Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6, 9472-9486 2024 Publisher: Learning Gate DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate © 2024 by the authors; licensee Learning Gate * Correspondence: fatma.elzhraa6590@gmail.com The excision theory for homology theory through A_∞-algebras Faten Ragab Karar1, Fatma Elzhraa Ahmed Mohammed2*, A. A. El Fattah3 1Department of Mathematics, Faculty of Science, Aswan University, Aswan, Egypt; fatenragab2020@yahoo.com (F.R.K.). 2,3Mathematics teacher in secondary school, Ministry of Education; fatma.elzhraa6590@gmail.com (F.A.M.) Ahmedfarrouk222@gmail.com (A.A.E.F.). Abstract: This paper investigated A_∞-algebras, which are generalizations of associative algebras that incorporate higher homotopy structures. We began by revisiting the fundamental definitions and properties of A_∞-algebras and their associated homological theories, providing a solid foundation for understanding these complex structures. The study included an in-depth analysis of simplicial homology as it relates to A_∞-algebras, focusing on significant results, particularly those concerning excision theory. In this context, we introduced new insights into the relationship between bar homology and simplicial homology, presenting a precise sequence elucidating the interaction between these two homological structures. Within this framework, we provided proofs for key results, such as the quantitative coherence of certain maps and the interchanging diagram that connects different homological categories. We address the specific failure of excision properties and its implications for long exact sequences in both homological and homotopical contexts. This paper offered a comprehensive overview of current developments in A_∞-algebra theory and simplicial cohomology, highlighting classical and contemporary insights into these sophisticated mathematical structures. By presenting detailed definitions, examples, and theorems, we strive to contribute to a deeper understanding of homology within the framework of advanced algebraic systems. Our analysis sheds light on existing theories and paves the way for future research in the field, providing a valuable resource for mathematicians interested in the interplay between algebra and topology. Keywords: A_∞-algebras, Excision, Hochschild, Homology. 1. Introduction As demonstrated by Riemann's solutions to problems involving surface connections, homology is pivotal in mathematical investigations. Green's theorem, which relates line integrals over a closed curve to double integrals over the enclosed plane region, underpins this principle. The theorem implies that certain integrals will yield identical values for any two homologous curves, an idea that permeates classical vector spaces and theoretical physics . Simplicial homology, a branch of algebraic topology, addresses the study of homology theory in associative algebras over a field. Gerhard Hochschild introduced this theory [1], initially focusing on algebras over a field. Henri Cartan and Samuel Eilenberg [2] later extended this work to more generalized rings . In [3], Stasheff developed the concepts of 𝒜∞-spaces and 𝒜∞-algebras during his study of topological spaces. These ideas emerged as generalizations of topological groups while maintaining continuous, albeit non-associative, multiplication. 