PUNCT
easat-4026	256	14	emphasizing	emphasize	VERB
easat-4026	256	15	the	the	DET
easat-4026	256	16	behavior	behavior	NOUN
easat-4026	256	17	of	of	ADP
easat-4026	256	18	these	these	DET
easat-4026	256	19	isomorphisms	isomorphism	NOUN
easat-4026	256	20	under	under	ADP
easat-4026	256	21	inclusions	inclusion	NOUN
easat-4026	256	22	of	of	ADP
easat-4026	256	23	subspaces	subspace	NOUN
easat-4026	256	24	.	.	PUNCT
easat-4026	257	1	3.2	3.2	NUM
easat-4026	257	2	.	.	PUNCT
easat-4026	257	3	definition	definition	NOUN
easat-4026	257	4	for	for	ADP
easat-4026	257	5	the	the	DET
easat-4026	257	6	space	space	NOUN
easat-4026	257	7	𝒳	𝒳	PROPN
easat-4026	257	8	and	and	CCONJ
easat-4026	257	9	ℰ	ℰ	PROPN
easat-4026	257	10	⊂	⊂	PROPN
easat-4026	257	11	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	257	12	where	where	SCONJ
easat-4026	257	13	ℰ	ℰ	PROPN
easat-4026	257	14	⊂	⊂	NOUN
easat-4026	257	15	𝒜	𝒜	PROPN
easat-4026	257	16	⊂	⊂	PROPN
easat-4026	257	17	𝒳	𝒳	PROPN
easat-4026	257	18	,	,	PUNCT
easat-4026	257	19	the	the	DET
easat-4026	257	20	simplicial	simplicial	ADJ
easat-4026	257	21	homology	homology	NOUN
easat-4026	257	22	isomorphisms	isomorphism	NOUN
easat-4026	257	23	induced	induce	VERB
easat-4026	257	24	by	by	ADP
easat-4026	257	25	the	the	DET
easat-4026	257	26	inclusion	inclusion	NOUN
easat-4026	257	27	(	(	PUNCT
easat-4026	257	28	𝒳\ℰ,𝒜\ℰ	𝒳\ℰ,𝒜\ℰ	NOUN
easat-4026	257	29	)	)	PUNCT
easat-4026	257	30	↪	↪	PROPN
easat-4026	257	31	(	(	PUNCT
easat-4026	257	32	𝒳,𝒜	𝒳,𝒜	NOUN
easat-4026	257	33	)	)	PUNCT
easat-4026	257	34	for	for	ADP
easat-4026	257	35	all	all	DET
easat-4026	257	36	𝑛	𝑛	PRON
easat-4026	257	37	is	be	AUX
easat-4026	257	38	:	:	PUNCT
easat-4026	257	39	ℋℋ𝑛(𝒳\ℰ,𝒜\ℰ	ℋℋ𝑛(𝒳\ℰ,𝒜\ℰ	ADJ
easat-4026	257	40	)	)	PUNCT
easat-4026	258	1	→	→	PUNCT
easat-4026	258	2	ℋℋ𝑛(𝒳,𝒜	ℋℋ𝑛(𝒳,𝒜	PROPN
easat-4026	258	3	)	)	PUNCT
easat-4026	258	4	.	.	PUNCT
easat-4026	259	1	by	by	ADP
easat-4026	259	2	setting	set	VERB
easat-4026	259	3	the	the	DET
easat-4026	259	4	space	space	NOUN
easat-4026	259	5	ℬ	ℬ	NOUN
easat-4026	259	6	=	=	SYM
easat-4026	259	7	𝒳\ℰ	𝒳\ℰ	NOUN
easat-4026	259	8	,	,	PUNCT
easat-4026	259	9	let	let	VERB
easat-4026	259	10	𝒳	𝒳	PRON
easat-4026	259	11	covered	cover	VERB
easat-4026	259	12	by	by	ADP
easat-4026	259	13	the	the	DET
easat-4026	259	14	interiors	interior	NOUN
easat-4026	259	15	of	of	ADP
easat-4026	259	16	the	the	DET
easat-4026	259	17	spaces	space	NOUN
easat-4026	259	18	𝒜,ℬ	𝒜,ℬ	VERB
easat-4026	259	19	for	for	ADP
easat-4026	259	20	𝒜	𝒜	NOUN
easat-4026	259	21	,	,	PUNCT
easat-4026	259	22	ℬ	ℬ	PROPN
easat-4026	259	23	⊂	⊂	PROPN
easat-4026	259	24	𝒳	𝒳	PROPN
easat-4026	259	25	,	,	PUNCT
easat-4026	259	26	then	then	ADV
easat-4026	259	27	the	the	DET
easat-4026	259	28	equivalent	equivalent	ADJ
easat-4026	259	29	statement	statement	NOUN
easat-4026	259	30	is	be	AUX
easat-4026	259	31	that	that	SCONJ
easat-4026	259	32	the	the	DET
easat-4026	259	33	following	follow	VERB
easat-4026	259	34	isomorphisms	isomorphism	NOUN
easat-4026	259	35	persuaded	persuade	VERB
easat-4026	259	36	by	by	ADP
easat-4026	259	37	the	the	DET
easat-4026	259	38	inclusion	inclusion	NOUN
easat-4026	259	39	(	(	PUNCT
easat-4026	259	40	ℬ,𝒜	ℬ,𝒜	NOUN
easat-4026	259	41	∩	∩	ADJ
easat-4026	259	42	ℬ	ℬ	NOUN
easat-4026	259	43	)	)	PUNCT
easat-4026	259	44	↪	↪	PROPN
easat-4026	259	45	(	(	PUNCT
easat-4026	259	46	𝒳,𝒜	𝒳,𝒜	NOUN
easat-4026	259	47	):	):	PUNCT
easat-4026	259	48	ℋℋ𝑛(ℬ,𝒜	ℋℋ𝑛(ℬ,𝒜	NOUN
easat-4026	259	49	∩	∩	ADJ
easat-4026	259	50	ℬ	ℬ	NOUN
easat-4026	259	51	)	)	PUNCT
easat-4026	259	52	→	→	SYM
easat-4026	259	53	ℋℋ𝑛(𝒳,𝒜	ℋℋ𝑛(𝒳,𝒜	NUM
easat-4026	259	54	)	)	PUNCT
easat-4026	259	55	∀𝑛.	∀𝑛.	PROPN
easat-4026	259	56	(	(	PUNCT
easat-4026	259	57	7	7	X
easat-4026	259	58	)	)	PUNCT
easat-4026	259	59	now	now	ADV
easat-4026	259	60	,	,	PUNCT
easat-4026	259	61	we	we	PRON
easat-4026	259	62	introduce	introduce	VERB
easat-4026	259	63	the	the	DET
easat-4026	259	64	bar	bar	NOUN
easat-4026	259	65	homology	homology	NOUN
easat-4026	259	66	of	of	ADP
easat-4026	259	67	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	259	68	with	with	ADP
easat-4026	259	69	coefficients	coefficient	NOUN
easat-4026	259	70	in	in	ADP
easat-4026	259	71	a	a	DET
easat-4026	259	72	module	module	NOUN
easat-4026	259	73	detailing	detail	VERB
easat-4026	259	74	the	the	DET
easat-4026	259	75	associated	associated	ADJ
easat-4026	259	76	boundary	boundary	ADJ
easat-4026	259	77	maps	map	NOUN
easat-4026	259	78	.	.	PUNCT
easat-4026	260	1	3.3	3.3	NUM
easat-4026	260	2	.	.	PUNCT
easat-4026	261	1	definition	definition	NOUN
easat-4026	261	2	assuming	assume	VERB
easat-4026	261	3	that	that	SCONJ
easat-4026	261	4	ℐ	ℐ	PRON
easat-4026	261	5	is	be	AUX
easat-4026	261	6	an	an	DET
easat-4026	261	7	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	261	8	,	,	PUNCT
easat-4026	261	9	which	which	PRON
easat-4026	261	10	is	be	AUX
easat-4026	261	11	not	not	PART
easat-4026	261	12	necessarily	necessarily	ADV
easat-4026	261	13	unitary	unitary	ADJ
easat-4026	261	14	and	and	CCONJ
easat-4026	261	15	make	make	VERB
easat-4026	261	16	ℛ	ℛ	PRON
easat-4026	261	17	a	a	DET
easat-4026	261	18	right	right	ADJ
easat-4026	261	19	ℐ-module	ℐ-module	PROPN
easat-4026	261	20	.	.	PUNCT
easat-4026	262	1	then	then	ADV
easat-4026	262	2	,	,	PUNCT
easat-4026	262	3	the	the	DET
easat-4026	262	4	complexes	complex	NOUN
easat-4026	262	5	'	'	PART
easat-4026	262	6	homology	homology	NOUN
easat-4026	262	7	ℋ𝐵∗	ℋ𝐵∗	PROPN
easat-4026	262	8	′(ℐ	′(ℐ	NOUN
easat-4026	262	9	,	,	PUNCT
easat-4026	262	10	ℛ	ℛ	PROPN
easat-4026	262	11	)	)	PUNCT
easat-4026	262	12	is	be	AUX
easat-4026	262	13	the	the	DET
easat-4026	262	14	bar	bar	NOUN
easat-4026	262	15	homology	homology	NOUN
easat-4026	262	16	of	of	ADP
easat-4026	262	17	ℐ	ℐ	PRON
easat-4026	262	18	via	via	ADP
easat-4026	262	19	coefficients	coefficient	NOUN
easat-4026	262	20	in	in	ADP
easat-4026	262	21	ℛ	ℛ	NOUN
easat-4026	262	22	:	:	PUNCT
easat-4026	262	23	(	(	PUNCT
easat-4026	262	24	ℛ	ℛ	PROPN
easat-4026	262	25	⊗	⊗	PROPN
easat-4026	262	26	ℐ⨂∗	ℐ⨂∗	NOUN
easat-4026	262	27	,	,	PUNCT
easat-4026	262	28	𝜌∗	𝜌∗	NOUN
easat-4026	262	29	′):=	′):=	PROPN
easat-4026	262	30	ℛ	ℛ	NOUN
easat-4026	262	31	𝜌1	𝜌1	NOUN
easat-4026	262	32	′	′	NUM
easat-4026	263	1	←	←	PROPN
easat-4026	263	2	ℛ⊗	ℛ⊗	PROPN
easat-4026	263	3	ℐ	ℐ	PROPN
easat-4026	263	4	𝜌2	𝜌2	ADJ
easat-4026	263	5	′	′	NUM
easat-4026	264	1	←	←	PROPN
easat-4026	264	2	ℛ	ℛ	PROPN
easat-4026	264	3	⊗	⊗	PROPN
easat-4026	264	4	ℐ	ℐ	PROPN
easat-4026	264	5	⊗	⊗	PROPN
easat-4026	264	6	ℐ	ℐ	PROPN
easat-4026	264	7	𝜌3	𝜌3	VERB
easat-4026	264	8	′	′	NUM
easat-4026	264	9	←	←	PROPN
easat-4026	265	1	ℛ⊗	ℛ⊗	PROPN
easat-4026	266	1	ℐ	ℐ	PROPN
easat-4026	266	2	⊗	⊗	PROPN
easat-4026	266	3	ℐ	ℐ	PROPN
easat-4026	266	4	⊗	⊗	PROPN
easat-4026	266	5	ℐ	ℐ	ADV
easat-4026	266	6	𝜌4	𝜌4	NOUN
easat-4026	266	7	′	′	NOUN
easat-4026	266	8	←	←	PROPN
easat-4026	266	9	…	…	PUNCT
easat-4026	266	10	such	such	ADJ
easat-4026	266	11	that	that	SCONJ
easat-4026	266	12	a	a	DET
easat-4026	266	13	tensor	tensor	NOUN
easat-4026	266	14	product	product	NOUN
easat-4026	266	15	obtained	obtain	VERB
easat-4026	266	16	over	over	ADP
easat-4026	266	17	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	266	18	and	and	CCONJ
easat-4026	266	19	a	a	DET
easat-4026	266	20	boundary	boundary	ADJ
easat-4026	266	21	map	map	NOUN
easat-4026	266	22	is	be	AUX
easat-4026	266	23	provided	provide	VERB
easat-4026	266	24	by	by	ADP
easat-4026	266	25	:	:	PUNCT
easat-4026	266	26	𝜌𝑛	𝜌𝑛	PROPN
easat-4026	266	27	′	′	NUM
easat-4026	266	28	(	(	PUNCT
easat-4026	266	29	𝒶0⊗	𝒶0⊗	PROPN
easat-4026	266	30	…	…	NUM
easat-4026	266	31	⊗𝒶𝑛	⊗𝒶𝑛	NUM
easat-4026	266	32	)	)	PUNCT
easat-4026	266	33	=	=	SYM
easat-4026	266	34	∑	∑	PUNCT
easat-4026	266	35	(	(	PUNCT
easat-4026	266	36	−1)𝑖𝒶0⊗	−1)𝑖𝒶0⊗	PROPN
easat-4026	266	37	…	…	SYM
easat-4026	266	38	⊗𝒶𝑖𝒶𝑖+1⊗	⊗𝒶𝑖𝒶𝑖+1⊗	NUM
easat-4026	266	39	…	…	SYM
easat-4026	266	40	⊗𝒶𝑛	⊗𝒶𝑛	NOUN
easat-4026	266	41	𝑛−1	𝑛−1	PROPN
easat-4026	266	42	𝑖=0	𝑖=0	PROPN
easat-4026	266	43	.	.	PUNCT
easat-4026	267	1	the	the	DET
easat-4026	267	2	following	follow	VERB
easat-4026	267	3	definition	definition	NOUN
easat-4026	267	4	defines	define	VERB
easat-4026	267	5	the	the	DET
easat-4026	267	6	simplicial	simplicial	ADJ
easat-4026	267	7	homology	homology	NOUN
easat-4026	267	8	of	of	ADP
easat-4026	267	9	complexes	complex	NOUN
easat-4026	267	10	and	and	CCONJ
easat-4026	267	11	explores	explore	NOUN
easat-4026	267	12	the	the	DET
easat-4026	267	13	boundary	boundary	ADJ
easat-4026	267	14	maps	map	NOUN
easat-4026	267	15	involved	involve	VERB
easat-4026	267	16	,	,	PUNCT
easat-4026	267	17	illustrating	illustrate	VERB
easat-4026	267	18	their	their	PRON
easat-4026	267	19	connection	connection	NOUN
easat-4026	267	20	to	to	ADP
easat-4026	267	21	the	the	DET
easat-4026	267	22	bar	bar	NOUN
easat-4026	267	23	homology	homology	NOUN
easat-4026	267	24	definitions	definition	NOUN
easat-4026	267	25	.	.	PUNCT
easat-4026	268	1	3.4	3.4	NUM
easat-4026	268	2	.	.	PUNCT
easat-4026	268	3	definition	definition	NOUN
easat-4026	268	4	the	the	DET
easat-4026	268	5	homology	homology	PROPN
easat-4026	268	6	ℋℋ∗(ℐ	ℋℋ∗(ℐ	PROPN
easat-4026	268	7	,	,	PUNCT
easat-4026	268	8	ℛ	ℛ	PROPN
easat-4026	268	9	)	)	PUNCT
easat-4026	268	10	of	of	ADP
easat-4026	268	11	complexes	complex	NOUN
easat-4026	268	12	represents	represent	VERB
easat-4026	268	13	the	the	DET
easat-4026	268	14	simplicial	simplicial	ADJ
easat-4026	268	15	homology	homology	NOUN
easat-4026	268	16	of	of	ADP
easat-4026	268	17	ℐ	ℐ	PRON
easat-4026	268	18	via	via	ADP
easat-4026	268	19	coefficients	coefficient	NOUN
easat-4026	268	20	in	in	ADP
easat-4026	268	21	ℛ	ℛ	NOUN
easat-4026	268	22	:	:	PUNCT
easat-4026	268	23	(	(	PUNCT
easat-4026	268	24	ℛ	ℛ	PROPN
easat-4026	268	25	⊗	⊗	PROPN
easat-4026	268	26	ℐ⨂∗	ℐ⨂∗	NOUN
easat-4026	268	27	,	,	PUNCT
easat-4026	268	28	𝜌∗	𝜌∗	NOUN
easat-4026	268	29	):	):	PUNCT
easat-4026	268	30	=	=	PUNCT
easat-4026	268	31	ℛ	ℛ	PROPN
easat-4026	268	32	𝜌1	𝜌1	PROPN
easat-4026	268	33	←	←	PROPN
easat-4026	268	34	ℛ	ℛ	PROPN
easat-4026	268	35	⊗	⊗	PROPN
easat-4026	269	1	ℐ	ℐ	PROPN
easat-4026	269	2	𝜌2	𝜌2	ADJ
easat-4026	269	3	←	←	PROPN
easat-4026	269	4	ℛ	ℛ	PROPN
easat-4026	269	5	⊗	⊗	PROPN
easat-4026	269	6	ℐ	ℐ	PROPN
easat-4026	269	7	⊗	⊗	PROPN
easat-4026	269	8	ℐ	ℐ	PROPN
easat-4026	269	9	𝜌3	𝜌3	PROPN
easat-4026	269	10	←	←	PROPN
easat-4026	269	11	ℛ	ℛ	PROPN
easat-4026	269	12	⊗	⊗	PROPN
easat-4026	269	13	ℐ	ℐ	PROPN
easat-4026	269	14	⊗	⊗	PROPN
easat-4026	270	1	ℐ	ℐ	PROPN
easat-4026	271	1	⊗	⊗	PROPN
easat-4026	271	2	ℐ	ℐ	PROPN
easat-4026	271	3	𝜌4	𝜌4	PROPN
easat-4026	271	4	←	←	NOUN
easat-4026	271	5	…	…	PUNCT
easat-4026	271	6	,	,	PUNCT
easat-4026	271	7	with	with	SCONJ
easat-4026	271	8	the	the	DET
easat-4026	271	9	boundary	boundary	ADJ
easat-4026	271	10	map	map	NOUN
easat-4026	271	11	provided	provide	VERB
easat-4026	271	12	by	by	ADP
easat-4026	271	13	:	:	PUNCT
easat-4026	271	14	9483	9483	NUM
easat-4026	271	15	edelweiss	edelweiss	PROPN
easat-4026	271	16	applied	apply	VERB
easat-4026	271	17	science	science	NOUN
easat-4026	271	18	and	and	CCONJ
easat-4026	271	19	technology	technology	NOUN
easat-4026	271	20	issn	issn	PROPN
easat-4026	271	21	:	:	PUNCT
easat-4026	271	22	2576	2576	NUM
easat-4026	271	23	-	-	SYM
easat-4026	271	24	8484	8484	NUM
easat-4026	271	25	vol	vol	NOUN
easat-4026	271	26	.	.	PROPN
easat-4026	271	27	8	8	NUM
easat-4026	271	28	,	,	PUNCT
easat-4026	271	29	no	no	INTJ
easat-4026	271	30	.	.	NOUN
easat-4026	272	1	6	6	NUM
easat-4026	272	2	:	:	SYM
easat-4026	272	3	9472	9472	NUM
easat-4026	272	4	-	-	SYM
easat-4026	272	5	9486	9486	NUM
easat-4026	272	6	,	,	PUNCT
easat-4026	272	7	2024	2024	NUM
easat-4026	272	8	doi	doi	NOUN
easat-4026	272	9	:	:	PUNCT
easat-4026	272	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	272	11	©	©	PROPN
easat-4026	272	12	2024	2024	NUM
easat-4026	272	13	by	by	ADP
easat-4026	272	14	the	the	DET
easat-4026	272	15	authors	author	NOUN
easat-4026	272	16	;	;	PUNCT
easat-4026	272	17	licensee	licensee	PROPN
easat-4026	272	18	learning	learning	NOUN
easat-4026	272	19	gate	gate	VERB
easat-4026	272	20	𝜌𝑛(𝒶0⊗	𝜌𝑛(𝒶0⊗	PROPN
easat-4026	272	21	…	…	SYM
easat-4026	272	22	⊗𝒶𝑛	⊗𝒶𝑛	NUM
easat-4026	272	23	)	)	PUNCT
easat-4026	272	24	=	=	SYM
easat-4026	272	25	𝜌𝑛	𝜌𝑛	ADP
easat-4026	272	26	′	′	NUM
easat-4026	272	27	(	(	PUNCT
easat-4026	272	28	𝒶0⊗	𝒶0⊗	PROPN
easat-4026	272	29	…	…	PUNCT
easat-4026	272	30	⊗𝒶𝑛	⊗𝒶𝑛	NUM
easat-4026	272	31	)	)	PUNCT
easat-4026	273	1	+	+	CCONJ
easat-4026	273	2	(	(	PUNCT
easat-4026	273	3	−1)𝑛𝒶𝑛𝒶0⊗𝒶1	−1)𝑛𝒶𝑛𝒶0⊗𝒶1	ADJ
easat-4026	273	4	⊗	⊗	ADJ
easat-4026	273	5	…	…	SYM
easat-4026	273	6	⊗𝒶𝑛−1	⊗𝒶𝑛−1	NOUN
easat-4026	273	7	.	.	PUNCT
easat-4026	274	1	we	we	PRON
easat-4026	274	2	present	present	VERB
easat-4026	274	3	a	a	DET
easat-4026	274	4	corollary	corollary	ADJ
easat-4026	274	5	relating	relate	VERB
easat-4026	274	6	to	to	ADP
easat-4026	274	7	𝐻-homology	𝐻-homology	PROPN
easat-4026	274	8	and	and	CCONJ
easat-4026	274	9	bar	bar	NOUN
easat-4026	274	10	homology	homology	NOUN
easat-4026	274	11	,	,	PUNCT
easat-4026	274	12	establishing	establish	VERB
easat-4026	274	13	the	the	DET
easat-4026	274	14	relationships	relationship	NOUN
easat-4026	274	15	between	between	ADP
easat-4026	274	16	different	different	ADJ
easat-4026	274	17	homological	homological	ADJ
easat-4026	274	18	constructs	construct	NOUN
easat-4026	274	19	for	for	ADP
easat-4026	274	20	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	274	21	and	and	CCONJ
easat-4026	274	22	discussing	discuss	VERB
easat-4026	274	23	exact	exact	ADJ
easat-4026	274	24	sequences	sequence	NOUN
easat-4026	274	25	that	that	PRON
easat-4026	274	26	arise	arise	VERB
easat-4026	274	27	.	.	PUNCT
easat-4026	275	1	3.5	3.5	NUM
easat-4026	275	2	.	.	PUNCT
easat-4026	275	3	corollary	corollary	VERB
easat-4026	275	4	the	the	DET
easat-4026	275	5	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	275	6	that	that	PRON
easat-4026	275	7	are	be	AUX
easat-4026	275	8	produced	produce	VERB
easat-4026	275	9	from	from	ADP
easat-4026	275	10	ℐ	ℐ	PRON
easat-4026	275	11	by	by	ADP
easat-4026	275	12	adding	add	VERB
easat-4026	275	13	the	the	DET
easat-4026	275	14	value	value	NOUN
easat-4026	275	15	of	of	ADP
easat-4026	275	16	unity	unity	NOUN
easat-4026	275	17	to	to	ADP
easat-4026	275	18	ℐ	ℐ	PRON
easat-4026	275	19	be	be	AUX
easat-4026	275	20	denoted	denote	VERB
easat-4026	275	21	by	by	ADP
easat-4026	275	22	ℐ̃	ℐ̃	PROPN