𝒜∞-algebras are essentially chain complexes with homotopy associative products, satisfying higher homotopy associativity conditions. A prime example of an 𝒜∞-algebra is the singular chain complex 𝒞●(𝒳), which illustrates their homotopy-invariant nature. Research into 𝒜∞-algebras has advanced significantly due to contributions from mathematicians such as Kadeishvili [4], Smirnov [5], and Prouté [6]. J. Huebschmann highlighted the relevance of homological perturbation theory and 𝒜∞-structures in homological algebra, especially within 9473 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate topological contexts [7]. Further advancements were made by John D. S. Jones and E. Getzler [8]. Kenji Fukaya's exploration of 𝒜∞-categories [9] and Kontsevich's influential 1994 lecture on categorical mirror symmetry contributed significantly to the field's development [10]. In [11], Keller integrated 𝒜∞-language into ring theory and representation theory, showing that the derived category of any Grothendieck category with a compact generator is equivalent to the derived category of an 𝒜∞-algebra. Additional studies by P. Seidel on 𝒜∞-structures concerning Lefschetz fibrations [12], and research by Alaa H. and Y. Gouda on simplicial cohomology for 𝒜∞- algebras [13,14] expanded our understanding of these structures. The lack of excision in specific contexts leads to long exact sequences in homological categories but not in homotopic categories. This paper presents various definitions of homology for 𝒜∞-algebras, examining the simplicial homology of these structures and providing reliable results for their excision theory. We explored the excision theory of simplicial homology for 𝒜∞-algebras, establishing the relationship between bar homology ℋ𝐵𝑛(ℐ) and simplicial homology ℋℋ𝑛(ℐ) through an exact sequence: … ← 𝐻𝑛−1(ℐ) ← ℋ𝐵𝑛−1(ℐ) ← ℋℋ𝑛(ℐ) ← 𝐻𝑛(ℐ) ← ℋ𝐵𝑛(ℐ) ← ℋℋ𝑛+1(ℐ) ← ….. Additionally, we proved that the quasi-isomorphisms for the following maps: 𝑖: (ℛ ⊗ ℐ⨂∗, 𝜌∗)⊗ 𝒢 ↪ (ℛ ⊗𝒜⨂∗, 𝜌∗)⊗ 𝒢 , 𝑖′: (ℛ ⊗ ℐ⨂∗, 𝜌∗ ′)⊗ 𝒢 ↪ (ℛ ⊗𝒜⨂∗, 𝜌∗ ′)⊗ 𝒢, are quasi-isomorphisms, as are the two maps: 𝜋: (ℬ ⊗𝒜⊗∗, 𝜌∗)⊗ 𝒢 → (ℬ ⊗ℬ⊗∗, 𝜌∗)⊗ 𝒢, 𝜋′: (ℬ ⊗𝒜⊗∗, 𝜌′∗)⊗ 𝒢 → (ℬ ⊗ℬ⊗∗, 𝜌′∗)⊗ 𝒢, These results enable us to construct a commutative diagram, elucidating the intricate relationships between these homological structures: 0 0 → → (ℐ ⊗𝒜⨂ ∗, 𝜌∗) ⊗ 𝒢 |𝑗 𝑘𝑒𝑟 (𝜋) → → (𝒜 ⊗𝒜⨂ ∗, 𝜌∗) ⊗ 𝒢 | = (𝒜 ⊗𝒜⨂ ∗, 𝜌∗) ⊗ 𝒢 → 𝜋 → (ℬ ⊗𝒜⨂ ∗, 𝜌∗) ⊗ 𝒢 |𝜋1 (ℬ ⊗𝒜⨂∗ , 𝜌∗) ⊗ 𝒢 → → 0 0 . 2. Homology theory of 𝓐∞-algebras In this section, we will explore the fundamentals and definitions related to the homology theory of 𝒜∞-algebras. We begin by introducing the basic definitions and concepts of 𝒜∞-algebras. Following this, we will present the simple homology of 𝒜∞-algebras. Firstly, we define an algebra over the field ℛ, focusing on properties. 2.1. Definition [2] Over the field ℛ, algebra is represented by the linear vector space 𝒳, which has the multiplication function 𝒯:𝒳 ×𝒳 → 𝒳, (𝓋,𝓊) ↦ 𝓋𝓊, that which 𝒯 is distributed and linear in the two variables. That applies for all 𝓋, 𝓊,𝓌 ∈ 𝒳, 𝛼 ∈ ℛ ▪ 𝓌(𝓋 +𝓊) = 𝓌𝓊 +𝓌𝓊. ▪ (𝓋 + 𝓊)𝓌 = 𝓋𝓌 +𝓊𝓌. ▪ 𝛼(𝓋𝓊) = (𝛼𝓋)𝓊 = 𝓋(𝛼𝓊). Following, we define graded vector spaces where we specify the vector space 𝒳 and related graded vector spaces 𝒳𝚤. We describe how to handle homogeneous elements in these spaces. 2.2. Definition [3] Suppose ℐ denotes the set of index. The vector space 𝒳 that has the grade ℐ, known as the ℐ-graded vector space, takes the following form 𝒳 = ⨁ 𝚤 ∈ ℐ 𝒳𝚤, hence for each 𝚤, and then 𝒳𝚤 would be a vector 9474 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate space. The elements 𝓍 ∈ 𝒳𝚤 are hence known as homogeneous elements with degree 𝚤 and denoted by 𝑑𝑒𝑔 𝓍 = 𝚤 or |𝓍| = 𝚤. Next, we introduce the tensor product of vector spaces 𝒳 and 𝒴 over a field 𝔉. We define the bilinear map and discuss the properties of these tensor spaces. 2.3. Definition [3] If the vector spaces 𝒳,𝒴 over a field 𝔉 with elements {𝓍𝚤}𝚤∈ℐ and {𝓎ℓ}ℓ∈ℒ; then, the tensor product 𝒳 ⊗𝒴 is defined as a vector space over 𝔉 with the symbols {𝓍𝚤 ⊗𝓎ℓ , ∀𝚤 ∈ ℐ , ℓ ∈ ℒ} as its basis. Additionally, we define the bilinear map 𝒳 × 𝒴 → 𝒳⊗𝒴 as the two vectors' combined tensor product 𝓍 = ∑ 𝒶𝚤𝓍𝚤𝚤 and 𝓎 = ∑ 𝒷ℓ𝓎ℓℓ , which is provided by: 𝒳⊗𝒴 = (∑ 𝒶𝚤𝓍𝚤𝚤 ) ⊗ (∑ 𝒷ℓ𝓎ℓℓ ) = ∑ 𝒶𝚤𝒷ℓ(𝓍𝚤 ⊗𝓎ℓ𝚤,ℓ ). In the following proposition, we will prove the existence of a unique linear map between the tensor products of vector spaces. We will demonstrate how to establish this map and its distinctive properties. 