easat-4026	275	23	=	=	SYM
easat-4026	276	1	𝑘	𝑘	DET
easat-4026	276	2	×	×	NOUN
easat-4026	276	3	ℐ.	ℐ.	PROPN
easat-4026	276	4	then	then	ADV
easat-4026	276	5	the	the	DET
easat-4026	276	6	𝐻-homology	𝐻-homology	PROPN
easat-4026	276	7	𝐻∗(ℐ	𝐻∗(ℐ	NOUN
easat-4026	276	8	)	)	PUNCT
easat-4026	276	9	is	be	AUX
easat-4026	276	10	defined	define	VERB
easat-4026	276	11	by	by	ADP
easat-4026	276	12	ℋ∗(ℐ):=	ℋ∗(ℐ):=	PROPN
easat-4026	276	13	ℋ∗(ℐ	ℋ∗(ℐ	PROPN
easat-4026	276	14	,	,	PUNCT
easat-4026	276	15	ℐ	ℐ	PROPN
easat-4026	276	16	)	)	PUNCT
easat-4026	276	17	,	,	PUNCT
easat-4026	276	18	the	the	DET
easat-4026	276	19	bar	bar	NOUN
easat-4026	276	20	homology	homology	NOUN
easat-4026	276	21	ℋ𝐵∗(ℐ	ℋ𝐵∗(ℐ	PROPN
easat-4026	276	22	)	)	PUNCT
easat-4026	276	23	is	be	AUX
easat-4026	276	24	defined	define	VERB
easat-4026	276	25	by	by	ADP
easat-4026	276	26	ℋ𝐵∗(ℐ):=	ℋ𝐵∗(ℐ):=	PROPN
easat-4026	276	27	ℋ𝐵∗	ℋ𝐵∗	PROPN
easat-4026	276	28	′(ℐ	′(ℐ	NOUN
easat-4026	276	29	,	,	PUNCT
easat-4026	276	30	ℐ	ℐ	PROPN
easat-4026	276	31	)	)	PUNCT
easat-4026	276	32	,	,	PUNCT
easat-4026	276	33	and	and	CCONJ
easat-4026	276	34	ℋℋ∗(ℐ	ℋℋ∗(ℐ	PROPN
easat-4026	276	35	):	):	PUNCT
easat-4026	276	36	=	=	SYM
easat-4026	276	37	ℋ̅∗(ℐ	ℋ̅∗(ℐ	PROPN
easat-4026	276	38	,	,	PUNCT
easat-4026	276	39	ℐ̃	ℐ̃	NUM
easat-4026	276	40	)	)	PUNCT
easat-4026	276	41	defines	define	VERB
easat-4026	276	42	the	the	DET
easat-4026	276	43	simplicial	simplicial	ADJ
easat-4026	276	44	homology	homology	NOUN
easat-4026	276	45	of	of	ADP
easat-4026	276	46	ℐ	ℐ	PROPN
easat-4026	276	47	,	,	PUNCT
easat-4026	276	48	where	where	SCONJ
easat-4026	276	49	ℋ̅𝑛(ℐ	ℋ̅𝑛(ℐ	NOUN
easat-4026	276	50	,	,	PUNCT
easat-4026	276	51	ℐ̃	ℐ̃	NUM
easat-4026	276	52	)	)	PUNCT
easat-4026	276	53	=	=	SYM
easat-4026	276	54	ℋ𝑛(ℐ	ℋ𝑛(ℐ	PROPN
easat-4026	276	55	,	,	PUNCT
easat-4026	276	56	ℐ̃	ℐ̃	NUM
easat-4026	276	57	)	)	PUNCT
easat-4026	276	58	,	,	PUNCT
easat-4026	276	59	∀	∀	X
easat-4026	276	60	𝑛	𝑛	VERB
easat-4026	276	61	>	>	X
easat-4026	276	62	0	0	NUM
easat-4026	276	63	and	and	CCONJ
easat-4026	276	64	ℋ̅0(ℐ	ℋ̅0(ℐ	NUM
easat-4026	276	65	,	,	PUNCT
easat-4026	276	66	ℐ̃	ℐ̃	NUM
easat-4026	276	67	)	)	PUNCT
easat-4026	276	68	=	=	SYM
easat-4026	276	69	ℋ0(ℐ	ℋ0(ℐ	NOUN
easat-4026	276	70	,	,	PUNCT
easat-4026	276	71	ℐ̃)/𝑘.	ℐ̃)/𝑘.	NOUN
easat-4026	276	72	let	let	VERB
easat-4026	276	73	the	the	DET
easat-4026	276	74	homology	homology	NOUN
easat-4026	276	75	ℋℋ∗(ℐ	ℋℋ∗(ℐ	PROPN
easat-4026	276	76	)	)	PUNCT
easat-4026	276	77	be	be	VERB
easat-4026	276	78	the	the	DET
easat-4026	276	79	double	double	ADJ
easat-4026	276	80	complex	complex	NOUN
easat-4026	276	81	'	'	PUNCT
easat-4026	276	82	homology	homology	NOUN
easat-4026	276	83	such	such	ADJ
easat-4026	276	84	that	that	PRON
easat-4026	276	85	:	:	PUNCT
easat-4026	276	86	𝒞𝒞(ℐ)|2|	𝒞𝒞(ℐ)|2|	X
easat-4026	276	87	≔	≔	NOUN
easat-4026	276	88	(	(	PUNCT
easat-4026	276	89	ℐ	ℐ	ADV
easat-4026	276	90	⊗	⊗	PROPN
easat-4026	276	91	ℐ⨂∗	ℐ⨂∗	X
easat-4026	276	92	,	,	PUNCT
easat-4026	276	93	𝜌∗	𝜌∗	NOUN
easat-4026	276	94	)	)	PUNCT
easat-4026	276	95	1−𝑡	1−𝑡	NUM
easat-4026	276	96	←	←	PROPN
easat-4026	276	97	(	(	PUNCT
easat-4026	276	98	ℐ	ℐ	ADV
easat-4026	276	99	⊗	⊗	PROPN
easat-4026	276	100	ℐ⨂∗	ℐ⨂∗	NOUN
easat-4026	276	101	,	,	PUNCT
easat-4026	276	102	−𝜌∗	−𝜌∗	PROPN
easat-4026	276	103	′	′	NUM
easat-4026	276	104	)	)	PUNCT
easat-4026	276	105	.	.	PUNCT
easat-4026	277	1	(	(	PUNCT
easat-4026	277	2	8)	8)	NUM
easat-4026	277	3	consequently	consequently	ADV
easat-4026	277	4	,	,	PUNCT
easat-4026	277	5	the	the	DET
easat-4026	277	6	exact	exact	ADJ
easat-4026	277	7	sequence	sequence	NOUN
easat-4026	277	8	exists	exist	VERB
easat-4026	277	9	as	as	SCONJ
easat-4026	277	10	follows	follow	VERB
easat-4026	277	11	:	:	PUNCT
easat-4026	277	12	…	…	PUNCT
easat-4026	278	1	←	←	PROPN
easat-4026	278	2	𝐻𝑛−1(ℐ	𝐻𝑛−1(ℐ	PROPN
easat-4026	278	3	)	)	PUNCT
easat-4026	278	4	←	←	PROPN
easat-4026	278	5	ℋ𝐵𝑛−1(ℐ	ℋ𝐵𝑛−1(ℐ	NUM
easat-4026	278	6	)	)	PUNCT
easat-4026	278	7	←	←	PROPN
easat-4026	278	8	ℋℋ𝑛(ℐ	ℋℋ𝑛(ℐ	NOUN
easat-4026	278	9	)	)	PUNCT
easat-4026	278	10	←	←	PROPN
easat-4026	278	11	𝐻𝑛(ℐ	𝐻𝑛(ℐ	PROPN
easat-4026	278	12	)	)	PUNCT
easat-4026	278	13	←	←	PROPN
easat-4026	278	14	ℋ𝐵𝑛(ℐ	ℋ𝐵𝑛(ℐ	PROPN
easat-4026	278	15	)	)	PUNCT
easat-4026	278	16	←	←	PROPN
easat-4026	278	17	ℋℋ𝑛+1(ℐ	ℋℋ𝑛+1(ℐ	PROPN
easat-4026	278	18	)	)	PUNCT
easat-4026	278	19	←	←	PROPN
easat-4026	278	20	…	…	PUNCT
easat-4026	278	21	.	.	PUNCT
easat-4026	279	1	the	the	DET
easat-4026	279	2	following	follow	VERB
easat-4026	279	3	definition	definition	NOUN
easat-4026	279	4	covers	cover	VERB
easat-4026	279	5	the	the	DET
easat-4026	279	6	concept	concept	NOUN
easat-4026	279	7	of	of	ADP
easat-4026	279	8	ℋ-unitarity	ℋ-unitarity	PROPN
easat-4026	279	9	in	in	ADP
easat-4026	279	10	𝒜∞-algebras	𝒜∞-algebras	NUM
easat-4026	279	11	,	,	PUNCT
easat-4026	279	12	focusing	focus	VERB
easat-4026	279	13	on	on	ADP
easat-4026	279	14	the	the	DET
easat-4026	279	15	conditions	condition	NOUN
easat-4026	279	16	under	under	ADP
easat-4026	279	17	which	which	PRON
easat-4026	279	18	a	a	DET
easat-4026	279	19	module	module	NOUN
easat-4026	279	20	ℛ	ℛ	NOUN
easat-4026	279	21	is	be	AUX
easat-4026	279	22	considered	consider	VERB
easat-4026	279	23	ℋ-unitary	ℋ-unitary	PROPN
easat-4026	279	24	.	.	PUNCT
easat-4026	279	25	3.6	3.6	NUM
easat-4026	279	26	.	.	PUNCT
easat-4026	280	1	definition	definition	NOUN
easat-4026	280	2	suppose	suppose	VERB
easat-4026	280	3	that	that	SCONJ
easat-4026	280	4	ℛ	ℛ	PROPN
easat-4026	280	5	is	be	AUX
easat-4026	280	6	an	an	DET
easat-4026	280	7	ℐ-bimodule	ℐ-bimodule	NOUN
easat-4026	280	8	since	since	SCONJ
easat-4026	280	9	ℐ	ℐ	PRON
easat-4026	280	10	is	be	AUX
easat-4026	280	11	an	an	DET
easat-4026	280	12	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	280	13	.	.	PUNCT
easat-4026	281	1	if	if	SCONJ
easat-4026	281	2	every	every	DET
easat-4026	281	3	𝒜∞-modules	𝒜∞-modules	PROPN
easat-4026	281	4	𝒢	𝒢	PROPN
easat-4026	281	5	has	have	VERB
easat-4026	281	6	an	an	DET
easat-4026	281	7	exact	exact	ADJ
easat-4026	281	8	complex	complex	NOUN
easat-4026	281	9	(	(	PUNCT
easat-4026	281	10	ℛ	ℛ	PROPN
easat-4026	281	11	⊗	⊗	PROPN
easat-4026	281	12	ℐ⨂∗	ℐ⨂∗	X
easat-4026	281	13	,	,	PUNCT
easat-4026	281	14	𝜌∗)⊗	𝜌∗)⊗	PUNCT
easat-4026	281	15	𝒢	𝒢	NOUN
easat-4026	281	16	,	,	PUNCT
easat-4026	281	17	we	we	PRON
easat-4026	281	18	can	can	AUX
easat-4026	281	19	deduce	deduce	VERB
easat-4026	281	20	that	that	SCONJ
easat-4026	281	21	ℛ	ℛ	PROPN
easat-4026	281	22	is	be	AUX
easat-4026	281	23	ℋ-unitary	ℋ-unitary	PROPN
easat-4026	281	24	.	.	PUNCT
easat-4026	282	1	one	one	NUM
easat-4026	282	2	states	state	VERB
easat-4026	282	3	that	that	SCONJ
easat-4026	282	4	ℐ	ℐ	PRON
easat-4026	282	5	is	be	AUX
easat-4026	282	6	ℋ-unital	ℋ-unital	ADJ
easat-4026	282	7	when	when	SCONJ
easat-4026	282	8	ℛ	ℛ	PROPN
easat-4026	282	9	=	=	SYM
easat-4026	282	10	ℐ	ℐ	PROPN
easat-4026	282	11	,	,	PUNCT
easat-4026	282	12	such	such	ADJ
easat-4026	282	13	ℛ	ℛ	PROPN
easat-4026	282	14	is	be	AUX
easat-4026	282	15	the	the	DET
easat-4026	282	16	left	left	ADJ
easat-4026	282	17	ℐ-module	ℐ-module	PROPN
easat-4026	282	18	.	.	PUNCT
easat-4026	283	1	it	it	PRON
easat-4026	283	2	follows	follow	VERB
easat-4026	283	3	logically	logically	ADV
easat-4026	283	4	that	that	SCONJ
easat-4026	283	5	ℛ⊗	ℛ⊗	PROPN
easat-4026	283	6	ℐ	ℐ	PRON
easat-4026	283	7	is	be	AUX
easat-4026	283	8	ℋ-unitary	ℋ-unitary	PROPN
easat-4026	283	9	,	,	PUNCT
easat-4026	283	10	if	if	SCONJ
easat-4026	283	11	ℐ	ℐ	PRON
easat-4026	283	12	is	be	AUX
easat-4026	283	13	ℋ-unital	ℋ-unital	PROPN
easat-4026	283	14	.	.	PUNCT
easat-4026	284	1	in	in	ADP
easat-4026	284	2	the	the	DET
easat-4026	284	3	following	following	NOUN
easat-4026	284	4	,	,	PUNCT
easat-4026	284	5	we	we	PRON
easat-4026	284	6	examine	examine	VERB
easat-4026	284	7	a	a	DET
easat-4026	284	8	theorem	theorem	NOUN
easat-4026	284	9	on	on	ADP
easat-4026	284	10	quasi	quasi	NOUN
easat-4026	284	11	-	-	NOUN
easat-4026	284	12	isomorphisms	isomorphism	NOUN
easat-4026	284	13	between	between	ADP
easat-4026	284	14	complexes	complex	NOUN
easat-4026	284	15	in	in	ADP
easat-4026	284	16	the	the	DET
easat-4026	284	17	context	context	NOUN
easat-4026	284	18	of	of	ADP
easat-4026	284	19	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	284	20	and	and	CCONJ
easat-4026	284	21	bimodules	bimodule	NOUN
easat-4026	284	22	,	,	PUNCT
easat-4026	284	23	proving	prove	VERB
easat-4026	284	24	the	the	DET
easat-4026	284	25	results	result	NOUN
easat-4026	284	26	under	under	ADP
easat-4026	284	27	specific	specific	ADJ
easat-4026	284	28	assumptions	assumption	NOUN
easat-4026	284	29	about	about	ADP
easat-4026	284	30	ℋ-unitarity	ℋ-unitarity	PROPN
easat-4026	284	31	.	.	PROPN
easat-4026	284	32	3.7	3.7	NUM
easat-4026	284	33	.	.	PUNCT
easat-4026	285	1	theorem	theorem	NOUN
easat-4026	285	2	assume	assume	VERB
easat-4026	285	3	that	that	SCONJ
easat-4026	285	4	the	the	DET
easat-4026	285	5	extension	extension	NOUN
easat-4026	285	6	of	of	ADP
easat-4026	285	7	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	285	8	is	be	AUX
easat-4026	285	9	given	give	VERB
easat-4026	285	10	by	by	ADP
easat-4026	285	11	0	0	NUM
easat-4026	285	12	→	→	SYM
easat-4026	285	13	ℐ	ℐ	PROPN
easat-4026	285	14	→	→	SYM
easat-4026	285	15	𝒜	𝒜	NOUN
easat-4026	285	16	→	→	SYM
easat-4026	285	17	ℬ	ℬ	NOUN
easat-4026	285	18	→	→	SYM
easat-4026	285	19	0	0	NUM
easat-4026	285	20	,	,	PUNCT
easat-4026	285	21	and	and	CCONJ
easat-4026	285	22	defines	define	VERB
easat-4026	285	23	𝒢	𝒢	PROPN
easat-4026	285	24	to	to	PART
easat-4026	285	25	be	be	AUX
easat-4026	285	26	a	a	DET
easat-4026	285	27	𝒜∞-modules	𝒜∞-modules	PROPN
easat-4026	285	28	and	and	CCONJ
easat-4026	285	29	ℛ	ℛ	PROPN
easat-4026	285	30	to	to	PART
easat-4026	285	31	be	be	AUX
easat-4026	285	32	an	an	DET
easat-4026	285	33	𝒜-bimodule	𝒜-bimodule	PROPN
easat-4026	285	34	.	.	PUNCT
easat-4026	286	1	then	then	ADV
easat-4026	286	2	we	we	PRON
easat-4026	286	3	can	can	AUX
easat-4026	286	4	say	say	VERB
easat-4026	286	5	that	that	SCONJ
easat-4026	286	6	the	the	DET
easat-4026	286	7	following	follow	VERB
easat-4026	286	8	canonical	canonical	ADJ
easat-4026	286	9	inclusions	inclusion	NOUN
easat-4026	286	10	:	:	PUNCT
easat-4026	286	11	𝑖	𝑖	X
easat-4026	286	12	:	:	PUNCT
easat-4026	286	13	(	(	PUNCT
easat-4026	286	14	ℛ	ℛ	PROPN
easat-4026	286	15	⊗	⊗	PROPN
easat-4026	286	16	ℐ⨂∗	ℐ⨂∗	X
easat-4026	286	17	,	,	PUNCT
easat-4026	286	18	𝜌∗)⊗	𝜌∗)⊗	PUNCT
easat-4026	286	19	𝒢	𝒢	PROPN
easat-4026	286	20	↪	↪	PROPN
easat-4026	286	21	(	(	PUNCT
easat-4026	286	22	ℛ	ℛ	NOUN
easat-4026	286	23	⊗𝒜⨂∗	⊗𝒜⨂∗	NOUN
easat-4026	286	24	,	,	PUNCT
easat-4026	286	25	𝜌∗)⊗	𝜌∗)⊗	PUNCT
easat-4026	287	1	𝒢	𝒢	NOUN
easat-4026	287	2	,	,	PUNCT
easat-4026	287	3	(	(	PUNCT
easat-4026	287	4	9	9	NUM
easat-4026	287	5	)	)	PUNCT
easat-4026	287	6	𝑖′	𝑖′	NUM
easat-4026	287	7	:	:	PUNCT
easat-4026	287	8	(	(	PUNCT
easat-4026	287	9	ℛ	ℛ	PROPN
easat-4026	287	10	⊗	⊗	PROPN
easat-4026	287	11	ℐ⨂∗	ℐ⨂∗	NOUN
easat-4026	287	12	,	,	PUNCT
easat-4026	287	13	𝜌∗	𝜌∗	NOUN
easat-4026	287	14	′)⊗	′)⊗	NOUN
easat-4026	287	15	𝒢	𝒢	PROPN
easat-4026	287	16	↪	↪	PROPN
easat-4026	287	17	(	(	PUNCT
easat-4026	287	18	ℛ	ℛ	NOUN
easat-4026	287	19	⊗𝒜⨂∗	⊗𝒜⨂∗	NOUN
easat-4026	287	20	,	,	PUNCT
easat-4026	287	21	𝜌∗	𝜌∗	NOUN
easat-4026	287	22	′)⊗	′)⊗	NOUN
easat-4026	287	23	𝒢	𝒢	NOUN
easat-4026	287	24	(	(	PUNCT
easat-4026	287	25	10	10	NUM
easat-4026	287	26	)	)	PUNCT
easat-4026	287	27	are	be	AUX
easat-4026	287	28	quasi	quasi	NOUN
easat-4026	287	29	-	-	NOUN
easat-4026	287	30	isomorphisms	isomorphism	NOUN
easat-4026	287	31	when	when	SCONJ
easat-4026	287	32	the	the	DET
easat-4026	287	33	ℐ-bimodule	ℐ-bimodule	PROPN
easat-4026	287	34	ℛ	ℛ	PROPN
easat-4026	287	35	is	be	AUX
easat-4026	287	36	ℋ-unitary	ℋ-unitary	PROPN
easat-4026	287	37	.	.	PUNCT
easat-4026	287	38	proof	proof	NOUN
easat-4026	287	39	:	:	PUNCT
easat-4026	287	40	by	by	ADP
easat-4026	287	41	considering	consider	VERB
easat-4026	287	42	the	the	DET
easat-4026	287	43	filtration	filtration	NOUN
easat-4026	287	44	𝐹0	𝐹0	NOUN
easat-4026	287	45	⊆	⊆	NUM
easat-4026	287	46	𝐹0	𝐹0	NOUN
easat-4026	287	47	⊆	⊆	NUM
easat-4026	287	48	⋯	⋯	PROPN
easat-4026	287	49	of	of	ADP
easat-4026	287	50	(	(	PUNCT
easat-4026	287	51	ℛ	ℛ	PROPN
easat-4026	287	52	⊗𝒜⨂∗	⊗𝒜⨂∗	NOUN
easat-4026	287	53	,	,	PUNCT
easat-4026	287	54	𝜌∗	𝜌∗	NOUN
easat-4026	287	55	)	)	PUNCT
easat-4026	287	56	and	and	CCONJ
easat-4026	287	57	using	use	VERB
easat-4026	287	58	[	[	X
easat-4026	287	59	32	32	NUM
easat-4026	287	60	]	]	PUNCT
easat-4026	287	61	,	,	PUNCT
easat-4026	287	62	[	[	X
easat-4026	287	63	33	33	NUM
easat-4026	287	64	]	]	PUNCT
easat-4026	287	65	,	,	PUNCT
easat-4026	287	66	assume	assume	VERB
easat-4026	287	67	that	that	SCONJ
easat-4026	287	68	𝒢	𝒢	PROPN
easat-4026	287	69	is	be	AUX
easat-4026	287	70	an	an	DET
easat-4026	287	71	𝒜∞-modules	𝒜∞-module	NOUN
easat-4026	287	72	,	,	PUNCT
easat-4026	287	73	such	such	ADJ
easat-4026	287	74	that	that	SCONJ
easat-4026	287	75	:	:	PUNCT
easat-4026	287	76	𝐹ℓ	𝐹ℓ	PROPN
easat-4026	287	77	≔	≔	NOUN
easat-4026	287	78	ℛ	ℛ	NOUN
easat-4026	287	79	𝜌1	𝜌1	NOUN
easat-4026	287	80	←	←	PROPN
easat-4026	287	81	ℛ	ℛ	PROPN
easat-4026	287	82	⊗𝒜	⊗𝒜	PROPN
easat-4026	287	83	𝜌2	𝜌2	ADJ
easat-4026	287	84	←	←	PROPN
easat-4026	287	85	ℛ⊗𝒜⊗2	ℛ⊗𝒜⊗2	PROPN
easat-4026	287	86	𝜌3	𝜌3	PROPN
easat-4026	287	87	←	←	PROPN
easat-4026	287	88	…	…	PUNCT
easat-4026	287	89	𝜌ℓ	𝜌ℓ	NOUN
easat-4026	287	90	←	←	PROPN
easat-4026	287	91	ℛ	ℛ	PROPN
easat-4026	287	92	⊗𝒜⊗𝑝	⊗𝒜⊗𝑝	PROPN
easat-4026	287	93	𝜌ℓ+1	𝜌ℓ+1	NUM
easat-4026	287	94	←	←	PROPN
easat-4026	287	95	ℛ	ℛ	PROPN
easat-4026	287	96	⊗	⊗	PROPN
easat-4026	287	97	ℐ	ℐ	PROPN
easat-4026	287	98	⊗𝒜⊗ℓ	⊗𝒜⊗ℓ	PROPN
easat-4026	287	99	𝜌ℓ+2	𝜌ℓ+2	NUM
easat-4026	287	100	←	←	PROPN
easat-4026	287	101	ℛ	ℛ	PROPN
easat-4026	287	102	⊗	⊗	PROPN
easat-4026	287	103	ℐ⊗2	ℐ⊗2	NOUN
easat-4026	287	104	⊗𝒜⊗ℓ	⊗𝒜⊗ℓ	NOUN
easat-4026	287	105	𝜌ℓ+3	𝜌ℓ+3	PRON
easat-4026	287	106	←	←	PROPN
easat-4026	287	107	…	…	PUNCT
easat-4026	287	108	for	for	ADP
easat-4026	287	109	all	all	DET
easat-4026	287	110	ℓ	ℓ	PROPN
easat-4026	287	111	≥	≥	NOUN
easat-4026	287	112	0	0	NUM
easat-4026	287	113	,	,	PUNCT
easat-4026	287	114	we	we	PRON
easat-4026	287	115	have	have	VERB
easat-4026	287	116	:	:	PUNCT
easat-4026	287	117	(	(	PUNCT
easat-4026	287	118	𝐹ℓ+1⊗	𝐹ℓ+1⊗	X
easat-4026	287	119	𝒢	𝒢	PROPN
easat-4026	287	120	𝐹ℓ	𝐹ℓ	PROPN
easat-4026	287	121	⊗𝒢	⊗𝒢	NOUN
easat-4026	287	122	)	)	PUNCT
easat-4026	287	123	∗	∗	NOUN