2.4. Proposition Assume that ℳ is the vector space. A unique linear map ℜ′:𝒳 ⊗𝒴 →ℳ exists for the bilinear map ℜ:𝒳 × 𝒴 →ℳ, such as ℜ = ℜ′ ∘ 𝜙 and 𝜙 is the normal incorporation of 𝒳 ×𝒴 in 𝒳 ⊗𝒴. Additionally, the universal characteristic is satisfied by the unique isomorphism in a vector space with a bilinear map. Proof: The relation ℜ′(𝓍𝚤 ⊗𝓎ℓ) = 𝛽(𝓍𝚤, 𝓎ℓ) is a basic 𝒳⊗𝒴 in ℜ′. This map is unique because it satisfies the requirement for ℜ = ℜ′ ∘ 𝜙. Now, let 𝒩 be the vector space with the bilinear map 𝜔:𝒳 × 𝒴 → 𝒩. So, for each bilinear map ℜ:𝒳 × 𝒴 →ℳ, there is a unique linear map ℜ′, for instance ℜ = ℜ′ ∘ 𝜙. Suppose that ℳ = 𝒳⊗𝒴 and ℜ is the map (𝓍, 𝓎) ⟼ 𝓍⊗𝓎. Therefore, through the general property of 𝒩, a unique linear map ℜ′:𝒩 → 𝒳⊗𝒴 exists as ℜ = ℜ′ ∘ 𝜙. Likewise, through the general property of 𝒳 ⊗𝒴, there exists the unique linear map ℰ′:𝒳 ⊗𝒴 → 𝒩 as 𝜔 = ℰ′ ∘ 𝜙 so 𝜙 = (ℜ′ ∘ ℰ′) ∘ 𝜙. Therefore, ℜ′ ∘ ℰ′ = 𝑖𝑑 and ℜ′ is unique isomorphism ℜ′:𝒩 ∼ → 𝒳⊗𝒴. In the following, we define a graded algebra over a field ℛ and how to deal with the resulting multiplicity of multiplication maps in this type of algebra. 2.5. Definition [14] Suppose ℳ is the graded vector space on the field ℛ, then the graded algebra on ℛ is defined as the algebra ℳ such as ℳ = ⊕ 𝚤 ∈ ℐ ℳ𝚤 , and the multiplication map for all 𝓂,𝓃 ∈ ℳ is given by: 𝑑𝑒𝑔 𝓂𝓃 = 𝑑𝑒𝑔 𝓂 + 𝑑𝑒𝑔 𝓃. After that, we will introduce the construction of a tensor algebra from a vector space ℳ, explaining how this construction produces a graded algebra defined by the tensor product. 2.6. Definition [15] If ℛ is a field, then the vector space ℳ provides the tensor algebra of ℳ by: 𝑇(ℳ) = ℛ⊕ℳ⊕ℳ⊗2⊕ℳ⊗3⊕… The tensor product's presumed multiplication appears as a graded algebra. In the following definition, we define linear morphisms between graded algebras using the tensor product and describe the degree-related properties of these morphisms. 9475 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate 2.7. Definition [15] Suppose that ℳ = ⊗𝑛∈𝑍 ℳ𝑛 is the ℤ-graded vector space. Then, the homeomorphisms of vector spaces are linear maps 𝑓:ℳ⊗𝑛 →ℳ. Since the degree of 𝑓, |𝑓| is given by 𝚤, when 𝑑𝑒𝑔 𝑓(𝓂) = (𝑑𝑒𝑔 𝓂) + 𝚤 for all 𝓂 ∈ ℳ . Obviously, |𝑓(𝓂1⊗𝓂2⊗ · · · ⊗𝓂𝑛)| = (|𝓂1| + |𝓂2| + · · · + |𝓂𝑛|) + 𝚤. We will provide an illustrative example of a graded algebra using a complex sequence over a specific field and explaine how to explore homology through this example. 2.8. Example Assume the complex over a field known as: 𝑉• : … → 𝑉𝑛+1 𝒹𝑛+1 → 𝑉𝑛 𝒹𝑛 → 𝑉𝑛−1 → … , hence the modules 𝑉𝑛 seem to be, in fact, vector spaces. So 𝑉 = ⨁𝑛𝑉𝑛 is known as the graded vector space, where 𝒹:𝑉⊗1 → 𝑉 have a degree −1. Following, we define morphisms between 𝒜∞-algebras and explain the conditions and properties necessary for these morphisms. 2.9. Definition [13] By assuming that 𝛽, 𝛾 are two linear maps of the graded vector spaces 𝒳,𝒴 respectively, such that 𝛽:𝒳 → 𝒳 , 𝛾: 𝒴 → 𝒴. Next, the linear maps' tensor product 𝛽 and 𝛾 is denoted by: 𝛽 ⊗ 𝛾 ∶ 𝒳 ⊗𝒴 → 𝒳⊗𝒴 (𝓍 ⊗𝓎) ⟼ (−1)|𝛾||𝓎|(𝛽(𝓍)⊗ 𝛾(𝓎)). (1) Remind that the degree of (𝛽 ⊗ 𝛾) is |𝛽| + |𝛾|. Remark: The Koszul sign rule states that if two symbols' positions 𝓈 and 𝓇 are switched, the outcome is multiplied by (−1)|𝓈||𝓇|, which is the actual cause of the change in sign in the previous expression. In the expression above, it is applied as (𝛽 ⊗ 𝛾)(𝓍 ⊗𝓎) = (−1)|𝛾||𝓎|(𝛽(𝓍)⊗ 𝛾(𝓎)), where the morphisms acting on the elements are represented by the symbols 𝓍 and 𝓎, which are swapped. In the following, we describe the concept of differential graded algebras (DGAs) and how to define them using the chain complex maps and Leibniz rules. 2.10. Definition [15] The differential graded algebra (DGA) is the complex ℳ with the degree +1 chain map 𝒹:ℳ⊗ℳ → ℳ so that 𝒹2 = 0, which is unital and associative. The differential graded algebra is simply a differential 𝒹 and a graded module equivalent. The graded Leibniz rule for all 𝓂,𝓃 ∈ ℳ is denoted by: 𝒹(𝓂𝓃) = 𝒹(𝓂)𝓃 + (−1)|𝓂|𝒹(𝓃). (2) Remark: (1) If 𝓂𝓃 = (−1)|𝓂||𝓃|𝓃𝓂, for all 𝓂,𝓃 ∈ ℳ, then the differential graded algebra ℳ is regarded as commutative. If 1/2 ∈ 𝑘, it is implied that 𝓂2 = 0 holds true if 𝓂 would have an odd degree. When ℳ is the commutative graded algebra, therefore the left ℳ-module, and automatically 𝑅 is a right ℳ-module, by the formula 𝑟𝓂 = (−1)|𝑟||𝓂|𝓂𝑟. (2) Similar definitions are provided for the terms differential graded algebra and derivation. Here, we will explain how to handle left models of differential graded algebras and use rules for dealing with derivatives in these models. 9476 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate 2.11. Definition [16] Assume that the differential graded algebra is (ℳ,𝒹). In addition, let 𝑅 be the left graded ℳ- module and 𝒹𝑅 be the differential of 𝑅, so the complex (𝑅, 𝒹𝑅)and the left multiplication ℳ⊗𝑅 → 𝑅 combined to create the left differential graded ℳ-module such that the Leibniz rule satisfies for all 𝓂 ∈ ℳ, 𝑟 ∈ 𝑅: 𝒹𝑅(𝓂𝑟) = 𝒹(𝓂)𝑟 + (−1)|𝓂|𝓂𝒹𝑅(𝑟). An ℳ-module with differential grading is merely a complex. Similar terms are used to define a right differential graded ℳ-module. After that, we define graded spaces using morphisms and methods to determine derivatives within these graded spaces. 2.12. Definition [16] Let 𝐻𝑜𝑚ℳ(𝑅, 𝑆) be a graded vector space that contains ℳ-homomorphisms as of 𝑅 to 𝑆. Graded modules: 𝐻𝑜𝑚ℳ(𝑅, 𝑆) ∶=⊕ 𝚤∈ℤ 𝐻𝑜𝑚ℳ 𝐺𝑟(𝑅, 𝑆)𝚤, using the differential 𝒹𝐻𝑜𝑚 determined to be: 𝒹𝐻𝑜𝑚(𝒽) = 𝒹𝑆 ∘ 𝒽 − (−1)|𝒽|𝒽 ∘ 𝒹𝑅 , ∀ 𝒽 ∈ 𝐻𝑜𝑚ℳ(𝑅, 𝑆). In particular, differential graded algebras are used to create the graded vector space 𝐻𝑜𝑚ℳ(𝑅, 𝑅), where 𝐻𝑜𝑚ℳ(𝐿, 𝑅) is a differential-graded module over 𝐻𝑜𝑚ℳ(𝑅, 𝑅) differential-graded algebras. Next, we will describe how to handle the tensor product of graded spaces and determine the graded map's properties in this context. 2.13. Definition [17] By the differential 𝒹⊗ and the complex ℳ, we can define the graded vector space for a tensor product 𝑅 ⊗ℳ 𝑆 over ℳ as: 𝒹⊗(𝑟 ⊗ℳ 𝑠) = 𝒹𝑅(𝑟) ⊗ℳ 𝑠 + (−1)|𝑟|𝑟 ⊗ℳ 𝒹𝑆(𝑠). There is an adjoint property between 𝐻𝑜𝑚ℳ and ⊗ℳ: 𝐻𝑜𝑚ℳ(𝐿 ⊗ℳ 𝑅, 𝑆) ≅ 𝐻𝑜𝑚ℳ(𝐿, 𝐻𝑜𝑚ℳ(𝑅, 𝑆)). Here, we will define 𝒜∞-algebras and how to handle their fundamental properties, such as the Stasheff identity. 2.14. Definition [18] A vector space with ℤ grades is an 𝒜∞-algebras on a field 𝑀 such that: 𝒜 =⊕ 𝓅∈ℤ 𝒜𝒫, supplied with graded maps that are homogenous 𝑀-linear mappings; 𝑟𝑛:𝒜 ⊗𝑛 → 𝒜, 𝑛 ≥ 1. of (|𝑟𝑛| = 2 − 𝑛)-degree, fulfilling the subsequent requirements of Stasheff identities: ∀ 𝑛 ∈ ℕ, (𝑆𝐿(𝑛)) ∑(−1)𝑚+𝑠𝑡𝑟𝓆(𝑖𝑑 ⊗𝑚 ⊗ 𝑟𝑠⊗ 𝑖𝑑⊗𝑡) = 0 (3) where the total is applied to all decompositions 𝑛 = 𝑚 + 𝑠 + 𝑡, 𝑚, 𝑡 ≥ 0 and 𝑠 ≥ 1, and 𝓆 = 𝑚 + 1 + 𝑡. Thus, 𝑖𝑑 refers to the identification map of 𝒜. Because of the Koszul sign rule, additional signs occur when these formulas get used on elements. 𝒜∞-algebras are known as strongly homotopy associative algebras. For example: 1) 𝑆𝐿(1) means that: 𝑟1 ∘ 𝑟1 = 0. where 𝑟1 has a degree of 1, which means that 𝑟1 is a derivative of 𝒜. 2) 𝑆𝐿(2) says that 𝑟1 is a derivation for 𝑟2, such that: 𝑟1 ∘ 𝑟2 = 𝑟2 ∘ ( 𝑟1⊗ 𝑖𝑑 + 𝑖𝑑 ⊗ 𝑟1) The degree of 𝑟2 is zero. 3) 𝑆𝐿(3) indicates that 𝑟2 is associative till the homotopy 𝑟3 can be re-written as: 9477 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate 𝑟2 ∘ (𝑖𝑑 ⊗ 𝑟2 − 𝑟2 ⊗ 𝑖𝑑) = 𝜕(𝑟3) = 𝑟1 ∘ 𝑟3 + 𝑟3 ∘ (𝑟1⊗ 𝑖𝑑 ⊗ 𝑖𝑑 + 𝑖𝑑 ⊗ 𝑟1⊗ 𝑖𝑑 + 𝑖𝑑 ⊗ 𝑖𝑑 ⊗ 𝑟1) where 𝜕 is the differential of 𝐻𝑜𝑚(𝒜⊗3,𝒜) induced by 𝑟1. 4) Pentagonal homotopy associative algebra (𝒜, 𝑟1, 𝑟2, 𝑟3): 𝑆𝐿(4): 𝑟2 ∘ (𝑖𝑑 ⊗ 𝑟3 + 𝑟3 ⊗ 𝑖𝑑) = 𝑟3 ∘ (𝑟2⊗ 𝑖𝑑 ⊗ 𝑖𝑑 − 𝑖𝑑 ⊗ 𝑟2 ⊗ 𝑖𝑑 + 𝑖𝑑 ⊗ 𝑖𝑑 ⊗ 𝑟2). We will present an example of an 𝒜∞-algebra with degree 0 and analyze its properties through specific assignments. 