easat-4026	287	124	=	=	SYM
easat-4026	287	125	(	(	PUNCT
easat-4026	287	126	ℛ	ℛ	PROPN
easat-4026	287	127	⊗	⊗	PROPN
easat-4026	287	128	ℐ⨂	ℐ⨂	VERB
easat-4026	287	129	∗−ℓ−1	∗−ℓ−1	PROPN
easat-4026	287	130	,	,	PUNCT
easat-4026	287	131	𝜌∗	𝜌∗	NOUN
easat-4026	287	132	′)⊗	′)⊗	PROPN
easat-4026	287	133	ℬ⊗𝒜⊗ℓ	ℬ⊗𝒜⊗ℓ	PROPN
easat-4026	287	134	⊗𝒢	⊗𝒢	PROPN
easat-4026	287	135	,	,	PUNCT
easat-4026	287	136	(	(	PUNCT
easat-4026	287	137	11	11	NUM
easat-4026	287	138	)	)	PUNCT
easat-4026	287	139	according	accord	VERB
easat-4026	287	140	to	to	ADP
easat-4026	287	141	theory	theory	NOUN
easat-4026	287	142	,	,	PUNCT
easat-4026	287	143	this	this	PRON
easat-4026	287	144	is	be	AUX
easat-4026	287	145	exact	exact	ADJ
easat-4026	287	146	.	.	PUNCT
easat-4026	288	1	taking	take	VERB
easat-4026	288	2	into	into	ADP
easat-4026	288	3	consideration	consideration	NOUN
easat-4026	288	4	the	the	DET
easat-4026	288	5	homology	homology	NOUN
easat-4026	288	6	of	of	ADP
easat-4026	288	7	long	long	ADJ
easat-4026	288	8	exact	exact	ADJ
easat-4026	288	9	sequence	sequence	NOUN
easat-4026	288	10	for	for	ADP
easat-4026	288	11	all	all	DET
easat-4026	288	12	𝑛	𝑛	DET
easat-4026	288	13	≥	≥	NOUN
easat-4026	288	14	0	0	NUM
easat-4026	288	15	connected	connect	VERB
easat-4026	288	16	to	to	ADP
easat-4026	288	17	0	0	NUM
easat-4026	288	18	→	→	SYM
easat-4026	288	19	𝐹𝑛	𝐹𝑛	PROPN
easat-4026	288	20	⊗𝒢	⊗𝒢	PROPN
easat-4026	288	21	→	→	SYM
easat-4026	288	22	𝐹𝑛+1	𝐹𝑛+1	X
easat-4026	288	23	⊗𝒢	⊗𝒢	PROPN
easat-4026	288	24	→	→	X
easat-4026	288	25	𝐹𝑛+1⊗𝒢	𝐹𝑛+1⊗𝒢	ADJ
easat-4026	288	26	𝐹𝑛⊗𝒢	𝐹𝑛⊗𝒢	ADJ
easat-4026	288	27	→	→	SYM
easat-4026	288	28	0	0	NUM
easat-4026	288	29	(	(	PUNCT
easat-4026	288	30	12	12	NUM
easat-4026	288	31	)	)	PUNCT
easat-4026	288	32	as	as	SCONJ
easat-4026	288	33	can	can	AUX
easat-4026	288	34	be	be	AUX
easat-4026	288	35	seen	see	VERB
easat-4026	288	36	,	,	PUNCT
easat-4026	288	37	the	the	DET
easat-4026	288	38	canonical	canonical	ADJ
easat-4026	288	39	map	map	NOUN
easat-4026	288	40	𝐹0	𝐹0	NOUN
easat-4026	288	41	→	→	PUNCT
easat-4026	288	42	𝐹ℓ	𝐹ℓ	PROPN
easat-4026	288	43	represents	represent	VERB
easat-4026	288	44	quasi	quasi	NOUN
easat-4026	288	45	-	-	NOUN
easat-4026	288	46	isomorphism	isomorphism	NOUN
easat-4026	288	47	for	for	ADP
easat-4026	288	48	every	every	DET
easat-4026	288	49	ℓ	ℓ	NOUN
easat-4026	288	50	,	,	PUNCT
easat-4026	288	51	𝑖	𝑖	X
easat-4026	288	52	is	be	AUX
easat-4026	288	53	also	also	ADV
easat-4026	288	54	a	a	DET
easat-4026	288	55	quasiisomorphism	quasiisomorphism	NOUN
easat-4026	288	56	consequently	consequently	ADV
easat-4026	288	57	.	.	PUNCT
easat-4026	289	1	in	in	ADP
easat-4026	289	2	a	a	DET
easat-4026	289	3	similar	similar	ADJ
easat-4026	289	4	demonstration	demonstration	NOUN
easat-4026	289	5	,	,	PUNCT
easat-4026	289	6	the	the	DET
easat-4026	289	7	same	same	ADJ
easat-4026	289	8	applies	apply	VERB
easat-4026	289	9	to	to	PART
easat-4026	289	10	𝑖′.	𝑖′.	VERB
easat-4026	289	11	remark	remark	NOUN
easat-4026	289	12	:	:	PUNCT
easat-4026	289	13	note	note	VERB
easat-4026	289	14	that	that	SCONJ
easat-4026	289	15	the	the	DET
easat-4026	289	16	theorem	theorem	NOUN
easat-4026	289	17	(	(	PUNCT
easat-4026	289	18	3.7	3.7	NUM
easat-4026	289	19	)	)	PUNCT
easat-4026	289	20	given	give	VERB
easat-4026	289	21	above	above	ADV
easat-4026	289	22	may	may	AUX
easat-4026	289	23	also	also	ADV
easat-4026	289	24	be	be	AUX
easat-4026	289	25	proved	prove	VERB
easat-4026	289	26	in	in	ADP
easat-4026	289	27	the	the	DET
easat-4026	289	28	case	case	NOUN
easat-4026	289	29	when	when	SCONJ
easat-4026	289	30	ℐ	ℐ	PRON
easat-4026	289	31	is	be	AUX
easat-4026	289	32	a	a	DET
easat-4026	289	33	right	right	ADJ
easat-4026	289	34	ideal	ideal	NOUN
easat-4026	289	35	of	of	ADP
easat-4026	289	36	𝒜	𝒜	NOUN
easat-4026	289	37	instead	instead	ADV
easat-4026	289	38	of	of	ADP
easat-4026	289	39	a	a	DET
easat-4026	289	40	two	two	NUM
easat-4026	289	41	-	-	PUNCT
easat-4026	289	42	sided	sided	ADJ
easat-4026	289	43	ideal	ideal	NOUN
easat-4026	289	44	.	.	PUNCT
easat-4026	290	1	9484	9484	NUM
easat-4026	290	2	edelweiss	edelweiss	PROPN
easat-4026	290	3	applied	apply	VERB
easat-4026	290	4	science	science	NOUN
easat-4026	290	5	and	and	CCONJ
easat-4026	290	6	technology	technology	NOUN
easat-4026	290	7	issn	issn	PROPN
easat-4026	290	8	:	:	PUNCT
easat-4026	290	9	2576	2576	NUM
easat-4026	290	10	-	-	SYM
easat-4026	290	11	8484	8484	NUM
easat-4026	290	12	vol	vol	NOUN
easat-4026	290	13	.	.	PROPN
easat-4026	290	14	8	8	NUM
easat-4026	290	15	,	,	PUNCT
easat-4026	290	16	no	no	INTJ
easat-4026	290	17	.	.	NOUN
easat-4026	291	1	6	6	NUM
easat-4026	291	2	:	:	SYM
easat-4026	291	3	9472	9472	NUM
easat-4026	291	4	-	-	SYM
easat-4026	291	5	9486	9486	NUM
easat-4026	291	6	,	,	PUNCT
easat-4026	291	7	2024	2024	NUM
easat-4026	291	8	doi	doi	NOUN
easat-4026	291	9	:	:	PUNCT
easat-4026	291	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	291	11	©	©	PROPN
easat-4026	291	12	2024	2024	NUM
easat-4026	291	13	by	by	ADP
easat-4026	291	14	the	the	DET
easat-4026	291	15	authors	author	NOUN
easat-4026	291	16	;	;	PUNCT
easat-4026	291	17	licensee	licensee	PROPN
easat-4026	291	18	learning	learning	NOUN
easat-4026	291	19	gate	gate	VERB
easat-4026	291	20	the	the	DET
easat-4026	291	21	following	follow	VERB
easat-4026	291	22	corollary	corollary	NOUN
easat-4026	291	23	provides	provide	VERB
easat-4026	291	24	further	further	ADJ
easat-4026	291	25	insight	insight	NOUN
easat-4026	291	26	into	into	ADP
easat-4026	291	27	quasi	quasi	NOUN
easat-4026	291	28	-	-	NOUN
easat-4026	291	29	isomorphisms	isomorphism	NOUN
easat-4026	291	30	in	in	ADP
easat-4026	291	31	the	the	DET
easat-4026	291	32	context	context	NOUN
easat-4026	291	33	of	of	ADP
easat-4026	291	34	extensions	extension	NOUN
easat-4026	291	35	of	of	ADP
easat-4026	291	36	𝒜∞-algebras	𝒜∞-algebras	NUM
easat-4026	291	37	,	,	PUNCT
easat-4026	291	38	focusing	focus	VERB
easat-4026	291	39	on	on	ADP
easat-4026	291	40	modules	module	NOUN
easat-4026	291	41	and	and	CCONJ
easat-4026	291	42	the	the	DET
easat-4026	291	43	conditions	condition	NOUN
easat-4026	291	44	for	for	ADP
easat-4026	291	45	ℋ-unitarity	ℋ-unitarity	PROPN
easat-4026	291	46	.	.	PUNCT
easat-4026	292	1	3.8	3.8	NUM
easat-4026	292	2	.	.	PUNCT
easat-4026	292	3	corollary	corollary	NOUN
easat-4026	292	4	suppose	suppose	VERB
easat-4026	292	5	that	that	SCONJ
easat-4026	292	6	0	0	NUM
easat-4026	292	7	→	→	SYM
easat-4026	292	8	ℐ	ℐ	PROPN
easat-4026	292	9	→	→	SYM
easat-4026	292	10	𝒜	𝒜	NOUN
easat-4026	292	11	→	→	SYM
easat-4026	292	12	ℬ	ℬ	NOUN
easat-4026	292	13	→	→	SYM
easat-4026	292	14	0	0	NUM
easat-4026	292	15	is	be	AUX
easat-4026	292	16	an	an	DET
easat-4026	292	17	extension	extension	NOUN
easat-4026	292	18	of	of	ADP
easat-4026	292	19	the	the	DET
easat-4026	292	20	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	292	21	where	where	SCONJ
easat-4026	292	22	ℐ	ℐ	PRON
easat-4026	292	23	⊂	⊂	X
easat-4026	292	24	𝒜	𝒜	NOUN
easat-4026	292	25	⊂	⊂	PROPN
easat-4026	292	26	ℬ	ℬ	PROPN
easat-4026	292	27	and	and	CCONJ
easat-4026	292	28	𝒢	𝒢	PROPN
easat-4026	292	29	is	be	AUX
easat-4026	292	30	a	a	DET
easat-4026	292	31	𝑘-module	𝑘-module	NOUN
easat-4026	292	32	and	and	CCONJ
easat-4026	292	33	use	use	NOUN
easat-4026	292	34	[	[	X
easat-4026	292	35	34	34	NUM
easat-4026	292	36	]	]	PUNCT
easat-4026	292	37	.	.	PUNCT
easat-4026	293	1	the	the	DET
easat-4026	293	2	canonical	canonical	ADJ
easat-4026	293	3	arrows	arrow	NOUN
easat-4026	293	4	:	:	PUNCT
easat-4026	293	5	𝜋	𝜋	NOUN
easat-4026	293	6	:	:	PUNCT
easat-4026	293	7	(	(	PUNCT
easat-4026	293	8	ℬ	ℬ	NOUN
easat-4026	293	9	⊗𝒜⊗∗	⊗𝒜⊗∗	NUM
easat-4026	293	10	,	,	PUNCT
easat-4026	293	11	𝜌∗)⊗	𝜌∗)⊗	PUNCT
easat-4026	293	12	𝒢	𝒢	NOUN
easat-4026	293	13	→	→	SYM
easat-4026	293	14	(	(	PUNCT
easat-4026	293	15	ℬ	ℬ	NOUN
easat-4026	293	16	⊗ℬ⊗∗	⊗ℬ⊗∗	NOUN
easat-4026	293	17	,	,	PUNCT
easat-4026	293	18	𝜌∗)⊗	𝜌∗)⊗	PUNCT
easat-4026	293	19	𝒢	𝒢	NOUN
easat-4026	293	20	,	,	PUNCT
easat-4026	293	21	𝜋′	𝜋′	NUM
easat-4026	293	22	:	:	PUNCT
easat-4026	293	23	(	(	PUNCT
easat-4026	293	24	ℬ	ℬ	NOUN
easat-4026	293	25	⊗𝒜⊗∗	⊗𝒜⊗∗	NUM
easat-4026	293	26	,	,	PUNCT
easat-4026	293	27	𝜌′∗)⊗	𝜌′∗)⊗	PROPN
easat-4026	293	28	𝒢	𝒢	PROPN
easat-4026	293	29	→	→	SYM
easat-4026	293	30	(	(	PUNCT
easat-4026	293	31	ℬ	ℬ	NOUN
easat-4026	293	32	⊗ℬ⊗∗	⊗ℬ⊗∗	NOUN
easat-4026	293	33	,	,	PUNCT
easat-4026	293	34	𝜌′∗)⊗	𝜌′∗)⊗	PROPN
easat-4026	293	35	𝒢	𝒢	PROPN
easat-4026	293	36	,	,	PUNCT
easat-4026	293	37	are	be	AUX
easat-4026	293	38	quasi	quasi	NOUN
easat-4026	293	39	-	-	NOUN
easat-4026	293	40	isomorphisms	isomorphism	NOUN
easat-4026	293	41	when	when	SCONJ
easat-4026	293	42	ℐ	ℐ	PRON
easat-4026	293	43	is	be	AUX
easat-4026	293	44	an	an	DET
easat-4026	293	45	ℋ-unital	ℋ-unital	PROPN
easat-4026	293	46	.	.	PUNCT
easat-4026	294	1	proof	proof	NOUN
easat-4026	294	2	:	:	PUNCT
easat-4026	294	3	we	we	PRON
easat-4026	294	4	must	must	AUX
easat-4026	294	5	take	take	VERB
easat-4026	294	6	into	into	ADP
easat-4026	294	7	consideration	consideration	NOUN
easat-4026	294	8	for	for	ADP
easat-4026	294	9	all	all	DET
easat-4026	294	10	ℓ	ℓ	PROPN
easat-4026	294	11	≥	≥	NOUN
easat-4026	294	12	0	0	NUM
easat-4026	294	13	,	,	PUNCT
easat-4026	294	14	the	the	DET
easat-4026	294	15	quotient	quotient	NOUN
easat-4026	294	16	complex	complex	PROPN
easat-4026	294	17	�	�	PROPN
easat-4026	294	18	̃	̃	PROPN
easat-4026	294	19	�	�	NOUN
easat-4026	294	20	ℓ	ℓ	NOUN
easat-4026	294	21	for	for	ADP
easat-4026	294	22	(	(	PUNCT
easat-4026	294	23	ℬ	ℬ	PROPN
easat-4026	294	24	⊗𝒜⊗∗	⊗𝒜⊗∗	NUM
easat-4026	294	25	,	,	PUNCT
easat-4026	294	26	𝜌∗	𝜌∗	NOUN
easat-4026	294	27	)	)	PUNCT
easat-4026	294	28	provided	provide	VERB
easat-4026	294	29	by	by	ADP
easat-4026	294	30	:	:	PUNCT
easat-4026	294	31	�	�	PROPN
easat-4026	294	32	̃	̃	PROPN
easat-4026	294	33	�	�	PROPN
easat-4026	294	34	ℓ	ℓ	NOUN
easat-4026	294	35	≔	≔	NOUN
easat-4026	294	36	ℬ	ℬ	NOUN
easat-4026	294	37	𝜌1	𝜌1	NOUN
easat-4026	294	38	←	←	PROPN
easat-4026	294	39	ℬ	ℬ	PROPN
easat-4026	294	40	⊗ℬ	⊗ℬ	NUM
easat-4026	294	41	𝜌2	𝜌2	ADJ
easat-4026	294	42	←	←	PROPN
easat-4026	294	43	ℬ	ℬ	NOUN
easat-4026	294	44	⊗ℬ⊗2	⊗ℬ⊗2	VERB
easat-4026	294	45	𝜌3	𝜌3	NOUN
easat-4026	294	46	←	←	PROPN
easat-4026	294	47	…	…	PUNCT
easat-4026	294	48	𝜌ℓ	𝜌ℓ	NOUN
easat-4026	294	49	←	←	PROPN
easat-4026	294	50	ℬ	ℬ	PROPN
easat-4026	294	51	⊗ℬ⊗ℓ	⊗ℬ⊗ℓ	PROPN
easat-4026	294	52	𝜌ℓ+1	𝜌ℓ+1	NUM
easat-4026	294	53	←	←	PROPN
easat-4026	294	54	ℬ	ℬ	PROPN
easat-4026	294	55	⊗ℬ⊗ℓ	⊗ℬ⊗ℓ	PROPN
easat-4026	294	56	⊗𝒜	⊗𝒜	ADP
easat-4026	294	57	𝜌ℓ+2	𝜌ℓ+2	PROPN
easat-4026	294	58	←	←	PROPN
easat-4026	294	59	ℬ	ℬ	PROPN
easat-4026	294	60	⊗	⊗	PROPN
easat-4026	294	61	ℬ⊗ℓ	ℬ⊗ℓ	NUM
easat-4026	294	62	⊗𝒜⊗2	⊗𝒜⊗2	PROPN
easat-4026	294	63	𝜌ℓ+3	𝜌ℓ+3	NUM
easat-4026	294	64	←	←	NOUN
easat-4026	294	65	…	…	PUNCT
easat-4026	294	66	.	.	PUNCT
easat-4026	295	1	to	to	PART
easat-4026	295	2	demonstrate	demonstrate	VERB
easat-4026	295	3	whether	whether	SCONJ
easat-4026	295	4	𝜋	𝜋	PRON
easat-4026	295	5	is	be	AUX
easat-4026	295	6	a	a	DET
easat-4026	295	7	quasi	quasi	NOUN
easat-4026	295	8	-	-	NOUN
easat-4026	295	9	isomorphism	isomorphism	ADJ
easat-4026	295	10	,	,	PUNCT
easat-4026	295	11	consider	consider	VERB
easat-4026	295	12	that	that	SCONJ
easat-4026	295	13	the	the	DET
easat-4026	295	14	canonical	canonical	ADJ
easat-4026	295	15	projections	projection	NOUN
easat-4026	295	16	𝜋ℓ	𝜋ℓ	VERB
easat-4026	295	17	:	:	PUNCT
easat-4026	295	18	�	�	PROPN
easat-4026	295	19	̃	̃	PROPN
easat-4026	295	20	�	�	PROPN
easat-4026	295	21	ℓ	ℓ	PROPN
easat-4026	295	22	⊗𝒢	⊗𝒢	PROPN
easat-4026	295	23	→	→	SYM
easat-4026	295	24	�	�	PROPN
easat-4026	295	25	̃	̃	PROPN
easat-4026	295	26	�	�	PROPN
easat-4026	295	27	ℓ+1⊗𝒢.	ℓ+1⊗𝒢.	VERB
easat-4026	295	28	given	give	VERB
easat-4026	295	29	that	that	PRON
easat-4026	295	30	ℬ(ℓ	ℬ(ℓ	NOUN
easat-4026	295	31	)	)	PUNCT
easat-4026	295	32	=	=	SYM
easat-4026	296	1	ℬ	ℬ	SYM
easat-4026	296	2	⊗ℬ⊗ℓ	⊗ℬ⊗ℓ	PROPN
easat-4026	296	3	⊗	⊗	PROPN
easat-4026	296	4	ℐ	ℐ	PROPN
easat-4026	296	5	,	,	PUNCT
easat-4026	296	6	the	the	DET
easat-4026	296	7	straightforward	straightforward	ADJ
easat-4026	296	8	calculation	calculation	NOUN
easat-4026	296	9	reveals	reveal	VERB
easat-4026	296	10	that	that	SCONJ
easat-4026	296	11	:	:	PUNCT
easat-4026	296	12	𝐾𝑒𝑟(𝜋ℓ	𝐾𝑒𝑟(𝜋ℓ	PROPN
easat-4026	296	13	)	)	PUNCT
easat-4026	296	14	=	=	PUNCT
easat-4026	296	15	(	(	PUNCT
easat-4026	296	16	ℬ(ℓ)⊗𝒜⊗∗−ℓ−1	ℬ(ℓ)⊗𝒜⊗∗−ℓ−1	X
easat-4026	296	17	,	,	PUNCT
easat-4026	296	18	𝜌∗	𝜌∗	NOUN
easat-4026	296	19	)	)	PUNCT
easat-4026	296	20	⊗	⊗	PROPN
easat-4026	296	21	𝒢.	𝒢.	PROPN
easat-4026	296	22	therefore	therefore	ADV
easat-4026	296	23	,	,	PUNCT
easat-4026	296	24	according	accord	VERB
easat-4026	296	25	to	to	ADP
easat-4026	296	26	theorem	theorem	NOUN
easat-4026	296	27	(	(	PUNCT
easat-4026	296	28	3.7	3.7	NUM
easat-4026	296	29	)	)	PUNCT
easat-4026	296	30	,	,	PUNCT
easat-4026	296	31	𝐾𝑒𝑟(𝜋ℓ	𝐾𝑒𝑟(𝜋ℓ	PROPN
easat-4026	296	32	)	)	PUNCT
easat-4026	296	33	is	be	AUX
easat-4026	296	34	quasi	quasi	ADJ
easat-4026	296	35	-	-	ADJ
easat-4026	296	36	isomorphic	isomorphic	ADJ
easat-4026	296	37	to	to	ADP
easat-4026	296	38	:	:	PUNCT
easat-4026	296	39	(	(	PUNCT
easat-4026	296	40	ℬ(ℓ)⊗	ℬ(ℓ)⊗	NUM
easat-4026	296	41	ℐ⊗∗−ℓ−1	ℐ⊗∗−ℓ−1	PROPN
easat-4026	296	42	,	,	PUNCT
easat-4026	296	43	𝜌∗	𝜌∗	NOUN
easat-4026	296	44	)	)	PUNCT
easat-4026	297	1	⊗	⊗	NOUN
easat-4026	298	1	𝒢	𝒢	NOUN
easat-4026	298	2	=	=	SYM
easat-4026	298	3	(	(	PUNCT
easat-4026	298	4	ℬ(ℓ)⊗	ℬ(ℓ)⊗	NUM
easat-4026	298	5	ℐ⊗∗−ℓ−1	ℐ⊗∗−ℓ−1	PROPN
easat-4026	298	6	,	,	PUNCT
easat-4026	298	7	𝜌∗	𝜌∗	NOUN
easat-4026	298	8	′	′	NOUN
easat-4026	298	9	)	)	PUNCT
easat-4026	298	10	⊗	⊗	PROPN
easat-4026	298	11	𝒢	𝒢	PROPN
easat-4026	298	12	,	,	PUNCT
easat-4026	298	13	(	(	PUNCT
easat-4026	298	14	13	13	NUM
easat-4026	298	15	)	)	PUNCT
easat-4026	298	16	which	which	PRON
easat-4026	298	17	is	be	AUX
easat-4026	298	18	exact	exact	ADJ
easat-4026	298	19	by	by	ADP
easat-4026	298	20	assumption	assumption	NOUN
easat-4026	298	21	.	.	PUNCT
easat-4026	299	1	for	for	ADP
easat-4026	299	2	𝜋′	𝜋′	NUM
easat-4026	299	3	,	,	PUNCT
easat-4026	299	4	a	a	DET
easat-4026	299	5	similar	similar	ADJ
easat-4026	299	6	proof	proof	NOUN
easat-4026	299	7	applies	apply	VERB
easat-4026	299	8	.	.	PUNCT
easat-4026	300	1	now	now	ADV
easat-4026	300	2	,	,	PUNCT
easat-4026	300	3	we	we	PRON
easat-4026	300	4	conclude	conclude	VERB
easat-4026	300	5	with	with	ADP
easat-4026	300	6	a	a	DET
easat-4026	300	7	theorem	theorem	NOUN