2.15. Example Let the associative algebras 𝒜 be an 𝒜∞-algebras focused at degree 0 through all multiplications 𝑟𝑛 = 0 for 𝑛 ≠ 2. As a result, the associative algebras make a subclass of 𝒜∞-algebras with the form (𝒜, 𝑟2). We will define morphisms between 𝒜∞-algebras and describe how to handle these morphisms' necessary conditions and properties. 2.16. Definition [19] Suppose that 𝒜 and ℬ are both 𝒜∞-algebras, then the family of 𝑀-linear graded maps is the morphism 𝒽:𝒜 → ℬ of 𝒜∞-algebras such that: 𝒽𝑛:𝒜 ⊗𝑛 → ℬ 𝑛 ≥ 1 of degree (1 − 𝑛), where the subsequent identities apply for each 𝑛 ≥ 1: ∑ (−1)𝑚+𝑠𝑡𝒽𝑚+1+𝑡 ∘ (𝑖𝑑 ⊗𝑚 ⊗𝑟𝑠 ⊗ 𝑖𝑑⊗𝑡) 𝑚+𝑠+𝑡=1 𝑚,𝑡≥0 𝑠≥1 = ∑ ∑ (−1)𝓆𝑟𝑗 ′(𝒽𝑖1⊗𝒽𝑖2 ⊗…⊗𝒽𝑖𝑗 ) 𝑖1+···+𝑖𝑗=𝑛 𝑛 𝑗=1 , (𝑀𝐿(𝑛)) where: 𝓆 = (𝑖𝑗−1 − 1) + 2(𝑖𝑗−2 − 1) +⋯+ (𝑗 − 2)(𝑖2 − 1) + (𝑗 − 1)(𝑖1 − 1). Assuming 𝒜 and ℬ are both unital 𝒜∞-algebras of stringent units 1𝒜 and 1ℬ respectively, so 𝒽 additionally needs to fulfill the following unital morphism requirements: 1) As required for ring morphisms, 𝒽1(1𝒜) = 1ℬ. 2) If 𝒶𝑖 = 1𝒜 , then 𝒽𝑛(𝒶1⊗…⊗𝒶𝑛) = 0 for all 𝑛 ≥ 2 , 𝑖 ∈ {1,… , 𝑛}. 3) A stringent morphism over 𝒜∞-algebras is 𝒽:𝒜 → ℬ when 𝒽𝑛 = 0 for all 𝑛 ≥ 2. 4) When 𝒽:𝒜 → ℬ is a stringent morphism, the morphism identification 𝑀𝐿(𝑛) changes to 𝒽1𝑟𝑛 = 𝑟𝑛(𝒽1⊗…⊗𝒽1). As a result, ring homeomorphisms and stringent morphisms of 𝒜∞-algebras tend to be identical. 5) We state that 𝒽 is a stringent isomorphism if 𝒽1 is a vector spaces' isomorphism and 𝒽 is a stringent morphism. It proves to be even more significant to take into consideration the homology of the 𝒜∞-algebras, similar to how it is conducted via chain complexes. Here, we will define strict morphisms and how to handle them within 𝒜∞-algebras, including the use of these definitions in analysis. 2.17. Definition [19] The morphism 𝒽 is considered stringent if 𝒽𝑖 = 0 for any 𝑖 ≠ 1. The stringent morphism 𝒽 that makes 𝒽1 the identity of 𝒜 is the identity morphism. While 𝒽:𝒜 → ℬ is a stringent morphism, the identity 𝑀𝐿(𝑛) becomes: 𝒽1𝑟𝑛 = 𝑟𝑛(𝒽1⊗⋯⊗𝒽1). 9478 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate In classical ring theory, homeomorphisms are similar to strict morphisms. Morphism 𝒽 called a strict isomorphism if it is 𝒽 strict and 𝒽1is an isomorphism of vector spaces. In this instance, the inverse morphism of 𝒽 is denoted by 𝒽1 −1: ℬ → 𝒜. We will explain how to classify 𝒜∞-algebras through equivalence classes and how to handle different models of algebras in the following. 2.18. Definition [20] Assume that 𝒜 and ℬ are 𝒜∞-algebras and that 𝒽:𝒜 → ℬ is a 𝒜∞-morphism. If 𝒽1 represents a quasi-isomorphism of complexes, then we can argue that 𝒽 is a quasi-isomorphism as well. It follows that the induced map ℋ(𝒽1):ℋ(𝒜•) → ℋ(ℬ•) is isomorphic. Here, we use the notation 𝒜 ≃ ℬ. We will classify two 𝒜∞-algebras as belonging to the same 𝒜∞-algebras if they are quasi-isomorphic. In other words, quasi-isomorphism classes of 𝒜∞-algebras will be taken into consideration. A model of 𝒜 is a representation of a class of 𝒜∞-algebras that are quasi-isomorphic to 𝒜 and satisfy some helpful properties. As an illustration, the term "minimal model of 𝒜" refers to a representative 𝒜∞-algebras containing a zero 𝑟1. As a result, we can examine 𝒜∞-algebras by taking into consideration alternative models of 𝒜. Now, we will show that 𝐸𝑥𝑡𝒜 ∗ (𝑀,𝑀) is an 𝒜∞-algebra and how to use different solutions in this context. 