easat-4026	300	8	establishing	establish	VERB
easat-4026	300	9	the	the	DET
easat-4026	300	10	equivalence	equivalence	NOUN
easat-4026	300	11	of	of	ADP
easat-4026	300	12	various	various	ADJ
easat-4026	300	13	conditions	condition	NOUN
easat-4026	300	14	related	relate	VERB
easat-4026	300	15	to	to	ADP
easat-4026	300	16	ℋunitarity	ℋunitarity	PROPN
easat-4026	300	17	,	,	PUNCT
easat-4026	300	18	excision	excision	NOUN
easat-4026	300	19	,	,	PUNCT
easat-4026	300	20	and	and	CCONJ
easat-4026	300	21	homology	homology	NOUN
easat-4026	300	22	requirements	requirement	NOUN
easat-4026	300	23	for	for	ADP
easat-4026	300	24	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	300	25	,	,	PUNCT
easat-4026	300	26	providing	provide	VERB
easat-4026	300	27	a	a	DET
easat-4026	300	28	comprehensive	comprehensive	ADJ
easat-4026	300	29	view	view	NOUN
easat-4026	300	30	of	of	ADP
easat-4026	300	31	their	their	PRON
easat-4026	300	32	interrelations	interrelation	NOUN
easat-4026	300	33	.	.	PUNCT
easat-4026	301	1	3.9	3.9	NUM
easat-4026	301	2	.	.	PUNCT
easat-4026	301	3	theorem	theorem	NOUN
easat-4026	301	4	let	let	VERB
easat-4026	301	5	the	the	DET
easat-4026	301	6	assumption	assumption	NOUN
easat-4026	301	7	that	that	SCONJ
easat-4026	301	8	ℐ	ℐ	PRON
easat-4026	301	9	is	be	AUX
easat-4026	301	10	a	a	DET
easat-4026	301	11	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	301	12	,	,	PUNCT
easat-4026	301	13	and	and	CCONJ
easat-4026	301	14	then	then	ADV
easat-4026	301	15	the	the	DET
easat-4026	301	16	next	next	ADJ
easat-4026	301	17	propositions	proposition	NOUN
easat-4026	301	18	are	be	AUX
easat-4026	301	19	equivalent	equivalent	ADJ
easat-4026	301	20	:	:	PUNCT
easat-4026	301	21	(	(	PUNCT
easat-4026	301	22	1	1	X
easat-4026	301	23	)	)	PUNCT
easat-4026	301	24	the	the	PRON
easat-4026	301	25	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	301	26	ℐ	ℐ	PRON
easat-4026	301	27	remains	remain	VERB
easat-4026	301	28	ℋ-unital	ℋ-unital	PROPN
easat-4026	301	29	.	.	PUNCT
easat-4026	302	1	(	(	PUNCT
easat-4026	302	2	2	2	X
easat-4026	302	3	)	)	PUNCT
easat-4026	302	4	the	the	DET
easat-4026	302	5	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	302	6	ℐ	ℐ	PRON
easat-4026	302	7	fulfills	fulfill	VERB
easat-4026	302	8	the	the	DET
easat-4026	302	9	𝐻-homology	𝐻-homology	PROPN
easat-4026	302	10	excision	excision	NOUN
easat-4026	302	11	.	.	PUNCT
easat-4026	303	1	(	(	PUNCT
easat-4026	303	2	3	3	X
easat-4026	303	3	)	)	PUNCT
easat-4026	303	4	the	the	DET
easat-4026	303	5	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	303	6	ℐ	ℐ	PRON
easat-4026	303	7	fulfills	fulfill	VERB
easat-4026	303	8	the	the	DET
easat-4026	303	9	excision	excision	NOUN
easat-4026	303	10	requirement	requirement	NOUN
easat-4026	303	11	of	of	ADP
easat-4026	303	12	bar	bar	NOUN
easat-4026	303	13	homology	homology	NOUN
easat-4026	303	14	.	.	PUNCT
easat-4026	304	1	(	(	PUNCT
easat-4026	304	2	4	4	X
easat-4026	304	3	)	)	PUNCT
easat-4026	304	4	the	the	DET
easat-4026	304	5	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	304	6	ℐ	ℐ	PRON
easat-4026	304	7	fulfills	fulfill	VERB
easat-4026	304	8	the	the	DET
easat-4026	304	9	excision	excision	NOUN
easat-4026	304	10	requirement	requirement	NOUN
easat-4026	304	11	of	of	ADP
easat-4026	304	12	simplicial	simplicial	ADJ
easat-4026	304	13	homology	homology	NOUN
easat-4026	304	14	.	.	PUNCT
easat-4026	305	1	proof	proof	NOUN
easat-4026	305	2	:	:	PUNCT
easat-4026	305	3	the	the	DET
easat-4026	305	4	proposition	proposition	NOUN
easat-4026	305	5	(	(	PUNCT
easat-4026	305	6	1	1	X
easat-4026	305	7	)	)	PUNCT
easat-4026	305	8	is	be	AUX
easat-4026	305	9	equivalent	equivalent	ADJ
easat-4026	305	10	to	to	ADP
easat-4026	305	11	(	(	PUNCT
easat-4026	305	12	2	2	NUM
easat-4026	305	13	)	)	PUNCT
easat-4026	305	14	when	when	SCONJ
easat-4026	305	15	we	we	PRON
easat-4026	305	16	let	let	VERB
easat-4026	305	17	0	0	NUM
easat-4026	305	18	→	→	SYM
easat-4026	305	19	ℐ	ℐ	PROPN
easat-4026	305	20	→	→	SYM
easat-4026	305	21	𝒜	𝒜	NOUN
easat-4026	305	22	→	→	SYM
easat-4026	305	23	ℬ	ℬ	NOUN
easat-4026	305	24	→	→	SYM
easat-4026	305	25	0	0	NUM
easat-4026	305	26	be	be	AUX
easat-4026	305	27	an	an	DET
easat-4026	305	28	𝒜∞algebras	𝒜∞algebras	NOUN
easat-4026	305	29	as	as	ADP
easat-4026	305	30	a	a	DET
easat-4026	305	31	pure	pure	ADJ
easat-4026	305	32	extension	extension	NOUN
easat-4026	305	33	and	and	CCONJ
easat-4026	305	34	𝒢	𝒢	NOUN
easat-4026	305	35	be	be	VERB
easat-4026	305	36	a	a	DET
easat-4026	305	37	𝑘-module	𝑘-module	NOUN
easat-4026	305	38	,	,	PUNCT
easat-4026	305	39	and	and	CCONJ
easat-4026	305	40	the	the	DET
easat-4026	305	41	canonical	canonical	ADJ
easat-4026	305	42	projection	projection	NOUN
easat-4026	305	43	given	give	VERB
easat-4026	305	44	by	by	ADP
easat-4026	305	45	:	:	PUNCT
easat-4026	305	46	𝜋	𝜋	NOUN
easat-4026	305	47	:	:	PUNCT
easat-4026	305	48	(	(	PUNCT
easat-4026	305	49	𝒜	𝒜	NOUN
easat-4026	305	50	⊗𝒜⨂∗	⊗𝒜⨂∗	NOUN
easat-4026	305	51	,	,	PUNCT
easat-4026	305	52	𝜌∗	𝜌∗	NOUN
easat-4026	305	53	)	)	PUNCT
easat-4026	305	54	⊗	⊗	PROPN
easat-4026	305	55	𝒢	𝒢	NOUN
easat-4026	305	56	→	→	SYM
easat-4026	305	57	(	(	PUNCT
easat-4026	305	58	ℬ	ℬ	NOUN
easat-4026	305	59	⊗ℬ⨂∗	⊗ℬ⨂∗	NOUN
easat-4026	305	60	,	,	PUNCT
easat-4026	305	61	𝜌∗	𝜌∗	NOUN
easat-4026	305	62	)	)	PUNCT
easat-4026	305	63	⊗	⊗	PROPN
easat-4026	305	64	𝒢.	𝒢.	PROPN
easat-4026	305	65	suppose	suppose	VERB
easat-4026	305	66	that	that	SCONJ
easat-4026	305	67	the	the	DET
easat-4026	305	68	commutation	commutation	NOUN
easat-4026	305	69	diagram	diagram	NOUN
easat-4026	305	70	of	of	ADP
easat-4026	305	71	short	short	ADJ
easat-4026	305	72	exact	exact	ADJ
easat-4026	305	73	sequences	sequence	NOUN
easat-4026	305	74	is	be	AUX
easat-4026	305	75	as	as	SCONJ
easat-4026	305	76	follows	follow	VERB
easat-4026	305	77	:	:	PUNCT
easat-4026	305	78	0	0	NUM
easat-4026	305	79	0	0	NUM
easat-4026	305	80	→	→	SYM
easat-4026	305	81	→	→	X
easat-4026	305	82	(	(	PUNCT
easat-4026	305	83	ℐ	ℐ	PRON
easat-4026	305	84	⊗𝒜⨂	⊗𝒜⨂	PROPN
easat-4026	305	85	∗	∗	NOUN
easat-4026	305	86	,	,	PUNCT
easat-4026	305	87	𝜌∗	𝜌∗	NOUN
easat-4026	305	88	)	)	PUNCT
easat-4026	306	1	⊗	⊗	PROPN
easat-4026	306	2	𝒢	𝒢	PROPN
easat-4026	306	3	|𝑗	|𝑗	NOUN
easat-4026	306	4	𝑘𝑒𝑟	𝑘𝑒𝑟	NOUN
easat-4026	306	5	(	(	PUNCT
easat-4026	306	6	𝜋	𝜋	NOUN
easat-4026	306	7	)	)	PUNCT
easat-4026	306	8	→	→	SYM
easat-4026	306	9	→	→	SYM
easat-4026	306	10	(	(	PUNCT
easat-4026	306	11	𝒜	𝒜	NOUN
easat-4026	306	12	⊗𝒜⨂	⊗𝒜⨂	PROPN
easat-4026	306	13	∗	∗	NOUN
easat-4026	306	14	,	,	PUNCT
easat-4026	306	15	𝜌∗	𝜌∗	NOUN
easat-4026	306	16	)	)	PUNCT
easat-4026	306	17	⊗	⊗	NOUN
easat-4026	307	1	𝒢	𝒢	NOUN
easat-4026	307	2	|	|	NOUN
easat-4026	307	3	=	=	SYM
easat-4026	307	4	(	(	PUNCT
easat-4026	307	5	𝒜	𝒜	NOUN
easat-4026	307	6	⊗𝒜⨂	⊗𝒜⨂	PROPN
easat-4026	307	7	∗	∗	NOUN
easat-4026	307	8	,	,	PUNCT
easat-4026	307	9	𝜌∗	𝜌∗	NOUN
easat-4026	307	10	)	)	PUNCT
easat-4026	307	11	⊗	⊗	PROPN
easat-4026	307	12	𝒢	𝒢	PROPN
easat-4026	307	13	→	→	SYM
easat-4026	307	14	𝜋	𝜋	X
easat-4026	307	15	→	→	PUNCT
easat-4026	307	16	(	(	PUNCT
easat-4026	307	17	ℬ	ℬ	SYM
easat-4026	307	18	⊗𝒜⨂	⊗𝒜⨂	PROPN
easat-4026	307	19	∗	∗	NOUN
easat-4026	307	20	,	,	PUNCT
easat-4026	307	21	𝜌∗	𝜌∗	NOUN
easat-4026	307	22	)	)	PUNCT
easat-4026	307	23	⊗	⊗	NOUN
easat-4026	308	1	𝒢	𝒢	NOUN
easat-4026	308	2	|𝜋1	|𝜋1	NOUN
easat-4026	308	3	(	(	PUNCT
easat-4026	308	4	ℬ	ℬ	PROPN
easat-4026	308	5	⊗𝒜⨂∗	⊗𝒜⨂∗	NOUN
easat-4026	308	6	,	,	PUNCT
easat-4026	308	7	𝜌∗	𝜌∗	NOUN
easat-4026	308	8	)	)	PUNCT
easat-4026	308	9	⊗	⊗	PROPN
easat-4026	308	10	𝒢	𝒢	NOUN
easat-4026	308	11	→	→	SYM
easat-4026	308	12	→	→	SYM
easat-4026	308	13	0	0	NUM
easat-4026	308	14	0	0	NUM
easat-4026	308	15	according	accord	VERB
easat-4026	308	16	to	to	ADP
easat-4026	308	17	corollary	corollary	ADJ
easat-4026	308	18	(	(	PUNCT
easat-4026	308	19	3.8	3.8	NUM
easat-4026	308	20	)	)	PUNCT
easat-4026	308	21	,	,	PUNCT
easat-4026	308	22	𝜋1	𝜋1	NOUN
easat-4026	308	23	indicates	indicate	VERB
easat-4026	308	24	a	a	DET
easat-4026	308	25	quasi	quasi	NOUN
easat-4026	308	26	-	-	NOUN
easat-4026	308	27	isomorphism	isomorphism	NOUN
easat-4026	308	28	.	.	PUNCT
easat-4026	309	1	as	as	ADP
easat-4026	309	2	a	a	DET
easat-4026	309	3	result	result	NOUN
easat-4026	309	4	,	,	PUNCT
easat-4026	309	5	𝑗	𝑗	X
easat-4026	309	6	is	be	AUX
easat-4026	309	7	as	as	ADV
easat-4026	309	8	well	well	ADV
easat-4026	309	9	.	.	PUNCT
easat-4026	310	1	we	we	PRON
easat-4026	310	2	continue	continue	VERB
easat-4026	310	3	the	the	DET
easat-4026	310	4	proof	proof	NOUN
easat-4026	310	5	using	use	VERB
easat-4026	310	6	the	the	DET
easat-4026	310	7	theorem	theorem	NOUN
easat-4026	310	8	(	(	PUNCT
easat-4026	310	9	3.7	3.7	NUM
easat-4026	310	10	)	)	PUNCT
easat-4026	310	11	.	.	PUNCT
easat-4026	311	1	the	the	DET
easat-4026	311	2	proposition	proposition	NOUN
easat-4026	311	3	(	(	PUNCT
easat-4026	311	4	1	1	X
easat-4026	311	5	)	)	PUNCT
easat-4026	311	6	is	be	AUX
easat-4026	311	7	equivalent	equivalent	ADJ
easat-4026	311	8	to	to	ADP
easat-4026	311	9	(	(	PUNCT
easat-4026	311	10	3	3	NUM
easat-4026	311	11	):	):	PUNCT
easat-4026	311	12	comparable	comparable	ADJ
easat-4026	311	13	to	to	ADP
easat-4026	311	14	(	(	PUNCT
easat-4026	311	15	1	1	NUM
easat-4026	311	16	)	)	PUNCT
easat-4026	311	17	⟹	⟹	NOUN
easat-4026	312	1	(	(	PUNCT
easat-4026	312	2	2	2	NUM
easat-4026	312	3	)	)	PUNCT
easat-4026	312	4	.	.	PUNCT
easat-4026	313	1	the	the	DET
easat-4026	313	2	proposition	proposition	NOUN
easat-4026	313	3	(	(	PUNCT
easat-4026	313	4	2	2	NUM
easat-4026	313	5	)	)	PUNCT
easat-4026	313	6	and	and	CCONJ
easat-4026	313	7	(	(	PUNCT
easat-4026	313	8	3	3	X
easat-4026	313	9	)	)	PUNCT
easat-4026	313	10	is	be	AUX
easat-4026	313	11	equivalent	equivalent	ADJ
easat-4026	313	12	to	to	ADP
easat-4026	313	13	(	(	PUNCT
easat-4026	313	14	4	4	NUM
easat-4026	313	15	):	):	PUNCT
easat-4026	313	16	the	the	DET
easat-4026	313	17	long	long	ADJ
easat-4026	313	18	exact	exact	ADJ
easat-4026	313	19	sequence	sequence	NOUN
easat-4026	313	20	:	:	PUNCT
easat-4026	313	21	⋯	⋯	PROPN
easat-4026	313	22	←	←	PROPN
easat-4026	313	23	𝐻𝑛−1(ℐ	𝐻𝑛−1(ℐ	PROPN
easat-4026	313	24	)	)	PUNCT
easat-4026	314	1	←	←	PROPN
easat-4026	314	2	ℋ𝐵𝑛−1(ℐ	ℋ𝐵𝑛−1(ℐ	NUM
easat-4026	314	3	)	)	PUNCT
easat-4026	314	4	←	←	PROPN
easat-4026	314	5	ℋℋ𝑛(ℐ	ℋℋ𝑛(ℐ	NOUN
easat-4026	314	6	)	)	PUNCT
easat-4026	314	7	←	←	PROPN
easat-4026	314	8	𝐻𝑛(ℐ	𝐻𝑛(ℐ	PROPN
easat-4026	314	9	)	)	PUNCT
easat-4026	314	10	←	←	PROPN
easat-4026	314	11	ℋ𝐵𝑛(ℐ	ℋ𝐵𝑛(ℐ	PROPN
easat-4026	314	12	)	)	PUNCT
easat-4026	314	13	←	←	PROPN
easat-4026	314	14	ℋℋ𝑛+1(ℐ	ℋℋ𝑛+1(ℐ	PROPN
easat-4026	314	15	)	)	PUNCT
easat-4026	314	16	←	←	PROPN
easat-4026	314	17	⋯	⋯	PROPN
easat-4026	314	18	gives	give	VERB
easat-4026	314	19	this	this	DET
easat-4026	314	20	simple	simple	ADJ
easat-4026	314	21	consequence	consequence	NOUN
easat-4026	314	22	.	.	PUNCT
easat-4026	315	1	the	the	DET
easat-4026	315	2	proposition	proposition	NOUN
easat-4026	315	3	(	(	PUNCT
easat-4026	315	4	2	2	X
easat-4026	315	5	)	)	PUNCT
easat-4026	315	6	is	be	AUX
easat-4026	315	7	equivalent	equivalent	ADJ
easat-4026	315	8	to	to	ADP
easat-4026	315	9	(	(	PUNCT
easat-4026	315	10	1	1	NUM
easat-4026	315	11	):	):	PUNCT
easat-4026	315	12	assume	assume	VERB
easat-4026	315	13	that	that	SCONJ
easat-4026	315	14	𝑘-algebra	𝑘-algebra	PROPN
easat-4026	315	15	𝒜	𝒜	NOUN
easat-4026	315	16	=	=	SYM
easat-4026	315	17	ℐ	ℐ	PROPN
easat-4026	315	18	⊕	⊕	NOUN
easat-4026	315	19	𝒢	𝒢	NOUN
easat-4026	315	20	with	with	ADP
easat-4026	315	21	𝑘-module	𝑘-module	NOUN
easat-4026	315	22	𝒢	𝒢	NOUN
easat-4026	315	23	in	in	ADP
easat-4026	315	24	addition	addition	NOUN
easat-4026	315	25	to	to	ADP
easat-4026	315	26	the	the	DET
easat-4026	315	27	canonical	canonical	ADJ
easat-4026	315	28	projection	projection	NOUN
easat-4026	315	29	:	:	PUNCT
easat-4026	315	30	𝜋	𝜋	NOUN
easat-4026	315	31	:	:	PUNCT
easat-4026	315	32	(	(	PUNCT
easat-4026	315	33	𝒜	𝒜	NOUN
easat-4026	315	34	⊗𝒜⨂	⊗𝒜⨂	PROPN
easat-4026	315	35	∗	∗	NOUN
easat-4026	315	36	,	,	PUNCT
easat-4026	315	37	𝜌∗	𝜌∗	NOUN
easat-4026	315	38	)	)	PUNCT
easat-4026	315	39	→	→	SYM
easat-4026	315	40	(	(	PUNCT
easat-4026	315	41	𝒢	𝒢	PROPN
easat-4026	315	42	⊗	⊗	PROPN
easat-4026	315	43	𝒢⨂∗	𝒢⨂∗	NOUN
easat-4026	315	44	,	,	PUNCT
easat-4026	315	45	𝜌∗	𝜌∗	NOUN
easat-4026	315	46	)	)	PUNCT
easat-4026	315	47	,	,	PUNCT
easat-4026	315	48	9485	9485	NUM
easat-4026	315	49	edelweiss	edelweiss	PROPN
easat-4026	315	50	applied	apply	VERB
easat-4026	315	51	science	science	NOUN
easat-4026	315	52	and	and	CCONJ
easat-4026	315	53	technology	technology	NOUN
easat-4026	315	54	issn	issn	PROPN
easat-4026	315	55	:	:	PUNCT
easat-4026	315	56	2576	2576	NUM
easat-4026	315	57	-	-	SYM
easat-4026	315	58	8484	8484	NUM
easat-4026	315	59	vol	vol	NOUN
easat-4026	315	60	.	.	PROPN
easat-4026	315	61	8	8	NUM
easat-4026	315	62	,	,	PUNCT
easat-4026	315	63	no	no	INTJ
easat-4026	315	64	.	.	NOUN
easat-4026	315	65	6	6	NUM
easat-4026	315	66	:	:	SYM
easat-4026	315	67	9472	9472	NUM
easat-4026	315	68	-	-	SYM
easat-4026	315	69	9486	9486	NUM
easat-4026	315	70	,	,	PUNCT
easat-4026	315	71	2024	2024	NUM
easat-4026	315	72	doi	doi	NOUN
easat-4026	315	73	:	:	PUNCT
easat-4026	315	74	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	315	75	©	©	PROPN
easat-4026	315	76	2024	2024	NUM
easat-4026	315	77	by	by	ADP
easat-4026	315	78	the	the	DET
easat-4026	315	79	authors	author	NOUN
easat-4026	315	80	;	;	PUNCT
easat-4026	315	81	licensee	licensee	PROPN
easat-4026	315	82	learning	learning	NOUN
easat-4026	315	83	gate	gate	NOUN
easat-4026	315	84	where	where	SCONJ
easat-4026	315	85	the	the	DET
easat-4026	315	86	product	product	NOUN
easat-4026	315	87	is	be	AUX
easat-4026	315	88	provided	provide	VERB
easat-4026	315	89	by	by	ADP
easat-4026	315	90	(	(	PUNCT
easat-4026	315	91	𝑢	𝑢	X
easat-4026	315	92	,	,	PUNCT
easat-4026	315	93	𝑣)(𝑢′	𝑣)(𝑢′	NUM
easat-4026	315	94	,	,	PUNCT
easat-4026	315	95	𝑣′	𝑣′	PROPN
easat-4026	315	96	)	)	PUNCT
easat-4026	316	1	=	=	SYM
easat-4026	316	2	(	(	PUNCT
easat-4026	316	3	𝑢𝑢′	𝑢𝑢′	X
easat-4026	316	4	,	,	PUNCT
easat-4026	316	5	0	0	NUM
easat-4026	316	6	)	)	PUNCT
easat-4026	316	7	.	.	PUNCT
easat-4026	317	1	given	give	VERB
easat-4026	317	2	that	that	DET
easat-4026	317	3	𝑘𝑒𝑟	𝑘𝑒𝑟	NOUN
easat-4026	317	4	(	(	PUNCT
easat-4026	317	5	𝜋	𝜋	NOUN
easat-4026	317	6	)	)	PUNCT
easat-4026	317	7	is	be	AUX
easat-4026	317	8	a	a	DET
easat-4026	317	9	direct	direct	ADJ
easat-4026	317	10	summation	summation	NOUN
easat-4026	317	11	of	of	ADP
easat-4026	317	12	the	the	DET
easat-4026	317	13	complex	complex	ADJ
easat-4026	317	14	𝒢	𝒢	PROPN
easat-4026	317	15	⊗	⊗	PROPN
easat-4026	317	16	(	(	PUNCT
easat-4026	317	17	ℐ	ℐ	ADV
easat-4026	317	18	⊗	⊗	PROPN
easat-4026	317	19	ℐ⨂	ℐ⨂	VERB
easat-4026	317	20	∗−1	∗−1	NOUN
easat-4026	317	21	,	,	PUNCT
easat-4026	317	22	𝜌∗	𝜌∗	NOUN
easat-4026	317	23	′	′	NOUN
easat-4026	317	24	)	)	PUNCT
easat-4026	317	25	⊕	⊕	PROPN
easat-4026	317	26	(	(	PUNCT
easat-4026	317	27	ℐ	ℐ	ADV
easat-4026	317	28	⊗	⊗	PROPN
easat-4026	317	29	ℐ⨂∗	ℐ⨂∗	NOUN
easat-4026	317	30	,	,	PUNCT
easat-4026	317	31	𝜌∗	𝜌∗	NOUN
easat-4026	317	32	′	′	NOUN
easat-4026	317	33	)	)	PUNCT
easat-4026	317	34	,	,	PUNCT
easat-4026	317	35	ℐ	ℐ	PRON
easat-4026	317	36	satisfy	satisfy	VERB
easat-4026	317	37	the	the	DET
easat-4026	317	38	excision	excision	NOUN
easat-4026	317	39	to	to	ADP
easat-4026	317	40	𝐻-homology	𝐻-homology	PROPN
easat-4026	317	41	.	.	PUNCT
easat-4026	318	1	in	in	ADP
easat-4026	318	2	such	such	DET
easat-4026	318	3	a	a	DET
easat-4026	318	4	case	case	NOUN
easat-4026	318	5	,	,	PUNCT
easat-4026	318	6	𝒢	𝒢	PROPN
easat-4026	318	7	⊗	⊗	PROPN
easat-4026	318	8	(	(	PUNCT
easat-4026	318	9	ℐ	ℐ	ADV
easat-4026	318	10	⊗	⊗	PROPN
easat-4026	318	11	ℐ⨂	ℐ⨂	VERB
easat-4026	318	12	∗−1	∗−1	NOUN
easat-4026	318	13	,	,	PUNCT
easat-4026	318	14	𝜌∗	𝜌∗	NOUN
easat-4026	318	15	′	′	NOUN
easat-4026	318	16	)	)	PUNCT
easat-4026	318	17	is	be	AUX
easat-4026	318	18	exact	exact	ADJ
easat-4026	318	19	.	.	PUNCT
easat-4026	319	1	the	the	DET
easat-4026	319	2	proposition	proposition	NOUN
easat-4026	319	3	(	(	PUNCT
easat-4026	319	4	3	3	X
easat-4026	319	5	)	)	PUNCT
easat-4026	319	6	is	be	AUX
easat-4026	319	7	equivalent	equivalent	ADJ
easat-4026	319	8	to	to	ADP
easat-4026	319	9	(	(	PUNCT
easat-4026	319	10	1	1	NUM
easat-4026	319	11	):	):	PUNCT
easat-4026	319	12	comparable	comparable	ADJ
easat-4026	319	13	to	to	ADP
easat-4026	319	14	(	(	PUNCT
easat-4026	319	15	2	2	NUM
easat-4026	319	16	)	)	PUNCT
easat-4026	319	17	⟹	⟹	NOUN
easat-4026	319	18	(	(	PUNCT
easat-4026	319	19	1	1	NUM
easat-4026	319	20	)	)	PUNCT
easat-4026	319	21	.	.	PUNCT
easat-4026	320	1	the	the	DET
easat-4026	320	2	proposition	proposition	NOUN
easat-4026	320	3	(	(	PUNCT
easat-4026	320	4	4	4	X
easat-4026	320	5	)	)	PUNCT
easat-4026	320	6	is	be	AUX
easat-4026	320	7	equivalent	equivalent	ADJ
easat-4026	320	8	to	to	ADP
easat-4026	320	9	(	(	PUNCT
easat-4026	320	10	1	1	NUM
easat-4026	320	11	):	):	PUNCT
easat-4026	320	12	allow	allow	VERB
easat-4026	320	13	𝒜	𝒜	NOUN
easat-4026	320	14	and	and	CCONJ
easat-4026	320	15	𝒢	𝒢	PROPN
easat-4026	320	16	are	be	AUX
easat-4026	320	17	being	be	AUX
easat-4026	320	18	in	in	ADP
easat-4026	320	19	(	(	PUNCT
easat-4026	320	20	2	2	NUM
easat-4026	320	21	)	)	PUNCT
easat-4026	320	22	⟹	⟹	NOUN
easat-4026	320	23	(	(	PUNCT
easat-4026	320	24	1	1	NUM
easat-4026	320	25	)	)	PUNCT
easat-4026	320	26	.	.	PUNCT
easat-4026	321	1	let	let	VERB
easat-4026	321	2	the	the	DET
easat-4026	321	3	canonical	canonical	ADJ
easat-4026	321	4	projection	projection	NOUN
easat-4026	321	5	given	give	VERB
easat-4026	321	6	by	by	ADP
easat-4026	321	7	�	�	PROPN
easat-4026	321	8	̅	̅	NOUN
easat-4026	321	9	�	�	NOUN
easat-4026	321	10	:	:	PUNCT
easat-4026	321	11	𝒞∗∗(𝒜	𝒞∗∗(𝒜	PROPN
easat-4026	321	12	)	)	PUNCT
easat-4026	321	13	→	→	SYM
easat-4026	321	14	𝒞∗∗(𝒢	𝒞∗∗(𝒢	PROPN
easat-4026	321	15	)	)	PUNCT
easat-4026	321	16	and	and	CCONJ
easat-4026	321	17	𝛽	𝛽	NOUN
easat-4026	321	18	be	be	AUX
easat-4026	321	19	the	the	DET
easat-4026	321	20	sub	sub	NOUN
easat-4026	321	21	-	-	ADJ
easat-4026	321	22	complex	complex	NOUN
easat-4026	321	23	of	of	ADP
easat-4026	321	24	𝑘𝑒𝑟	𝑘𝑒𝑟	NOUN
easat-4026	321	25	(	(	PUNCT
easat-4026	321	26	𝜋	𝜋	NOUN
easat-4026	321	27	)	)	PUNCT
easat-4026	321	28	produced	produce	VERB
easat-4026	321	29	by	by	ADP
easat-4026	321	30	the	the	DET
easat-4026	321	31	components	component	NOUN
easat-4026	321	32	(	(	PUNCT
easat-4026	321	33	𝒶0⊗⋯⊗𝒶𝑛	𝒶0⊗⋯⊗𝒶𝑛	PROPN
easat-4026	321	34	,	,	PUNCT
easat-4026	321	35	𝒶	𝒶	NOUN
easat-4026	321	36	′	′	NUM
easat-4026	321	37	0⊗⋯⊗𝒶′𝑛−1	0⊗⋯⊗𝒶′𝑛−1	NUM
easat-4026	321	38	)	)	PUNCT
easat-4026	321	39	that	that	PRON
easat-4026	321	40	include	include	VERB
easat-4026	321	41	some	some	DET
easat-4026	321	42	𝒶𝑖	𝒶𝑖	NOUN
easat-4026	321	43	with	with	ADP
easat-4026	321	44	some	some	PRON
easat-4026	321	45	𝒶′n	𝒶′n	ADJ
easat-4026	321	46	in	in	ADP
easat-4026	321	47	𝒢.	𝒢.	PROPN
easat-4026	321	48	so	so	ADV
easat-4026	321	49	𝛽	𝛽	NOUN
easat-4026	321	50	is	be	AUX
easat-4026	321	51	exact	exact	ADJ
easat-4026	322	1	such	such	ADJ
easat-4026	322	2	that	that	SCONJ
easat-4026	322	3	𝑘𝑒𝑟(	𝑘𝑒𝑟(	PROPN
easat-4026	322	4	�	�	PROPN
easat-4026	322	5	̅	̅	NOUN
easat-4026	322	6	�	�	NOUN
easat-4026	322	7	)	)	PUNCT
easat-4026	322	8	=	=	SYM
easat-4026	322	9	𝒞∗∗(ℐ	𝒞∗∗(ℐ	PROPN
easat-4026	322	10	)	)	PUNCT
easat-4026	322	11	⊕	⊕	PROPN
easat-4026	322	12	𝛽	𝛽	PROPN
easat-4026	322	13	and	and	CCONJ
easat-4026	322	14	ℐ	ℐ	PRON
easat-4026	322	15	fulfills	fulfill	VERB
easat-4026	322	16	excision	excision	NOUN
easat-4026	322	17	instead	instead	ADV
easat-4026	322	18	of	of	ADP
easat-4026	322	19	simplicial	simplicial	ADJ
easat-4026	322	20	homology	homology	NOUN
easat-4026	322	21	.	.	PUNCT
easat-4026	323	1	consider	consider	VERB
easat-4026	323	2	the	the	DET
easat-4026	323	3	case	case	NOUN
easat-4026	323	4	when	when	SCONJ
easat-4026	323	5	ℐ	ℐ	PRON
easat-4026	323	6	can	can	AUX
easat-4026	323	7	not	not	PART
easat-4026	323	8	be	be	AUX
easat-4026	323	9	ℋ-unital	ℋ-unital	PROPN
easat-4026	323	10	.	.	PUNCT
easat-4026	323	11	suppose	suppose	VERB
easat-4026	323	12	𝑥	𝑥	X
easat-4026	323	13	∈	∈	PROPN
easat-4026	323	14	𝒢	𝒢	PROPN
easat-4026	323	15	⊗	⊗	PROPN
easat-4026	323	16	ℐ⨂𝑛	ℐ⨂𝑛	NOUN
easat-4026	323	17	represents	represent	VERB
easat-4026	323	18	a	a	DET
easat-4026	323	19	cycle	cycle	NOUN
easat-4026	323	20	that	that	PRON
easat-4026	323	21	does	do	AUX
easat-4026	323	22	not	not	PART
easat-4026	323	23	represent	represent	VERB
easat-4026	323	24	a	a	DET
easat-4026	323	25	boundary	boundary	NOUN
easat-4026	323	26	for	for	ADP
easat-4026	323	27	𝜌𝑛	𝜌𝑛	NOUN
easat-4026	323	28	′	′	NUM
easat-4026	323	29	,	,	PUNCT
easat-4026	323	30	it	it	PRON
easat-4026	323	31	is	be	AUX
easat-4026	323	32	obvious	obvious	ADJ
easat-4026	323	33	that	that	SCONJ
easat-4026	323	34	(	(	PUNCT
easat-4026	323	35	0	0	NUM
easat-4026	323	36	,	,	PUNCT
easat-4026	323	37	𝑁(𝑥	𝑁(𝑥	NOUN
easat-4026	323	38	)	)	PUNCT
easat-4026	323	39	)	)	PUNCT
easat-4026	323	40	in	in	ADP
easat-4026	323	41	is	be	AUX
easat-4026	323	42	a	a	DET
easat-4026	323	43	cycle	cycle	NOUN
easat-4026	323	44	of	of	ADP
easat-4026	323	45	degree	degree	NOUN
easat-4026	323	46	𝑛	𝑛	PROPN
easat-4026	323	47	+	+	NUM
easat-4026	323	48	1	1	NUM
easat-4026	323	49	that	that	PRON
easat-4026	323	50	is	be	AUX
easat-4026	323	51	not	not	PART
easat-4026	323	52	a	a	DET
easat-4026	323	53	boundary	boundary	NOUN
easat-4026	323	54	,	,	PUNCT
easat-4026	323	55	which	which	PRON
easat-4026	323	56	is	be	AUX
easat-4026	323	57	in	in	ADP
easat-4026	323	58	direct	direct	ADJ
easat-4026	323	59	opposition	opposition	NOUN
easat-4026	323	60	to	to	ADP
easat-4026	323	61	the	the	DET
easat-4026	323	62	exactness	exactness	NOUN
easat-4026	323	63	of	of	ADP
easat-4026	323	64	𝛽.	𝛽.	NOUN
easat-4026	323	65	3.7	3.7	NUM
easat-4026	323	66	.	.	PUNCT
easat-4026	324	1	concluding	conclude	VERB
easat-4026	324	2	remarks	remark	VERB
easat-4026	324	3	the	the	DET
easat-4026	324	4	core	core	NOUN
easat-4026	324	5	focus	focus	NOUN
easat-4026	324	6	was	be	AUX
easat-4026	324	7	on	on	ADP
easat-4026	324	8	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-4026	324	9	,	,	PUNCT
easat-4026	324	10	defined	define	VERB
easat-4026	324	11	by	by	ADP
easat-4026	324	12	stasheff	stasheff	NOUN
easat-4026	324	13	identities	identity	NOUN
easat-4026	324	14	that	that	PRON
easat-4026	324	15	ensure	ensure	VERB
easat-4026	324	16	homotopy	homotopy	NOUN
easat-4026	324	17	associativity	associativity	NOUN
easat-4026	324	18	.	.	PUNCT
easat-4026	325	1	we	we	PRON
easat-4026	325	2	examined	examine	VERB
easat-4026	325	3	various	various	ADJ
easat-4026	325	4	examples	example	NOUN
easat-4026	325	5	and	and	CCONJ
easat-4026	325	6	their	their	PRON
easat-4026	325	7	applications	application	NOUN
easat-4026	325	8	,	,	PUNCT
easat-4026	325	9	highlighting	highlight	VERB
easat-4026	325	10	their	their	PRON
easat-4026	325	11	versatility	versatility	NOUN
easat-4026	325	12	.	.	PUNCT
easat-4026	326	1	morphisms	morphism	NOUN
easat-4026	326	2	between	between	ADP
easat-4026	326	3	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	326	4	were	be	AUX
easat-4026	326	5	also	also	ADV
easat-4026	326	6	analyzed	analyze	VERB
easat-4026	326	7	,	,	PUNCT
easat-4026	326	8	including	include	VERB
easat-4026	326	9	strict	strict	ADJ
easat-4026	326	10	morphisms	morphism	NOUN
easat-4026	326	11	and	and	CCONJ
easat-4026	326	12	quasiisomorphisms	quasiisomorphisms	NOUN
easat-4026	326	13	.	.	PUNCT
easat-4026	327	1	our	our	PRON
easat-4026	327	2	study	study	NOUN
easat-4026	327	3	further	far	ADV
easat-4026	327	4	explored	explore	VERB
easat-4026	327	5	the	the	DET
easat-4026	327	6	homology	homology	NOUN
easat-4026	327	7	of	of	ADP
easat-4026	327	8	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-4026	327	9	,	,	PUNCT
easat-4026	327	10	specifically	specifically	ADV
easat-4026	327	11	simplicial	simplicial	ADJ
easat-4026	327	12	homology	homology	NOUN
easat-4026	327	13	,	,	PUNCT
easat-4026	327	14	and	and	CCONJ
easat-4026	327	15	detailed	detail	VERB
easat-4026	327	16	the	the	DET
easat-4026	327	17	relationship	relationship	NOUN
easat-4026	327	18	between	between	ADP
easat-4026	327	19	bar	bar	NOUN
easat-4026	327	20	homology	homology	NOUN
easat-4026	327	21	ℋ𝐵𝑛−1(ℐ	ℋ𝐵𝑛−1(ℐ	NUM
easat-4026	327	22	)	)	PUNCT
easat-4026	327	23	and	and	CCONJ
easat-4026	327	24	simplicial	simplicial	ADJ
easat-4026	327	25	homology	homology	NOUN
easat-4026	327	26	ℋℋ𝑛(ℐ	ℋℋ𝑛(ℐ	NOUN
easat-4026	327	27	)	)	PUNCT
easat-4026	327	28	as	as	ADP
easat-4026	327	29	an	an	DET
easat-4026	327	30	exact	exact	ADJ
easat-4026	327	31	sequence	sequence	NOUN
easat-4026	327	32	.	.	PUNCT
easat-4026	328	1	we	we	PRON
easat-4026	328	2	demonstrated	demonstrate	VERB
easat-4026	328	3	several	several	ADJ
easat-4026	328	4	quasi	quasi	NOUN
easat-4026	328	5	-	-	NOUN
easat-4026	328	6	isomorphisms	isomorphisms	X
easat-4026	328	7	and	and	CCONJ
easat-4026	328	8	presented	present	VERB
easat-4026	328	9	a	a	DET
easat-4026	328	10	commutative	commutative	ADJ
easat-4026	328	11	diagram	diagram	NOUN
easat-4026	328	12	illustrating	illustrate	VERB
easat-4026	328	13	these	these	DET
easat-4026	328	14	relationships	relationship	NOUN
easat-4026	328	15	.	.	PUNCT
easat-4026	329	1	in	in	ADP
easat-4026	329	2	summary	summary	NOUN
easat-4026	329	3	,	,	PUNCT
easat-4026	329	4	the	the	DET
easat-4026	329	5	homology	homology	NOUN
easat-4026	329	6	theory	theory	NOUN
easat-4026	329	7	of	of	ADP
easat-4026	329	8	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-4026	329	9	extends	extend	VERB
easat-4026	329	10	traditional	traditional	ADJ
easat-4026	329	11	algebraic	algebraic	ADJ
easat-4026	329	12	concepts	concept	NOUN
easat-4026	329	13	into	into	ADP
easat-4026	329	14	a	a	DET
easat-4026	329	15	homotopical	homotopical	ADJ
easat-4026	329	16	framework	framework	NOUN
easat-4026	329	17	,	,	PUNCT
easat-4026	329	18	offering	offer	VERB
easat-4026	329	19	a	a	DET
easat-4026	329	20	deep	deep	ADJ
easat-4026	329	21	understanding	understanding	NOUN
easat-4026	329	22	of	of	ADP
easat-4026	329	23	their	their	PRON
easat-4026	329	24	foundational	foundational	ADJ
easat-4026	329	25	aspects	aspect	NOUN
easat-4026	329	26	and	and	CCONJ
easat-4026	329	27	broader	broad	ADJ
easat-4026	329	28	implications	implication	NOUN
easat-4026	329	29	in	in	ADP
easat-4026	329	30	mathematical	mathematical	ADJ
easat-4026	329	31	research	research	NOUN
easat-4026	329	32	.	.	PUNCT
easat-4026	330	1	copyright	copyright	NOUN
easat-4026	330	2	:	:	PUNCT
easat-4026	330	3	©	©	PROPN
easat-4026	330	4	2024	2024	NUM
easat-4026	330	5	by	by	ADP
easat-4026	330	6	the	the	DET
easat-4026	330	7	authors	author	NOUN
easat-4026	330	8	.	.	PUNCT
easat-4026	331	1	this	this	DET
easat-4026	331	2	article	article	NOUN
easat-4026	331	3	is	be	AUX
easat-4026	331	4	an	an	DET
easat-4026	331	5	open	open	ADJ
easat-4026	331	6	access	access	NOUN
easat-4026	331	7	article	article	NOUN
easat-4026	331	8	distributed	distribute	VERB
easat-4026	331	9	under	under	ADP
easat-4026	331	10	the	the	DET
easat-4026	331	11	terms	term	NOUN
easat-4026	331	12	and	and	CCONJ
easat-4026	331	13	conditions	condition	NOUN
easat-4026	331	14	of	of	ADP
easat-4026	331	15	the	the	DET
easat-4026	331	16	creative	creative	ADJ
easat-4026	331	17	commons	common	NOUN
easat-4026	331	18	attribution	attribution	NOUN
easat-4026	331	19	(	(	PUNCT
easat-4026	331	20	cc	cc	NOUN
easat-4026	331	21	by	by	ADP
easat-4026	331	22	)	)	PUNCT
easat-4026	331	23	license	license	NOUN
easat-4026	331	24	(	(	PUNCT
easat-4026	331	25	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-4026	331	26	)	)	PUNCT
easat-4026	331	27	.	.	PUNCT
easat-4026	332	1	references	reference	NOUN
easat-4026	332	2	[	[	X
easat-4026	332	3	1	1	NUM
easat-4026	332	4	]	]	PUNCT
easat-4026	332	5	g.	g.	PROPN
easat-4026	332	6	hochschild	hochschild	PROPN
easat-4026	332	7	,	,	PUNCT
easat-4026	332	8	“	"	PUNCT
easat-4026	332	9	on	on	ADP
easat-4026	332	10	the	the	DET
easat-4026	332	11	cohomology	cohomology	NOUN
easat-4026	332	12	groups	group	NOUN
easat-4026	332	13	of	of	ADP
easat-4026	332	14	an	an	DET
easat-4026	332	15	associative	associative	ADJ
easat-4026	332	16	algebra	algebra	NOUN
easat-4026	332	17	”	"	PUNCT
easat-4026	332	18	,	,	PUNCT
easat-4026	332	19	annals	annal	NOUN
easat-4026	332	20	of	of	ADP
easat-4026	332	21	mathematics	mathematic	NOUN
easat-4026	332	22	,	,	PUNCT
easat-4026	332	23	jstor	jstor	NOUN
easat-4026	332	24	,	,	PUNCT
easat-4026	332	25	vol	vol	NOUN
easat-4026	332	26	.	.	PROPN
easat-4026	333	1	46	46	NUM
easat-4026	333	2	,	,	PUNCT
easat-4026	333	3	no	no	INTJ
easat-4026	333	4	.	.	NOUN
easat-4026	333	5	1	1	NUM
easat-4026	333	6	,	,	PUNCT
easat-4026	333	7	pp	pp	ADJ
easat-4026	333	8	.	.	PUNCT
easat-4026	334	1	58–67	58–67	NUM
easat-4026	334	2	.	.	PUNCT
easat-4026	335	1	1945	1945	NUM
easat-4026	335	2	.	.	PUNCT