2.19. Theorem [21] Assuming 𝒜 is the algebra over 𝑀, then 𝐸𝑥𝑡𝒜 ∗ (𝑀,𝑀) is an 𝒜∞-algebras. To demonstrate this, we take the projective resolution 𝒫 of 𝑀𝒜 . H Consequently, the differential graded algebra with homology determined by the Yoneda algebra 𝐸𝑥𝑡𝒜 ∗ (𝑀,𝑀) is the morphism complex ℬ = 𝐻𝑜𝑚𝒜(𝒫,𝒫). We will provide an illustrative example of applications of 𝒜∞-algebras and how to use them in specific cases of tensor products and maps. 2.20. Example The map 𝑟2 is produced by multiplying 𝑅 and the mappings 𝑟𝑛 = 0 for all 𝑛 ≠ 2, The graded space 𝒜 = 𝑅[𝔷]/(ℰ2) has a trivial 𝒜∞-structures when 𝑅 is an extraordinary algebra for 𝑁 ≥ 1 and 𝔷 is an undetermined of degree 2 − 𝑁. We define the distorted multiplication and the linear map 𝒽: 𝑅⊗𝑁 → 𝑅 as follows: 𝑟𝑛 ′ = { 𝑟𝑛 𝑛 ≠ 𝑁 𝑟𝑁 + ℰ𝒽 𝑛 = 𝑁 . The given 𝑟𝑛 ′ thus becomes an 𝒜∞-algebras if and only if 𝒽 is a simplicial co-cycle over 𝑅. Next, we will define 𝒜∞-algebras using left models and how to handle fundamental properties of these models. 2.21. Definition [11] Consider 𝒜 as an 𝒜∞-algebras. The ℤ-graded vector space 𝑅 is defined as the left 𝒜∞-module of 𝒜, equipped with maps: 𝑟𝑛 𝑅:𝒜⊗𝑛−1⊗𝑅 → 𝑅, 𝑛 ≥ 1 of degree (2 − 𝑛), fulfilling the identical Stasheff identities 𝑆𝐿(𝑛): ∑(−1)𝑚+𝑠𝑡𝑟𝑚+1+𝑡(𝑖𝑑 ⊗𝑚 ⊗ 𝑟𝑠 ⊗ 𝑖𝑑⊗𝑡) = 0, as in the definition of 𝒜∞-algebras. We will describe how to define morphisms between left models of algebras and how to construct these morphisms to meet specified conditions. 9479 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate 2.22. Definition [8] Assume the morphism of the left 𝒜∞-modules 𝒽:𝑅 → 𝑆 is defined as the family of the graded maps: 𝒽𝑛:𝒜 ⊗𝑛−1⊗𝑅 → 𝑆 of degree (1 − 𝑛) for all 𝑛 ≥ 1. When 𝒽1 = 𝑖𝑑𝑅 and 𝒽𝑖 = 0 for all 𝑖 ≥ 2, the identity morphism 𝒽: 𝑅 → 𝑅 is produced. The composition of two morphisms 𝒽: 𝑅 → 𝑆 and ℊ: 𝐿 → 𝑅 is given by: (𝒽 ∘ ℊ)𝑛 = ∑𝒽1+𝓌 ∘ (𝑖𝑑⊗𝓌 ⊗ℊ𝓋), (4) where the sum is taken over all decompositions 𝑛 = 𝓋 +𝓌. We now classify nested categories and apply functorial transformations between 𝒜∞-algebras. 2.23. Theorem [8] Assume that 𝒽:𝒜 → ℬ is a quasi-isomorphism of 𝒜∞-algebras. Then, the equivalent functor for triangulated categories that maps ℬ to 𝒜 is: 𝒽∗:𝒜∞(ℬ) → 𝒜∞(𝒜). This theorem shows how to establish correspondences between nested categories of 𝒜∞-algebras and how to use these correspondences in practical applications. 2.24. Theorem [22] Let (𝒜, 𝒹𝒜) be a differential graded algebra, and let ℭ𝒹ℊ(𝒜) e the category of differential graded modules with morphisms between these modules, such that 𝒜𝒹ℊ(𝒜) is the category. Then, the equivalence for the triangulated categories: 𝒜𝒹ℊ(𝒜) → 𝒜∞(𝒜) is produced by the inclusion functor ℭ𝒹ℊ(𝒜) → ℭ∞(𝒜). Next we define 𝒜∞-algebras using graded algebra properties and examine how to utilize these definitions in analysis. 2.25. Definition [14] Assuming ℛ is a commutative ring and 𝒜 = ∑ 𝒜𝑛 𝑛∈ℤ is a graded ℛ-module. Therefore, the collection of multiplication maps 𝑟𝑛:𝒜 ⨂𝑛 → 𝒜 of degree (2 − 𝑛) is an 𝒜∞-structures on 𝒜, such that: ∑ (−1)𝑚𝑠+𝑡𝑟𝑚+1+𝑡 (1⨂𝑚 ⊗ 𝑟𝑠 ⊗1⨂𝑡) 𝑚+𝑠+𝑡=𝑛 = 0, for all 𝑛. Now, we discuss constructing 𝒜∞-algebras using homology and applying this structure in various contexts. 2.26. Theorem [23] Assume (𝒞, Ω, 𝜂) is a differential graded algebra, and ℋ∗(𝒞) is projective over 𝑀 in each degree. Then there exists an 𝒜∞-structure 𝑟 on ℋ∗(𝒞) such that: (i) 𝑟1 = 0 (minimal case), (ii) 𝑟2 is induced by 𝜂, (iii) (𝐻∗(𝒞), 𝑟) is quasi-isomorphic to (𝒞, 𝛺, 𝜂). Next, we introduce the Maurer-Cartan equation, which allows us to extend 𝒜∞-algebras and analyze different applications. 2.27. Definition [24] Let ℋ be an 𝒜∞-algebra, and let 𝒶 ∈ ℋ1, we say 𝒶 satisfies the Maurer-Cartan equation if: ∑±𝑟𝑛(𝒶⨂𝑛) 𝑛∈ℕ = 0. The twisted 𝒜∞-algebras ℋ𝒶 is then defined by: 9480 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate 𝑟𝑛 𝒶(𝒾1,⋯ , 𝒾𝑛) = ∑ ±𝑟𝑛+ℓ(𝒶⨂ℓ1 , 𝒾1, 𝒶⨂ℓ2 , 𝒾2,⋯ , 𝒾𝑛, 𝒶⨂ℓ𝑛+1) ℓ1+⋯+ℓ𝑛+1∈ℕ0 , where ℓ = ℓ1 +⋯+ ℓ𝑛+1. These formulas also apply to 𝒜∞-bimodules. Hence, if 𝒩 is an 𝒜∞-bimodules over ℋ then, 𝒩𝒶 is an 𝒜∞-bimodules over ℋ𝒶. Now, we define simple homology for 𝒜∞-algebras using complexes and their sequences. 