easat-4026	336	1	https://doi.org/10.2307/1969145	https://doi.org/10.2307/1969145	NUM
easat-4026	336	2	.	.	PUNCT
easat-4026	337	1	[	[	X
easat-4026	337	2	2	2	X
easat-4026	337	3	]	]	PUNCT
easat-4026	337	4	h.	h.	PROPN
easat-4026	337	5	cartan	cartan	PROPN
easat-4026	337	6	,	,	PUNCT
easat-4026	337	7	and	and	CCONJ
easat-4026	337	8	s.	s.	PROPN
easat-4026	337	9	eilenberg	eilenberg	PROPN
easat-4026	337	10	,	,	PUNCT
easat-4026	337	11	“	"	PUNCT
easat-4026	337	12	homological	homological	ADJ
easat-4026	337	13	algebra	algebra	NOUN
easat-4026	337	14	”	"	PUNCT
easat-4026	337	15	,	,	PUNCT
easat-4026	337	16	princeton	princeton	PROPN
easat-4026	337	17	mathematical	mathematical	PROPN
easat-4026	337	18	,	,	PUNCT
easat-4026	337	19	princeton	princeton	PROPN
easat-4026	337	20	university	university	PROPN
easat-4026	337	21	press	press	NOUN
easat-4026	337	22	,	,	PUNCT
easat-4026	337	23	vol	vol	NOUN
easat-4026	337	24	.	.	PROPN
easat-4026	337	25	19	19	NUM
easat-4026	337	26	,	,	PUNCT
easat-4026	337	27	1956	1956	NUM
easat-4026	337	28	.	.	PUNCT
easat-4026	338	1	doi.org/10.1515/9781400883844	doi.org/10.1515/9781400883844	NOUN
easat-4026	338	2	,	,	PUNCT
easat-4026	338	3	isbn	isbn	ADJ
easat-4026	338	4	978	978	NUM
easat-4026	338	5	-	-	SYM
easat-4026	338	6	0	0	NUM
easat-4026	338	7	-	-	NUM
easat-4026	338	8	691	691	NUM
easat-4026	338	9	04991	04991	NUM
easat-4026	338	10	-	-	SYM
easat-4026	338	11	5	5	NUM
easat-4026	338	12	,	,	PUNCT
easat-4026	338	13	mr	mr	PROPN
easat-4026	338	14	0077480	0077480	NUM
easat-4026	338	15	.	.	PUNCT
easat-4026	339	1	[	[	X
easat-4026	339	2	3	3	X
easat-4026	339	3	]	]	X
easat-4026	339	4	j.	j.	PROPN
easat-4026	339	5	d.	d.	PROPN
easat-4026	339	6	stasheff	stasheff	PROPN
easat-4026	339	7	,	,	PUNCT
easat-4026	339	8	“	"	PUNCT
easat-4026	339	9	homotopy	homotopy	VERB
easat-4026	339	10	associativity	associativity	NOUN
easat-4026	339	11	of	of	ADP
easat-4026	339	12	h	h	NOUN
easat-4026	339	13	-	-	PUNCT
easat-4026	339	14	spaces	space	NOUN
easat-4026	339	15	.	.	PUNCT
easat-4026	340	1	ii	ii	PROPN
easat-4026	340	2	”	"	PUNCT
easat-4026	340	3	,	,	PUNCT
easat-4026	340	4	transactions	transaction	NOUN
easat-4026	340	5	of	of	ADP
easat-4026	340	6	the	the	DET
easat-4026	340	7	american	american	PROPN
easat-4026	340	8	mathematical	mathematical	PROPN
easat-4026	340	9	society	society	NOUN
easat-4026	340	10	,	,	PUNCT
easat-4026	340	11	vol.108	vol.108	INTJ
easat-4026	340	12	,	,	PUNCT
easat-4026	340	13	no	no	INTJ
easat-4026	340	14	.	.	NOUN
easat-4026	340	15	2	2	NUM
easat-4026	340	16	,	,	PUNCT
easat-4026	340	17	pp	pp	ADJ
easat-4026	340	18	.	.	PUNCT
easat-4026	341	1	293	293	NUM
easat-4026	341	2	-	-	SYM
easat-4026	341	3	312	312	NUM
easat-4026	341	4	,	,	PUNCT
easat-4026	341	5	1963	1963	NUM
easat-4026	341	6	.	.	PUNCT
easat-4026	342	1	https://doi.org/10.2307/1993609	https://doi.org/10.2307/1993609	NOUN
easat-4026	342	2	.	.	PUNCT
easat-4026	343	1	[	[	X
easat-4026	343	2	4	4	X
easat-4026	343	3	]	]	X
easat-4026	343	4	t.v	t.v	PROPN
easat-4026	343	5	.	.	PROPN
easat-4026	343	6	kadeishvili	kadeishvili	PROPN
easat-4026	343	7	,	,	PUNCT
easat-4026	343	8	"	"	PUNCT
easat-4026	343	9	the	the	DET
easat-4026	343	10	𝐴∞-algebra	𝐴∞-algebra	PROPN
easat-4026	343	11	structure	structure	NOUN
easat-4026	343	12	and	and	CCONJ
easat-4026	343	13	the	the	DET
easat-4026	343	14	hochschild	hochschild	ADJ
easat-4026	343	15	and	and	CCONJ
easat-4026	343	16	harrison	harrison	NOUN
easat-4026	343	17	cohomologies	cohomologie	NOUN
easat-4026	343	18	,	,	PUNCT
easat-4026	343	19	"	"	PUNCT
easat-4026	343	20	proc	proc	NOUN
easat-4026	343	21	.	.	PUNCT
easat-4026	344	1	of	of	ADP
easat-4026	344	2	a.	a.	PROPN
easat-4026	344	3	razmadze	razmadze	PROPN
easat-4026	344	4	math	math	PROPN
easat-4026	344	5	.	.	PUNCT
easat-4026	345	1	inst	inst	PROPN
easat-4026	345	2	.	.	PROPN
easat-4026	345	3	,	,	PUNCT
easat-4026	345	4	vol	vol	NOUN
easat-4026	345	5	.	.	PROPN
easat-4026	345	6	91	91	NUM
easat-4026	345	7	,	,	PUNCT
easat-4026	345	8	1988	1988	NUM
easat-4026	345	9	.	.	PUNCT
easat-4026	346	1	https://doi.org/10.48550/arxiv.math/0210331	https://doi.org/10.48550/arxiv.math/0210331	PROPN
easat-4026	346	2	.	.	PUNCT
easat-4026	347	1	[	[	X
easat-4026	347	2	5	5	X
easat-4026	347	3	]	]	PUNCT
easat-4026	347	4	b.	b.	PROPN
easat-4026	347	5	keller	keller	PROPN
easat-4026	347	6	,	,	PUNCT
easat-4026	347	7	“	"	PUNCT
easat-4026	347	8	𝐴-infinity	𝐴-infinity	NOUN
easat-4026	347	9	algebras	algebra	NOUN
easat-4026	347	10	,	,	PUNCT
easat-4026	347	11	modules	module	NOUN
easat-4026	347	12	and	and	CCONJ
easat-4026	347	13	functor	functor	PROPN
easat-4026	347	14	categories	category	NOUN
easat-4026	347	15	”	"	PUNCT
easat-4026	347	16	,	,	PUNCT
easat-4026	347	17	contemporary	contemporary	ADJ
easat-4026	347	18	mathematices	mathematice	NOUN
easat-4026	347	19	,	,	PUNCT
easat-4026	347	20	vol.406	vol.406	VERB
easat-4026	347	21	,	,	PUNCT
easat-4026	347	22	pp	pp	PROPN
easat-4026	347	23	.	.	PUNCT
easat-4026	348	1	67	67	NUM
easat-4026	348	2	-	-	SYM
easat-4026	348	3	94	94	NUM
easat-4026	348	4	,	,	PUNCT
easat-4026	348	5	2006	2006	NUM
easat-4026	348	6	.	.	PUNCT
easat-4026	349	1	http://dx.doi.org/10.1090/conm/406/07654	http://dx.doi.org/10.1090/conm/406/07654	PROPN
easat-4026	349	2	.	.	PUNCT
easat-4026	349	3	arxiv	arxiv	PROPN
easat-4026	349	4	:	:	PUNCT
easat-4026	349	5	math/0510508	math/0510508	NOUN
easat-4026	349	6	.	.	PUNCT
easat-4026	350	1	[	[	X
easat-4026	350	2	6	6	NUM
easat-4026	350	3	]	]	PUNCT
easat-4026	350	4	a.	a.	NOUN
easat-4026	350	5	prouté	prouté	NOUN
easat-4026	350	6	,	,	PUNCT
easat-4026	350	7	“	"	PUNCT
easat-4026	350	8	algèbres	algèbre	NOUN
easat-4026	350	9	différentielles	différentielle	NOUN
easat-4026	350	10	fortement	fortement	X
easat-4026	350	11	homotopiquement	homotopiquement	NOUN
easat-4026	350	12	associatives(a	associatives(a	PROPN
easat-4026	350	13	indice	indice	PROPN
easat-4026	350	14	l’infini	l’infini	NOUN
easat-4026	350	15	-	-	NOUN
easat-4026	350	16	algèbres	algèbre	NOUN
easat-4026	350	17	)	)	PUNCT
easat-4026	350	18	,	,	PUNCT
easat-4026	350	19	”	"	PUNCT
easat-4026	350	20	ph.d	ph.d	PROPN
easat-4026	350	21	.	.	PUNCT
easat-4026	351	1	thesis	thesis	NOUN
easat-4026	351	2	,	,	PUNCT
easat-4026	351	3	1986	1986	NUM
easat-4026	351	4	.	.	PUNCT
easat-4026	352	1	[	[	X
easat-4026	352	2	7	7	X
easat-4026	352	3	]	]	X
easat-4026	352	4	j.	j.	PROPN
easat-4026	352	5	hübschmann	hübschmann	PROPN
easat-4026	352	6	,	,	PUNCT
easat-4026	352	7	“	"	PUNCT
easat-4026	352	8	the	the	DET
easat-4026	352	9	homotopy	homotopy	NOUN
easat-4026	352	10	type	type	NOUN
easat-4026	352	11	of	of	ADP
easat-4026	352	12	𝐹𝜓𝑞.	𝐹𝜓𝑞.	PROPN
easat-4026	352	13	the	the	DET
easat-4026	352	14	complex	complex	ADJ
easat-4026	352	15	and	and	CCONJ
easat-4026	352	16	symplectic	symplectic	ADJ
easat-4026	352	17	cases	case	NOUN
easat-4026	352	18	.	.	PUNCT
easat-4026	353	1	applications	application	NOUN
easat-4026	353	2	of	of	ADP
easat-4026	353	3	algebraic	algebraic	PROPN
easat-4026	353	4	𝐾-theory	𝐾-theory	PROPN
easat-4026	353	5	to	to	ADP
easat-4026	353	6	algebraic	algebraic	ADJ
easat-4026	353	7	geometry	geometry	NOUN
easat-4026	353	8	and	and	CCONJ
easat-4026	353	9	number	number	NOUN
easat-4026	353	10	theory	theory	NOUN
easat-4026	353	11	,	,	PUNCT
easat-4026	353	12	"	"	PUNCT
easat-4026	353	13	part	part	NOUN
easat-4026	353	14	i	i	PROPN
easat-4026	353	15	,	,	PUNCT
easat-4026	353	16	ii	ii	PROPN
easat-4026	353	17	(	(	PUNCT
easat-4026	353	18	boulder	boulder	PROPN
easat-4026	353	19	,	,	PUNCT
easat-4026	353	20	colo	colo	PROPN
easat-4026	353	21	.	.	PROPN
easat-4026	353	22	,	,	PUNCT
easat-4026	353	23	1983	1983	NUM
easat-4026	353	24	)	)	PUNCT
easat-4026	353	25	,	,	PUNCT
easat-4026	353	26	487–518	487–518	NUM
easat-4026	353	27	.	.	PUNCT
easat-4026	354	1	amer	amer	PROPN
easat-4026	354	2	.	.	PUNCT
easat-4026	354	3	math	math	PROPN
easat-4026	354	4	.	.	PUNCT
easat-4026	355	1	soc	soc	PROPN
easat-4026	355	2	.	.	PUNCT
easat-4026	355	3	,	,	PUNCT
easat-4026	355	4	providence	providence	NOUN
easat-4026	355	5	,	,	PUNCT
easat-4026	355	6	ri	ri	NOUN
easat-4026	355	7	,	,	PUNCT
easat-4026	355	8	1986	1986	NUM
easat-4026	355	9	.	.	PUNCT
easat-4026	356	1	[	[	X
easat-4026	356	2	8	8	NUM
easat-4026	356	3	]	]	X
easat-4026	356	4	e.	e.	PROPN
easat-4026	356	5	getzler	getzler	PROPN
easat-4026	356	6	,	,	PUNCT
easat-4026	356	7	and	and	CCONJ
easat-4026	356	8	john	john	PROPN
easat-4026	356	9	d.	d.	PROPN
easat-4026	356	10	s.	s.	PROPN
easat-4026	356	11	jones	jones	PROPN
easat-4026	356	12	.	.	PUNCT
easat-4026	357	1	“	"	PUNCT
easat-4026	357	2	𝐴∞-algebras	𝐴∞-algebras	PROPN
easat-4026	357	3	and	and	CCONJ
easat-4026	357	4	the	the	DET
easat-4026	357	5	cyclic	cyclic	ADJ
easat-4026	357	6	bar	bar	NOUN
easat-4026	357	7	complex	complex	NOUN
easat-4026	357	8	.	.	PUNCT
easat-4026	357	9	"	"	PUNCT
easat-4026	358	1	illinois	illinois	PROPN
easat-4026	358	2	j.	j.	PROPN
easat-4026	358	3	math	math	PROPN
easat-4026	358	4	.	.	PUNCT
easat-4026	358	5	,	,	PUNCT
easat-4026	358	6	vol	vol	NOUN
easat-4026	358	7	.	.	PROPN
easat-4026	359	1	34	34	NUM
easat-4026	359	2	,	,	PUNCT
easat-4026	359	3	no	no	INTJ
easat-4026	359	4	.	.	NOUN
easat-4026	359	5	2	2	NUM
easat-4026	359	6	,	,	PUNCT
easat-4026	359	7	pp	pp	ADJ
easat-4026	359	8	.	.	PUNCT
easat-4026	360	1	256	256	NUM
easat-4026	360	2	283	283	NUM
easat-4026	360	3	,	,	PUNCT
easat-4026	360	4	summer	summer	NOUN
easat-4026	360	5	1990	1990	NUM
easat-4026	360	6	.	.	PUNCT
easat-4026	361	1	https://doi.org/10.1215/ijm/1255988267	https://doi.org/10.1215/ijm/1255988267	PROPN
easat-4026	362	1	[	[	X
easat-4026	362	2	9	9	NUM
easat-4026	362	3	]	]	PUNCT
easat-4026	362	4	k.	k.	PROPN
easat-4026	362	5	fukaya	fukaya	PROPN
easat-4026	362	6	,	,	PUNCT
easat-4026	362	7	“	"	PUNCT
easat-4026	362	8	morse	morse	NOUN
easat-4026	362	9	homotopy	homotopy	NOUN
easat-4026	362	10	,	,	PUNCT
easat-4026	362	11	𝐴∞-category	𝐴∞-category	PROPN
easat-4026	362	12	,	,	PUNCT
easat-4026	362	13	and	and	CCONJ
easat-4026	362	14	floer	floer	NOUN
easat-4026	362	15	homologies	homology	NOUN
easat-4026	362	16	,	,	PUNCT
easat-4026	362	17	"	"	PUNCT
easat-4026	362	18	lecture	lecture	NOUN
easat-4026	362	19	notes	note	NOUN
easat-4026	362	20	ser	ser	PROPN
easat-4026	362	21	.	.	PROPN
easat-4026	363	1	vol	vol	NOUN
easat-4026	363	2	.	.	PROPN
easat-4026	364	1	18	18	NUM
easat-4026	364	2	,	,	PUNCT
easat-4026	364	3	pp	pp	ADJ
easat-4026	364	4	.	.	PUNCT
easat-4026	365	1	1	1	NUM
easat-4026	365	2	-	-	SYM
easat-4026	365	3	102	102	NUM
easat-4026	365	4	,	,	PUNCT
easat-4026	365	5	1993	1993	NUM
easat-4026	365	6	.	.	PUNCT
easat-4026	366	1	[	[	X
easat-4026	366	2	10	10	NUM
easat-4026	366	3	]	]	PUNCT
easat-4026	366	4	m.	m.	NOUN
easat-4026	366	5	kontsevich	kontsevich	PROPN
easat-4026	366	6	,	,	PUNCT
easat-4026	366	7	"	"	PUNCT
easat-4026	366	8	homological	homological	ADJ
easat-4026	366	9	algebra	algebra	NOUN
easat-4026	366	10	of	of	ADP
easat-4026	366	11	mirror	mirror	NOUN
easat-4026	366	12	symmetry	symmetry	NOUN
easat-4026	366	13	,	,	PUNCT
easat-4026	366	14	"	"	PUNCT
easat-4026	366	15	proceedings	proceeding	NOUN
easat-4026	366	16	of	of	ADP
easat-4026	366	17	the	the	DET
easat-4026	366	18	international	international	ADJ
easat-4026	366	19	congress	congress	PROPN
easat-4026	366	20	of	of	ADP
easat-4026	366	21	mathematicians	mathematician	NOUN
easat-4026	366	22	,	,	PUNCT
easat-4026	366	23	pages	page	NOUN
easat-4026	366	24	120–139	120–139	NUM
easat-4026	366	25	,	,	PUNCT
easat-4026	366	26	birkh¨auser	birkh¨auser	NOUN
easat-4026	366	27	,	,	PUNCT
easat-4026	366	28	1995	1995	NUM
easat-4026	366	29	,	,	PUNCT
easat-4026	366	30	alggeom/9411018	alggeom/9411018	PROPN
easat-4026	366	31	.	.	PUNCT
easat-4026	367	1	https://doi.org/10.48550/arxiv.alggeom/9411018	https://doi.org/10.48550/arxiv.alggeom/9411018	PROPN
easat-4026	367	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-4026	367	3	https://doi.org/10.2307/1969145	https://doi.org/10.2307/1969145	PROPN
easat-4026	367	4	https://doi.org/10.48550/arxiv.math/0210331	https://doi.org/10.48550/arxiv.math/0210331	PROPN
easat-4026	367	5	http://dx.doi.org/10.1090/conm/406/07654	http://dx.doi.org/10.1090/conm/406/07654	PROPN
easat-4026	367	6	https://doi.org/10.1215/ijm/1255988267	https://doi.org/10.1215/ijm/1255988267	ADJ
easat-4026	367	7	https://doi.org/10.48550/arxiv.alg-geom/9411018	https://doi.org/10.48550/arxiv.alg-geom/9411018	PROPN
easat-4026	367	8	https://doi.org/10.48550/arxiv.alg-geom/9411018	https://doi.org/10.48550/arxiv.alg-geom/9411018	PROPN
easat-4026	367	9	9486	9486	NUM
easat-4026	367	10	edelweiss	edelweiss	PROPN
easat-4026	367	11	applied	apply	VERB
easat-4026	367	12	science	science	NOUN
easat-4026	367	13	and	and	CCONJ
easat-4026	367	14	technology	technology	NOUN
easat-4026	367	15	issn	issn	PROPN
easat-4026	367	16	:	:	PUNCT
easat-4026	367	17	2576	2576	NUM
easat-4026	367	18	-	-	SYM
easat-4026	367	19	8484	8484	NUM
easat-4026	367	20	vol	vol	NOUN
easat-4026	367	21	.	.	PROPN
easat-4026	367	22	8	8	NUM
easat-4026	367	23	,	,	PUNCT
easat-4026	367	24	no	no	INTJ
easat-4026	367	25	.	.	NOUN
easat-4026	368	1	6	6	NUM
easat-4026	368	2	:	:	SYM
easat-4026	368	3	9472	9472	NUM
easat-4026	368	4	-	-	SYM
easat-4026	368	5	9486	9486	NUM
easat-4026	368	6	,	,	PUNCT
easat-4026	368	7	2024	2024	NUM
easat-4026	368	8	doi	doi	NOUN
easat-4026	368	9	:	:	PUNCT
easat-4026	368	10	10.55214/25768484.v8i6.4026	10.55214/25768484.v8i6.4026	PROPN
easat-4026	368	11	©	©	PROPN
easat-4026	368	12	2024	2024	NUM
easat-4026	368	13	by	by	ADP
easat-4026	368	14	the	the	DET
easat-4026	368	15	authors	author	NOUN
easat-4026	368	16	;	;	PUNCT
easat-4026	368	17	licensee	licensee	PROPN
easat-4026	368	18	learning	learning	NOUN
easat-4026	368	19	gate	gate	NOUN
easat-4026	369	1	[	[	X
easat-4026	369	2	11	11	NUM
easat-4026	369	3	]	]	PUNCT
easat-4026	369	4	b.	b.	PROPN
easat-4026	369	5	keller	keller	PROPN
easat-4026	369	6	,	,	PUNCT
easat-4026	369	7	"	"	PUNCT
easat-4026	369	8	introduction	introduction	NOUN
easat-4026	369	9	to	to	ADP
easat-4026	369	10	a	a	DET
easat-4026	369	11	-	-	PUNCT
easat-4026	369	12	infinity	infinity	NOUN
easat-4026	369	13	algebras	algebra	NOUN
easat-4026	369	14	and	and	CCONJ
easat-4026	369	15	modules	module	NOUN
easat-4026	369	16	,	,	PUNCT
easat-4026	369	17	"	"	PUNCT
easat-4026	369	18	homology	homology	PROPN
easat-4026	369	19	homotopy	homotopy	PROPN
easat-4026	369	20	appl	appl	PROPN
easat-4026	369	21	.	.	PROPN
easat-4026	369	22	,	,	PUNCT
easat-4026	369	23	vol	vol	NOUN
easat-4026	369	24	.	.	PROPN
easat-4026	369	25	1	1	NUM
easat-4026	369	26	,	,	PUNCT
easat-4026	369	27	999	999	NUM
easat-4026	369	28	,	,	PUNCT
easat-4026	369	29	vol	vol	NOUN
easat-4026	369	30	.	.	PROPN
easat-4026	369	31	2	2	NUM
easat-4026	369	32	,	,	PUNCT
easat-4026	369	33	2001	2001	NUM
easat-4026	369	34	.	.	PUNCT
easat-4026	370	1	https://doi.org/10.48550/arxiv.math/9910179	https://doi.org/10.48550/arxiv.math/9910179	NOUN
easat-4026	370	2	.	.	PUNCT
easat-4026	371	1	[	[	X
easat-4026	371	2	12	12	NUM
easat-4026	371	3	]	]	PUNCT
easat-4026	371	4	p.	p.	NOUN
easat-4026	371	5	seidel	seidel	PROPN
easat-4026	371	6	,	,	PUNCT
easat-4026	371	7	"	"	PUNCT
easat-4026	371	8	fukaya	fukaya	NOUN
easat-4026	371	9	𝐴∞-structures	𝐴∞-structure	NOUN
easat-4026	371	10	associated	associate	VERB
easat-4026	371	11	to	to	ADP
easat-4026	371	12	lefschetz	lefschetz	ADJ
easat-4026	371	13	fibrations	fibration	NOUN
easat-4026	371	14	.	.	PUNCT
easat-4026	372	1	i	i	PRON
easat-4026	372	2	,	,	PUNCT
easat-4026	372	3	"	"	PUNCT
easat-4026	372	4	journal	journal	NOUN
easat-4026	372	5	of	of	ADP
easat-4026	372	6	symplectic	symplectic	ADJ
easat-4026	372	7	geometry	geometry	NOUN
easat-4026	372	8	,	,	PUNCT
easat-4026	372	9	vol	vol	NOUN
easat-4026	372	10	.	.	PROPN
easat-4026	372	11	10	10	NUM
easat-4026	372	12	,	,	PUNCT
easat-4026	372	13	no	no	INTJ
easat-4026	372	14	.	.	NOUN
easat-4026	372	15	3	3	NUM
easat-4026	372	16	,	,	PUNCT
easat-4026	372	17	pp	pp	ADJ
easat-4026	372	18	.	.	PUNCT
easat-4026	373	1	325	325	NUM
easat-4026	373	2	-	-	SYM
easat-4026	373	3	388	388	NUM
easat-4026	373	4	,	,	PUNCT
easat-4026	373	5	september	september	PROPN
easat-4026	373	6	2012	2012	NUM
easat-4026	373	7	.	.	PUNCT
easat-4026	374	1	https://doi.org/10.48550/arxiv.0912.3932	https://doi.org/10.48550/arxiv.0912.3932	PROPN
easat-4026	374	2	.	.	PUNCT
easat-4026	375	1	[	[	X
easat-4026	375	2	13	13	NUM
easat-4026	375	3	]	]	PUNCT
easat-4026	375	4	a.	a.	NOUN
easat-4026	375	5	h.	h.	PROPN
easat-4026	375	6	noreldeen	noreldeen	PROPN
easat-4026	375	7	,	,	PUNCT
easat-4026	375	8	and	and	CCONJ
easat-4026	375	9	y.	y.	PROPN
easat-4026	375	10	gh	gh	PROPN
easat-4026	375	11	.	.	PUNCT
easat-4026	376	1	gouda	gouda	NOUN
easat-4026	376	2	,	,	PUNCT
easat-4026	376	3	“	"	PUNCT
easat-4026	376	4	on	on	ADP
easat-4026	376	5	the	the	DET
easat-4026	376	6	simplicial	simplicial	ADJ
easat-4026	376	7	cohomology	cohomology	NOUN
easat-4026	376	8	theory	theory	NOUN
easat-4026	376	9	of	of	ADP
easat-4026	376	10	algebra	algebra	PROPN
easat-4026	376	11	”	"	PUNCT
easat-4026	376	12	,	,	PUNCT
easat-4026	376	13	life	life	NOUN
easat-4026	376	14	science	science	NOUN
easat-4026	376	15	journal	journal	PROPN
easat-4026	376	16	,	,	PUNCT
easat-4026	376	17	vol	vol	NOUN
easat-4026	376	18	.	.	PROPN
easat-4026	377	1	10	10	NUM
easat-4026	377	2	,	,	PUNCT
easat-4026	377	3	no	no	INTJ
easat-4026	377	4	.	.	NOUN
easat-4026	377	5	3	3	NUM
easat-4026	377	6	,	,	PUNCT
easat-4026	377	7	pp	pp	ADJ
easat-4026	377	8	.	.	PUNCT
easat-4026	378	1	2639	2639	NUM
easat-4026	378	2	-	-	SYM
easat-4026	378	3	2644	2644	NUM
easat-4026	378	4	,	,	PUNCT
easat-4026	378	5	2013	2013	NUM
easat-4026	378	6	.	.	PUNCT
easat-4026	379	1	doi	doi	NOUN
easat-4026	379	2	:	:	PUNCT
easat-4026	379	3	10.7537	10.7537	NUM
easat-4026	379	4	/	/	SYM
easat-4026	379	5	marslsj100313.380	marslsj100313.380	NOUN
easat-4026	379	6	[	[	X
easat-4026	379	7	14	14	NUM
easat-4026	379	8	]	]	PUNCT
easat-4026	379	9	a.	a.	PROPN
easat-4026	379	10	h.	h.	PROPN
easat-4026	379	11	noreldeen	noreldeen	PROPN
easat-4026	379	12	,	,	PUNCT
easat-4026	379	13	“	"	PUNCT
easat-4026	379	14	on	on	ADP
easat-4026	379	15	the	the	DET
easat-4026	379	16	hochschild	hochschild	ADJ
easat-4026	379	17	cohomology	cohomology	NOUN
easat-4026	379	18	theory	theory	NOUN
easat-4026	379	19	of	of	ADP
easat-4026	379	20	a∞-algebra	a∞-algebra	PROPN
easat-4026	379	21	”	"	PUNCT
easat-4026	379	22	,	,	PUNCT
easat-4026	379	23	sci	sci	PROPN
easat-4026	379	24	.	.	PUNCT
easat-4026	379	25	afr	afr	PROPN
easat-4026	379	26	.	.	PROPN
easat-4026	379	27	,	,	PUNCT
easat-4026	379	28	vol	vol	NOUN
easat-4026	379	29	.	.	PROPN
easat-4026	379	30	5	5	NUM
easat-4026	379	31	,	,	PUNCT
easat-4026	379	32	2019	2019	NUM
easat-4026	379	33	,	,	PUNCT
easat-4026	379	34	e00115	e00115	PROPN
easat-4026	379	35	.	.	PUNCT
easat-4026	379	36	https://doi.org/10.1016/j.sciaf.2019.e00115	https://doi.org/10.1016/j.sciaf.2019.e00115	PUNCT
easat-4026	380	1	[	[	X
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easat-4026	381	8	.	.	PUNCT
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easat-4026	383	20	information	information	NOUN
easat-4026	383	21	sciences	sciences	PROPN
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easat-4026	384	6	,	,	PUNCT
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easat-4026	384	8	-	-	SYM
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easat-4026	384	10	,	,	PUNCT
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easat-4026	384	12	.	.	PUNCT
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easat-4026	386	2	17	17	NUM
easat-4026	386	3	]	]	PUNCT
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easat-4026	386	7	,	,	PUNCT
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easat-4026	386	22	(	(	PUNCT
easat-4026	386	23	co)homology	co)homology	NOUN
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easat-4026	389	1	appl	appl	PROPN
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easat-4026	389	11	,	,	PUNCT
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easat-4026	389	13	.	.	PUNCT
easat-4026	390	1	issn	issn	PROPN
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easat-4026	390	4	-	-	SYM
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easat-4026	390	6	.	.	PUNCT
easat-4026	391	1	https://api.semanticscholar.org/corpusid:236770019	https://api.semanticscholar.org/corpusid:236770019	X
easat-4026	391	2	.	.	PUNCT
easat-4026	392	1	[	[	X
easat-4026	392	2	18	18	NUM
easat-4026	392	3	]	]	PUNCT
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easat-4026	392	5	h.	h.	PROPN
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easat-4026	392	17	,	,	PUNCT
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easat-4026	392	23	mathematical	mathematical	ADJ
easat-4026	392	24	sciences	science	NOUN
easat-4026	392	25	,	,	PUNCT
easat-4026	392	26	vol	vol	NOUN
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easat-4026	392	29	,	,	PUNCT
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easat-4026	392	34	,	,	PUNCT
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easat-4026	392	37	,	,	PUNCT
easat-4026	392	38	2012	2012	NUM
easat-4026	392	39	.	.	PUNCT
easat-4026	393	1	https://doi.org/10.1155/2012/368527	https://doi.org/10.1155/2012/368527	PROPN
easat-4026	394	1	[	[	X
easat-4026	394	2	19	19	NUM
easat-4026	394	3	]	]	PUNCT
easat-4026	394	4	a.	a.	PROPN
easat-4026	394	5	h.	h.	PROPN
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easat-4026	394	9	“	"	PUNCT
easat-4026	394	10	perturbation	perturbation	NOUN
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easat-4026	394	12	a	a	DET
easat-4026	394	13	-	-	PUNCT
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easat-4026	394	17	,	,	PUNCT
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easat-4026	394	19	mathematics	mathematics	PROPN
easat-4026	394	20	&	&	CCONJ
easat-4026	394	21	information	information	NOUN
easat-4026	394	22	sciences	sciences	PROPN
easat-4026	394	23	,	,	PUNCT
easat-4026	394	24	vol	vol	NOUN
easat-4026	394	25	.	.	PROPN
easat-4026	395	1	14	14	NUM
easat-4026	395	2	,	,	PUNCT
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easat-4026	395	5	3	3	NUM
easat-4026	395	6	,	,	PUNCT
easat-4026	395	7	pp	pp	ADJ
easat-4026	395	8	.	.	PUNCT
easat-4026	396	1	447	447	NUM
easat-4026	396	2	-	-	NUM
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easat-4026	396	4	,	,	PUNCT
easat-4026	396	5	2020	2020	NUM
easat-4026	396	6	,	,	PUNCT
easat-4026	396	7	http://dx.doi.org/10.18576/amis/140311	http://dx.doi.org/10.18576/amis/140311	NOUN
easat-4026	396	8	.	.	PUNCT
easat-4026	397	1	[	[	X
easat-4026	397	2	20	20	NUM
easat-4026	397	3	]	]	PUNCT
easat-4026	397	4	s.	s.	PROPN
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easat-4026	397	7	"	"	PUNCT
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easat-4026	397	13	"	"	PUNCT
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easat-4026	397	20	.	.	PROPN
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easat-4026	398	2	.	.	PUNCT
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easat-4026	398	4	.	.	PUNCT
easat-4026	399	1	math	math	NOUN
easat-4026	399	2	.	.	PUNCT
easat-4026	400	1	soc	soc	PROPN
easat-4026	400	2	.	.	PUNCT
easat-4026	400	3	,	,	PUNCT
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easat-4026	401	2	21	21	NUM
easat-4026	401	3	]	]	X
easat-4026	401	4	y.	y.	PROPN
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easat-4026	401	8	,	,	PUNCT
easat-4026	401	9	a.	a.	PROPN
easat-4026	401	10	h.	h.	PROPN
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easat-4026	401	12	and	and	CCONJ
easat-4026	401	13	mahmoud	mahmoud	PROPN
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easat-4026	401	35	(	(	PUNCT
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easat-4026	401	42	,	,	PUNCT
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easat-4026	401	45	,	,	PUNCT
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easat-4026	401	49	,	,	PUNCT
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easat-4026	401	51	.	.	PUNCT
easat-4026	402	1	e	e	X
easat-4026	402	2	-	-	PUNCT
easat-4026	402	3	issn	issn	ADJ
easat-4026	402	4	:	:	PUNCT
easat-4026	402	5	2321	2321	NUM
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easat-4026	402	7	4767	4767	NUM
easat-4026	402	8	p	p	NOUN
easat-4026	402	9	-	-	PUNCT
easat-4026	402	10	issn	issn	NOUN
easat-4026	402	11	:	:	PUNCT
easat-4026	402	12	2321	2321	NUM
easat-4026	402	13	-	-	SYM
easat-4026	402	14	4759	4759	NUM
easat-4026	402	15	.	.	PUNCT
easat-4026	403	1	https://www.ijmsi.org/papers/volume.5.issue.1/c05012331.pdf	https://www.ijmsi.org/papers/volume.5.issue.1/c05012331.pdf	PUNCT
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easat-4026	404	2	22	22	NUM
easat-4026	404	3	]	]	PUNCT
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easat-4026	404	12	spectral	spectral	ADJ
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easat-4026	404	14	”	"	PUNCT
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easat-4026	404	16	sbornik	sbornik	ADJ
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easat-4026	404	18	,	,	PUNCT
easat-4026	404	19	volume	volume	NOUN
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easat-4026	404	21	,	,	PUNCT
easat-4026	404	22	issue	issue	NOUN
easat-4026	404	23	1	1	NUM
easat-4026	404	24	,	,	PUNCT
easat-4026	404	25	pp	pp	ADJ
easat-4026	404	26	.	.	PUNCT
easat-4026	404	27	119142	119142	NUM
easat-4026	404	28	,	,	PUNCT
easat-4026	404	29	2002	2002	NUM
easat-4026	404	30	.	.	PUNCT
easat-4026	405	1	https://doi.org/10.1070/sm2002v193n01abeh000623	https://doi.org/10.1070/sm2002v193n01abeh000623	VERB
easat-4026	405	2	.	.	PUNCT
easat-4026	406	1	[	[	X
easat-4026	406	2	23	23	NUM
easat-4026	406	3	]	]	PUNCT
easat-4026	406	4	a.	a.	PROPN
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easat-4026	406	7	,	,	PUNCT
easat-4026	406	8	“	"	PUNCT
easat-4026	406	9	on	on	ADP
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easat-4026	406	11	cohomology	cohomology	NOUN
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easat-4026	406	14	banach	banach	NOUN
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easat-4026	406	16	”	"	PUNCT
easat-4026	406	17	,	,	PUNCT
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easat-4026	406	21	,	,	PUNCT
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easat-4026	406	23	.	.	PROPN
easat-4026	406	24	13	13	NUM
easat-4026	406	25	,	,	PUNCT
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easat-4026	406	28	10	10	NUM
easat-4026	406	29	,	,	PUNCT
easat-4026	406	30	2019	2019	NUM
easat-4026	406	31	.	.	PUNCT
easat-4026	407	1	issn	issn	PROPN
easat-4026	407	2	1913	1913	NUM
easat-4026	407	3	-	-	SYM
easat-4026	407	4	1844	1844	NUM
easat-4026	407	5	e	e	NOUN
easat-4026	407	6	-	-	PROPN
easat-4026	407	7	issn	issn	PROPN
easat-4026	407	8	1913	1913	NUM
easat-4026	407	9	-	-	SYM
easat-4026	407	10	1852	1852	NUM
easat-4026	407	11	.	.	PUNCT
easat-4026	408	1	https://doi.org/10.5539/mas.v13n10p1	https://doi.org/10.5539/mas.v13n10p1	ADJ
easat-4026	409	1	[	[	X
easat-4026	409	2	24	24	NUM
easat-4026	409	3	]	]	PUNCT
easat-4026	409	4	a.	a.	NOUN
easat-4026	409	5	h.	h.	PROPN
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easat-4026	409	7	,	,	PUNCT
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easat-4026	409	9	.	.	PROPN
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easat-4026	409	15	,	,	PUNCT
easat-4026	409	16	“	"	PUNCT
easat-4026	409	17	the	the	DET
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easat-4026	409	21	algebraic	algebraic	PROPN
easat-4026	409	22	(	(	PUNCT
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easat-4026	409	24	”	"	PUNCT
easat-4026	409	25	,	,	PUNCT
easat-4026	409	26	mathematics	mathematic	NOUN
easat-4026	409	27	and	and	CCONJ
easat-4026	409	28	statistics	statistic	NOUN
easat-4026	409	29	,	,	PUNCT
easat-4026	409	30	vol	vol	NOUN
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easat-4026	409	33	,	,	PUNCT
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easat-4026	409	35	.	.	NOUN
easat-4026	409	36	5	5	NUM
easat-4026	409	37	,	,	PUNCT
easat-4026	409	38	pp	pp	ADJ
easat-4026	409	39	.	.	PUNCT
easat-4026	410	1	639	639	NUM
easat-4026	410	2	647	647	NUM
easat-4026	410	3	,	,	PUNCT
easat-4026	410	4	2021	2021	NUM
easat-4026	410	5	.	.	PUNCT
easat-4026	411	1	doi	doi	NOUN
easat-4026	411	2	:	:	PUNCT
easat-4026	411	3	10.13189	10.13189	NUM
easat-4026	411	4	/	/	SYM
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easat-4026	411	6	.	.	PUNCT
easat-4026	412	1	[	[	X
easat-4026	412	2	25	25	NUM
easat-4026	412	3	]	]	PUNCT
easat-4026	412	4	m.	m.	NOUN
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easat-4026	412	6	,	,	PUNCT