2.28. Definition [25] Let a space 𝒳 be an 𝒜∞-algebra, and let (𝒳∗ , 𝒹∗) = {𝒳𝑛 , 𝒹𝑛} represent a chain complex such that: … → 𝒳𝑛+1 𝒹𝑛+1 → 𝒳𝑛 𝒹𝑛 → 𝒳𝑛−1 𝒹𝑛−1 → … 𝒹1 → 𝒳0 𝒹0 → 0 Then, the 𝑛𝑡ℎ homology of an 𝒜∞-algebra 𝒳 is defined as: ℋ𝑛(𝒳) = 𝑘𝑒𝑟(𝒹𝑛) 𝐼𝑚(𝒹𝑛+1) . (5) Since 𝐼𝑚 𝒹𝑛+1 ⊂ 𝑘𝑒𝑟 𝒹𝑛, and ℋ𝑛(𝒳) = 𝑛−cycles 𝑛−boundaries . Finally, we define 𝒜∞-algebras using corrections for algebra models and explain how to handle derived structures. 2.29. Definition [22] Let ℬ be a unital associative algebra, 𝑅 a right ℬ-module, and 𝒫 → 𝑅 the projective resolution. Consider 𝒜 = 𝐻𝑜𝑚ℬ(𝒫,𝒫) represent the differential graded endomorphism algebras of 𝒫, with its nth component consisting of the morphisms of graded items of degree 𝑛 and its differential being the super commutator corresponding to the differential of 𝒫. Since 𝒜 is specifically an 𝒜∞-algebra, it includes a minimal model. The homology ℋ∗𝒜, as an algebra of 𝑟2, is now isomorphic to the Yoneda algebra 𝐸𝑥𝑡ℬ ∗ (𝑅, 𝑅). Next, we explain simple homology for 𝒜∞-algebras and how to use these definitions in various applications. 2.30. Definition [26] Assume that (𝒜 =⊕𝑗∈ℤ 𝒜𝑗 , (𝜂𝑛)𝑛∈ℕ) is an 𝒜∞-algebras over 𝑀 and let (𝒮 =⊕𝑗∈ℤ 𝒮𝑗, (𝜇𝑚,𝑠 𝒮 )𝑚,𝑠∈ℕ0) be an 𝒜∞-bimodules over 𝒜. We further denote the index of 𝑟 ∈ 𝒮 by 𝜇(𝑟) ∶= 𝜇𝒮(𝑟). Consider the graded 𝑀-bimodule: ℋℋ∗(𝒜, 𝒮):= ∞ ⊕ 𝑛 = 0 𝒮 ⊗𝒜⊗𝑛 , with grading on ℋℋ∗(𝒜, 𝒮) is given by: ℋℋ𝑗(𝒜, 𝒮) =⊕𝑛∈ℕ0 ⊕𝑗=𝑛−𝑗0−𝑗1−⋯−𝑗𝑛 𝒮𝑗0 ⊗𝒜𝑗1 ⊗… ⊗𝒜𝑗𝑛 . (6) which corresponds to the simplicial homology of 𝒜∞-algebras. After that, we discuss how to handle homology degrees in graded spaces and explain how to determine these degrees using advanced definitions. 2.31. Definition [27] For 𝒶 ∈ ℋℋ∗(𝒜, 𝒮) we write 𝑑𝑒𝑔(𝒶) = 𝑗 if and only if 𝒶 ∈ ℋℋ𝑗(𝒜, 𝒮) and call it the degree of 𝒶. Note that for all 𝓈 ∈ 𝒮, 𝑛 ∈ ℕ0 and 𝒶1, 𝒶2, … , 𝒶𝑛 ∈ 𝒜 the degree of 𝓈 ⊗𝒶1 ⊗…⊗𝒶𝑛 is explicitly given by: deg(𝓈 ⊗ 𝒶1 ⊗…⊗𝒶𝑛) = 𝑛 − 𝜂(𝓈) −∑𝜂(𝒶𝑗) 𝑛 𝑗=1 = −𝜂(𝓈) −∑‖𝒶𝑗‖ 𝑛 𝑗=1 . 9481 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate Remark: The degree on ℋℋ∗(𝒜, 𝒮) is best understood in terms of the shifted 𝒜∞-algebras 𝒜 [1]. We may identify ℋℋ∗(𝒜, 𝒮) as a group, with ∞ ⊕ 𝑛 = 0 𝒮 ⊗𝒜⊗𝑛. The degree on ℋℋ∗(𝒜, 𝒮) then coincides with the usual product degree of this tensor algebras. Next, we describe how to deal with the relative homology of algebras and how to use suitable models for analysis in this context. 2.32. Definition [28] There is a map for the relative homology of 𝒜 modulo ℐ if 𝒜 is an 𝒜∞-algebras and ℐ is an ideal, where 𝒜 → 𝒜/ℐ is 𝒜∞-split, such that: 𝔷:ℋℋ𝑛(ℐ) → ℋℋ𝑛(𝒜/ℐ), If this map is an isomorphism, it is claimed that the ideal ℐ is the excision of simplicial homology. This leads to the following exact sequence: … → ℋℋ𝑛(ℐ) → ℋℋ𝑛(𝒜) → ℋℋ𝑛(𝒜/ℐ) → ℋℋ𝑛−1(ℐ) → ℋℋ𝑛−1(𝒜) → … Finally, the exact sequence for 𝒜∞-algebras with ideal ℐ includes a boundary map 𝛿, which is clearly defined in terms of relative cycles. 2.33. Proposition [29] Given 𝒜 be 𝒜∞-algebras and ℐ-ideal since ℐ ⊂ 𝒜, we have the following exact sequence: ℋℋ𝑛(ℐ) 𝑖∗ → ℋℋ𝑛(𝒜) 𝛿(−1) 𝑗∗ ℋℋ𝑛(𝒜/ℐ) Actually, in this application of relative homology, the boundary map 𝛿 contains an obvious description: The (𝑛 − 1)-homology class provided by [𝛿𝜑] ∈ ℋ𝑛−1(𝒜) is 𝛿[𝜑] if 𝜑 ∈ 𝒞𝑛(𝒴,𝒜) denotes a relative cycle. 3. Main Result This text explores essential topics in algebraic topology and homology related to 𝒜∞-algebras. It covers how isomorphisms between homology groups are preserved under specific conditions, defines simplicial and bar homology with module coefficients, and discusses ℋ-unitarity. The discussion also includes the relationships between different homological constructs and the conditions required for quasi-isomorphisms, providing a comprehensive view of these concepts and their interconnections. The frist theorem discusses the excision theorem for 𝒜∞-algebras, demonstrating how isomorphisms between homology groups persist under specific conditions and the inclusion maps. It includes proof strategies involving chain complexes and homotopy equivalence. 