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easat-4026	412	8	a.	a.	PROPN
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easat-4026	412	10	quota	quota	PROPN
easat-4026	412	11	,	,	PUNCT
easat-4026	412	12	a.	a.	PROPN
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easat-4026	412	15	,	,	PUNCT
easat-4026	412	16	“	"	PUNCT
easat-4026	412	17	the	the	DET
easat-4026	412	18	relative	relative	NOUN
easat-4026	412	19	(	(	PUNCT
easat-4026	412	20	co)homology	co)homology	NOUN
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easat-4026	412	25	”	"	PUNCT
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easat-4026	412	28	and	and	CCONJ
easat-4026	412	29	statistics	statistic	NOUN
easat-4026	412	30	,	,	PUNCT
easat-4026	412	31	vol	vol	NOUN
easat-4026	412	32	.	.	PROPN
easat-4026	412	33	10	10	NUM
easat-4026	412	34	,	,	PUNCT
easat-4026	412	35	no	no	INTJ
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easat-4026	412	37	3	3	NUM
easat-4026	412	38	,	,	PUNCT
easat-4026	412	39	pp	pp	ADJ
easat-4026	412	40	.	.	PUNCT
easat-4026	413	1	468	468	NUM
easat-4026	413	2	-	-	SYM
easat-4026	413	3	476	476	NUM
easat-4026	413	4	,	,	PUNCT
easat-4026	413	5	2022	2022	NUM
easat-4026	413	6	.	.	PUNCT
easat-4026	414	1	doi	doi	NOUN
easat-4026	414	2	:	:	PUNCT
easat-4026	414	3	10.13189	10.13189	NUM
easat-4026	414	4	/	/	SYM
easat-4026	414	5	ms.2022.100302	ms.2022.100302	NOUN
easat-4026	414	6	.	.	PUNCT
easat-4026	415	1	[	[	X
easat-4026	415	2	26	26	NUM
easat-4026	415	3	]	]	X
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easat-4026	415	6	,	,	PUNCT
easat-4026	415	7	(	(	PUNCT
easat-4026	415	8	2020	2020	NUM
easat-4026	415	9	)	)	PUNCT
easat-4026	415	10	.	.	PUNCT
easat-4026	416	1	“	"	PUNCT
easat-4026	416	2	long	long	ADJ
easat-4026	416	3	exact	exact	ADJ
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easat-4026	416	5	of	of	ADP
easat-4026	416	6	k	k	NOUN
easat-4026	416	7	-	-	PUNCT
easat-4026	416	8	groups	group	NOUN
easat-4026	416	9	”	"	PUNCT
easat-4026	416	10	,	,	PUNCT
easat-4026	416	11	topology	topology	NOUN
easat-4026	416	12	and	and	CCONJ
easat-4026	416	13	k	k	NOUN
easat-4026	416	14	-	-	NOUN
easat-4026	416	15	theory	theory	NOUN
easat-4026	416	16	.	.	PUNCT
easat-4026	417	1	lecture	lecture	NOUN
easat-4026	417	2	notes	note	NOUN
easat-4026	417	3	in	in	ADP
easat-4026	417	4	mathematics	mathematic	NOUN
easat-4026	417	5	,	,	PUNCT
easat-4026	417	6	vol	vol	NOUN
easat-4026	417	7	2262	2262	NUM
easat-4026	417	8	.	.	PUNCT
easat-4026	418	1	springer	springer	NOUN
easat-4026	418	2	,	,	PUNCT
easat-4026	418	3	cham	cham	PROPN
easat-4026	418	4	.	.	PUNCT
easat-4026	418	5	,	,	PUNCT
easat-4026	418	6	2020	2020	NUM
easat-4026	418	7	.	.	PUNCT
easat-4026	419	1	https://doi.org/10.1007/978-3-030-43996-5_36	https://doi.org/10.1007/978-3-030-43996-5_36	ADV
easat-4026	419	2	.	.	PUNCT
easat-4026	420	1	[	[	X
easat-4026	420	2	27	27	NUM
easat-4026	420	3	]	]	PUNCT
easat-4026	420	4	a.	a.	PROPN
easat-4026	420	5	h.	h.	PROPN
easat-4026	420	6	noreldeen	noreldeen	PROPN
easat-4026	420	7	,	,	PUNCT
easat-4026	420	8	“	"	PUNCT
easat-4026	420	9	on	on	ADP
easat-4026	420	10	the	the	DET
easat-4026	420	11	(	(	PUNCT
easat-4026	420	12	co)homology	co)homology	NOUN
easat-4026	420	13	with	with	ADP
easat-4026	420	14	inner	inner	ADJ
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easat-4026	420	16	of	of	ADP
easat-4026	420	17	schemes	scheme	NOUN
easat-4026	420	18	”	"	PUNCT
easat-4026	420	19	,	,	PUNCT
easat-4026	420	20	life	life	NOUN
easat-4026	420	21	science	science	NOUN
easat-4026	420	22	journal	journal	PROPN
easat-4026	420	23	,	,	PUNCT
easat-4026	420	24	vol	vol	NOUN
easat-4026	420	25	.	.	PROPN
easat-4026	421	1	11	11	NUM
easat-4026	421	2	,	,	PUNCT
easat-4026	421	3	no	no	INTJ
easat-4026	421	4	.	.	NOUN
easat-4026	421	5	12	12	NUM
easat-4026	421	6	,	,	PUNCT
easat-4026	421	7	pp	pp	ADJ
easat-4026	421	8	.	.	PUNCT
easat-4026	422	1	698	698	NUM
easat-4026	422	2	-	-	SYM
easat-4026	422	3	703	703	NUM
easat-4026	422	4	,	,	PUNCT
easat-4026	422	5	2014	2014	NUM
easat-4026	422	6	.	.	PUNCT
easat-4026	423	1	doi	doi	NOUN
easat-4026	423	2	:	:	PUNCT
easat-4026	423	3	10.7537	10.7537	NUM
easat-4026	423	4	/	/	SYM
easat-4026	423	5	marslsj111214.131	marslsj111214.131	NOUN
easat-4026	423	6	[	[	X
easat-4026	423	7	28	28	NUM
easat-4026	423	8	]	]	X
easat-4026	423	9	a.	a.	PROPN
easat-4026	423	10	h.	h.	PROPN
easat-4026	423	11	noreldeen	noreldeen	PROPN
easat-4026	423	12	,	,	PUNCT
easat-4026	423	13	“	"	PUNCT
easat-4026	423	14	excision	excision	NOUN
easat-4026	423	15	theory	theory	NOUN
easat-4026	423	16	in	in	ADP
easat-4026	423	17	the	the	DET
easat-4026	423	18	dihedral	dihedral	ADJ
easat-4026	423	19	and	and	CCONJ
easat-4026	423	20	reflexive	reflexive	ADJ
easat-4026	423	21	(	(	PUNCT
easat-4026	423	22	co)homology	co)homology	NOUN
easat-4026	423	23	of	of	ADP
easat-4026	423	24	algebras	algebra	NOUN
easat-4026	423	25	”	"	PUNCT
easat-4026	423	26	,	,	PUNCT
easat-4026	423	27	cogent	cogent	NOUN
easat-4026	423	28	mathematics	mathematic	NOUN
easat-4026	423	29	and	and	CCONJ
easat-4026	423	30	amp	amp	PROPN
easat-4026	423	31	;	;	PUNCT
easat-4026	423	32	statistics	statistic	NOUN
easat-4026	423	33	,	,	PUNCT
easat-4026	423	34	vol	vol	NOUN
easat-4026	423	35	.	.	PROPN
easat-4026	423	36	7	7	NUM
easat-4026	424	1	no	no	NOUN
easat-4026	424	2	.	.	NOUN
easat-4026	424	3	1	1	NUM
easat-4026	424	4	,	,	PUNCT
easat-4026	424	5	2020	2020	NUM
easat-4026	424	6	https://doi.org/10.1080/25742558.2020.1868135	https://doi.org/10.1080/25742558.2020.1868135	PUNCT
easat-4026	425	1	[	[	X
easat-4026	425	2	29	29	NUM
easat-4026	425	3	]	]	X
easat-4026	425	4	w.	w.	PROPN
easat-4026	425	5	johannes	johannes	PROPN
easat-4026	425	6	,	,	PUNCT
easat-4026	425	7	“	"	PUNCT
easat-4026	425	8	abstract	abstract	ADJ
easat-4026	425	9	excision	excision	NOUN
easat-4026	425	10	and	and	CCONJ
easat-4026	425	11	ℓ1	ℓ1	NOUN
easat-4026	425	12	-	-	PUNCT
easat-4026	425	13	homology	homology	NOUN
easat-4026	425	14	”	"	PUNCT
easat-4026	425	15	,	,	PUNCT
easat-4026	425	16	universität	universität	NOUN
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easat-4026	425	18	,	,	PUNCT
easat-4026	425	19	93040	93040	NUM
easat-4026	425	20	regensburg	regensburg	PROPN
easat-4026	425	21	,	,	PUNCT
easat-4026	425	22	germany	germany	PROPN
easat-4026	425	23	,	,	PUNCT
easat-4026	425	24	vol	vol	NOUN
easat-4026	425	25	.	.	PROPN
easat-4026	425	26	1	1	NUM
easat-4026	425	27	,	,	PUNCT
easat-4026	425	28	2022	2022	NUM
easat-4026	425	29	.	.	PUNCT
easat-4026	426	1	https://doi.org/10.48550/arxiv.2203.06120	https://doi.org/10.48550/arxiv.2203.06120	PROPN
easat-4026	427	1	[	[	X
easat-4026	427	2	30	30	NUM
easat-4026	427	3	]	]	PUNCT
easat-4026	427	4	m.	m.	NOUN
easat-4026	427	5	wodzicki	wodzicki	PROPN
easat-4026	427	6	,	,	PUNCT
easat-4026	427	7	“	"	PUNCT
easat-4026	427	8	excision	excision	NOUN
easat-4026	427	9	in	in	ADP
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easat-4026	427	11	homology	homology	NOUN
easat-4026	427	12	and	and	CCONJ
easat-4026	427	13	in	in	ADP
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easat-4026	427	15	algebraic	algebraic	ADJ
easat-4026	427	16	k	k	NOUN
easat-4026	427	17	-	-	NOUN
easat-4026	427	18	theory	theory	NOUN
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easat-4026	427	24	,	,	PUNCT
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easat-4026	427	27	129	129	NUM
easat-4026	427	28	,	,	PUNCT
easat-4026	427	29	no	no	INTJ
easat-4026	427	30	.	.	NOUN
easat-4026	427	31	3	3	NUM
easat-4026	427	32	,	,	PUNCT
easat-4026	427	33	pp	pp	ADJ
easat-4026	427	34	.	.	PUNCT
easat-4026	428	1	591–639	591–639	NUM
easat-4026	428	2	,	,	PUNCT
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easat-4026	428	4	.	.	PUNCT
easat-4026	429	1	https://doi.org/10.2307/1971518	https://doi.org/10.2307/1971518	NOUN
easat-4026	429	2	.	.	PUNCT
easat-4026	430	1	[	[	X
easat-4026	430	2	31	31	NUM
easat-4026	430	3	]	]	PUNCT
easat-4026	430	4	m.	m.	NOUN
easat-4026	430	5	wodzicki	wodzicki	PROPN
easat-4026	430	6	,	,	PUNCT
easat-4026	430	7	“	"	PUNCT
easat-4026	430	8	excision	excision	NOUN
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easat-4026	430	15	algebraic	algebraic	ADJ
easat-4026	430	16	k	k	NOUN
easat-4026	430	17	-	-	NOUN
easat-4026	430	18	theory	theory	NOUN
easat-4026	430	19	”	"	PUNCT
easat-4026	430	20	,	,	PUNCT
easat-4026	430	21	annals	annal	NOUN
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easat-4026	430	26	.	.	PROPN
easat-4026	430	27	129	129	NUM
easat-4026	430	28	,	,	PUNCT
easat-4026	430	29	no	no	INTJ
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easat-4026	430	31	3	3	NUM
easat-4026	430	32	,	,	PUNCT
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easat-4026	430	34	,	,	PUNCT
easat-4026	430	35	pp	pp	ADP
easat-4026	430	36	.	.	PUNCT
easat-4026	431	1	591–639	591–639	NUM
easat-4026	431	2	,	,	PUNCT
easat-4026	431	3	2024	2024	NUM
easat-4026	431	4	.	.	PUNCT
easat-4026	432	1	https://doi.org/10.2307/1971518	https://doi.org/10.2307/1971518	NOUN
easat-4026	432	2	.	.	PUNCT
easat-4026	433	1	[	[	X
easat-4026	433	2	32	32	NUM
easat-4026	433	3	]	]	PUNCT
easat-4026	433	4	p.	p.	NOUN
easat-4026	433	5	megan	megan	NOUN
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easat-4026	433	7	“	"	PUNCT
easat-4026	433	8	an	an	DET
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easat-4026	433	11	for	for	ADP
easat-4026	433	12	persistent	persistent	ADJ
easat-4026	433	13	homology	homology	NOUN
easat-4026	433	14	”	"	PUNCT
easat-4026	433	15	,	,	PUNCT
easat-4026	433	16	arxiv:1910.03348v1	arxiv:1910.03348v1	PROPN
easat-4026	433	17	,	,	PUNCT
easat-4026	433	18	2019	2019	NUM
easat-4026	433	19	.	.	PUNCT
easat-4026	434	1	https://doi.org/10.48550/arxiv.1910.03348	https://doi.org/10.48550/arxiv.1910.03348	PROPN
easat-4026	434	2	.	.	PUNCT
easat-4026	435	1	[	[	X
easat-4026	435	2	33	33	NUM
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easat-4026	435	7	,	,	PUNCT
easat-4026	435	8	a.h	a.h	PROPN
easat-4026	435	9	.	.	PROPN
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easat-4026	435	11	,	,	PUNCT
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easat-4026	435	15	,	,	PUNCT
easat-4026	435	16	samar	samar	PROPN
easat-4026	435	17	a.a	a.a	PROPN
easat-4026	435	18	.	.	PROPN
easat-4026	435	19	quota	quota	PROPN
easat-4026	435	20	,	,	PUNCT
easat-4026	435	21	“	"	PUNCT
easat-4026	435	22	on	on	ADP
easat-4026	435	23	the	the	DET
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easat-4026	435	25	homology	homology	NOUN
easat-4026	435	26	theory	theory	NOUN
easat-4026	435	27	of	of	ADP
easat-4026	435	28	algebras	algebra	NOUN
easat-4026	435	29	”	"	PUNCT
easat-4026	435	30	,	,	PUNCT
easat-4026	435	31	scientific	scientific	ADJ
easat-4026	435	32	african	african	PROPN
easat-4026	435	33	,	,	PUNCT
easat-4026	435	34	vo	vo	NOUN
easat-4026	435	35	.	.	PROPN
easat-4026	435	36	25	25	NUM
easat-4026	435	37	,	,	PUNCT
easat-4026	435	38	september	september	PROPN
easat-4026	435	39	2024	2024	NUM
easat-4026	435	40	,	,	PUNCT
easat-4026	435	41	e02288	e02288	PROPN
easat-4026	435	42	.	.	PUNCT
easat-4026	436	1	https://doi.org/10.1016/j.sciaf.2024.e02288	https://doi.org/10.1016/j.sciaf.2024.e02288	ADJ
easat-4026	436	2	.	.	PUNCT
easat-4026	437	1	[	[	X
easat-4026	437	2	34	34	NUM
easat-4026	437	3	]	]	PUNCT
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easat-4026	437	9	a.	a.	PROPN
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easat-4026	437	11	quota	quota	PROPN
easat-4026	437	12	,	,	PUNCT
easat-4026	437	13	o.	o.	PROPN
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easat-4026	437	16	,	,	PUNCT
easat-4026	437	17	w.	w.	PROPN
easat-4026	437	18	m.	m.	PROPN
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easat-4026	437	21	“	"	PUNCT
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easat-4026	437	26	cohomology	cohomology	NOUN
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easat-4026	437	28	operator	operator	NOUN
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easat-4026	437	30	”	"	PUNCT
easat-4026	437	31	,	,	PUNCT
easat-4026	437	32	scientific	scientific	ADJ
easat-4026	437	33	african	african	ADJ
easat-4026	437	34	,	,	PUNCT
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easat-4026	437	36	,	,	PUNCT
easat-4026	437	37	september	september	PROPN
easat-4026	437	38	2024	2024	NUM
easat-4026	437	39	,	,	PUNCT
easat-4026	437	40	e02325	e02325	PROPN
easat-4026	437	41	.	.	PUNCT
easat-4026	438	1	https://doi.org/10.1016/j.sciaf.2024.e02325	https://doi.org/10.1016/j.sciaf.2024.e02325	PROPN
easat-4026	438	2	https://doi.org/10.48550/arxiv.math/9910179	https://doi.org/10.48550/arxiv.math/9910179	VERB
easat-4026	439	1	https://doi.org/10.48550/arxiv.0912.3932	https://doi.org/10.48550/arxiv.0912.3932	PROPN
easat-4026	439	2	https://doi.org/10.1016/j.sciaf.2019.e00115	https://doi.org/10.1016/j.sciaf.2019.e00115	NOUN
easat-4026	440	1	https://doi.org/10.1007/978-81-322-2843-1_10	https://doi.org/10.1007/978-81-322-2843-1_10	NOUN
easat-4026	440	2	http://dx.doi.org/10.18576/amis/140415	http://dx.doi.org/10.18576/amis/140415	NOUN
easat-4026	440	3	https://api.semanticscholar.org/corpusid:236770019	https://api.semanticscholar.org/corpusid:236770019	NUM
easat-4026	440	4	https://doi.org/10.1155/2012/368527	https://doi.org/10.1155/2012/368527	PROPN
easat-4026	440	5	http://dx.doi.org/10.18576/amis/140311	http://dx.doi.org/10.18576/amis/140311	NOUN
easat-4026	440	6	https://www.ijmsi.org/papers/volume.5.issue.1/c05012331.pdf	https://www.ijmsi.org/papers/volume.5.issue.1/c05012331.pdf	PROPN
easat-4026	440	7	https://doi.org/10.1070/sm2002v193n01abeh000623	https://doi.org/10.1070/sm2002v193n01abeh000623	PROPN
easat-4026	440	8	https://doi.org/10.1007/978-3-030-43996-5_36	https://doi.org/10.1007/978-3-030-43996-5_36	PROPN
easat-4026	440	9	https://doi.org/10.2307/1971518	https://doi.org/10.2307/1971518	VERB
easat-4026	440	10	https://doi.org/10.48550/arxiv.1910.03348	https://doi.org/10.48550/arxiv.1910.03348	PROPN
easat-4026	440	11	https://doi.org/10.48550/arxiv.1910.03348	https://doi.org/10.48550/arxiv.1910.03348	PROPN
easat-4026	440	12	https://doi.org/10.1016/j.sciaf.2024.e02288	https://doi.org/10.1016/j.sciaf.2024.e02288	NOUN
easat-4026	440	13	https://doi.org/10.1016/j.sciaf.2024.e02325	https://doi.org/10.1016/j.sciaf.2024.e02325	X