3.1. Theorem (Excision Theory) Suppose that ℰ is a subset of 𝒜∞-algebras such that ℰ ⊂ 𝒜 ⊂ 𝒳. Then, for all 𝑛 the isomorphisms ℋ𝑛(𝒳\ℰ,𝒜\ℰ) → ℋ𝑛(𝒳,𝒜) that given by the inclusion (𝒳\ℰ,𝒜\ℰ) ↪ (𝒳,𝒜). If 𝒳 is covered by the interiors of the spaces 𝒜, ℬ such that 𝒜, ℬ ⊂ 𝒳, then the inclusion (ℬ,𝒜 ∩ ℬ) ↪ (𝒳,𝒜) is the equivalent statement that persuades the isomorphisms ℋ𝑛(ℬ,𝒜 ∩ ℬ) → ℋ𝑛(𝒳,𝒜) for all 𝑛, where the space ℬ is given by ℬ = 𝒳\ℰ. Proof: By using [30], [31] and consider 𝒳 as the combination of 𝒜 and ℬ through interiors covering 𝒳. Next, there are maps of natural inclusion. 9482 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate 𝒞•(𝒳) 𝚤 ↑ 𝒞•(𝒜) + 𝒞•(ℬ) 𝒞•(𝒜) 𝒞•(ℬ) 𝒞•(𝒜 ∩ ℬ) We would obtain: 𝒞•(𝒳)/𝒞•(𝒜) = 𝒞•(ℬ)/𝒞•(𝒜 ∩ ℬ), where the map 𝜄 remains an isomorphism, which leads to the desired result. However, there are terrible simplices that can have non-empty intersections with (𝒜 −𝒜 ∩ ℬ) and (ℬ −𝒜 ∩ ℬ); therefore, the fact that the map 𝜄 is not an isomorphism is an issue. Assisted by the chain map 𝜉: 𝒞•(𝒳) → 𝒞•(𝒜) + 𝒞•(ℬ), we would desire to demonstrate how to split up terrible simplices into little good ones without changing the homology. We demonstrate that 𝒞•(𝒜) + 𝒞•(ℬ) be the distorted retracting of 𝒞•(𝒳), indicating that 𝜉 ∘ 𝚤 = 𝐼𝑑 and 𝚤 ∘ 𝜉 = 𝒹𝔇+𝔇𝒹 for particular chain homotopy 𝔇. In order to retain the sub-complexes 𝒞•(𝒜) and 𝒞•(ℬ), we choose 𝔇, indicating that we achieve the equivalence: 𝒞•(𝒳)/𝒞•(𝒜) → 𝒞•(ℬ)/𝒞•(𝒜 ∩ ℬ) as chain homotopy equivalence. Next, we define simplicial homology isomorphisms in the context of 𝒜∞-algebras, emphasizing the behavior of these isomorphisms under inclusions of subspaces. 3.2. Definition For the space 𝒳 and ℰ ⊂ 𝒜∞-algebras where ℰ ⊂ 𝒜 ⊂ 𝒳, the simplicial homology isomorphisms induced by the inclusion (𝒳\ℰ,𝒜\ℰ) ↪ (𝒳,𝒜) for all 𝑛 is: ℋℋ𝑛(𝒳\ℰ,𝒜\ℰ) → ℋℋ𝑛(𝒳,𝒜). By setting the space ℬ = 𝒳\ℰ, let 𝒳 covered by the interiors of the spaces 𝒜,ℬ for 𝒜, ℬ ⊂ 𝒳, then the equivalent statement is that the following isomorphisms persuaded by the inclusion (ℬ,𝒜 ∩ ℬ) ↪ (𝒳,𝒜): ℋℋ𝑛(ℬ,𝒜 ∩ ℬ) → ℋℋ𝑛(𝒳,𝒜) ∀𝑛. (7) Now, we introduce the bar homology of 𝒜∞-algebras with coefficients in a module detailing the associated boundary maps. 3.3. Definition Assuming that ℐ is an 𝒜∞-algebras, which is not necessarily unitary and make ℛ a right ℐ-module. Then, the complexes' homology ℋ𝐵∗ ′(ℐ, ℛ) is the bar homology of ℐ via coefficients in ℛ: (ℛ ⊗ ℐ⨂∗, 𝜌∗ ′):= ℛ 𝜌1 ′ ← ℛ⊗ ℐ 𝜌2 ′ ← ℛ ⊗ ℐ ⊗ ℐ 𝜌3 ′ ← ℛ⊗ ℐ ⊗ ℐ ⊗ ℐ 𝜌4 ′ ← … such that a tensor product obtained over 𝒜∞-algebras and a boundary map is provided by: 𝜌𝑛 ′ (𝒶0⊗…⊗𝒶𝑛) = ∑ (−1)𝑖𝒶0⊗…⊗𝒶𝑖𝒶𝑖+1⊗…⊗𝒶𝑛 𝑛−1 𝑖=0 . The following definition defines the simplicial homology of complexes and explores the boundary maps involved, illustrating their connection to the bar homology definitions. 3.4. Definition The homology ℋℋ∗(ℐ, ℛ) of complexes represents the simplicial homology of ℐ via coefficients in ℛ: (ℛ ⊗ ℐ⨂∗, 𝜌∗): = ℛ 𝜌1 ← ℛ ⊗ ℐ 𝜌2 ← ℛ ⊗ ℐ ⊗ ℐ 𝜌3 ← ℛ ⊗ ℐ ⊗ ℐ ⊗ ℐ 𝜌4 ← … , with the boundary map provided by: 9483 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate 𝜌𝑛(𝒶0⊗…⊗𝒶𝑛) = 𝜌𝑛 ′ (𝒶0⊗…⊗𝒶𝑛) + (−1)𝑛𝒶𝑛𝒶0⊗𝒶1 ⊗…⊗𝒶𝑛−1. We present a corollary relating to 𝐻-homology and bar homology, establishing the relationships between different homological constructs for 𝒜∞-algebras and discussing exact sequences that arise. 3.5. Corollary The 𝒜∞-algebras that are produced from ℐ by adding the value of unity to ℐ be denoted by ℐ̃ = 𝑘 × ℐ. Then the 𝐻-homology 𝐻∗(ℐ) is defined by ℋ∗(ℐ):= ℋ∗(ℐ, ℐ), the bar homology ℋ𝐵∗(ℐ) is defined by ℋ𝐵∗(ℐ):= ℋ𝐵∗ ′(ℐ, ℐ), and ℋℋ∗(ℐ): = ℋ̅∗(ℐ, ℐ̃) defines the simplicial homology of ℐ, where ℋ̅𝑛(ℐ, ℐ̃) = ℋ𝑛(ℐ, ℐ̃), ∀ 𝑛 > 0 and ℋ̅0(ℐ, ℐ̃) = ℋ0(ℐ, ℐ̃)/𝑘. Let the homology ℋℋ∗(ℐ) be the double complex' homology such that: 𝒞𝒞(ℐ)|2| ≔ (ℐ ⊗ ℐ⨂∗, 𝜌∗) 1−𝑡 ← (ℐ ⊗ ℐ⨂∗, −𝜌∗ ′). (8) Consequently, the exact sequence exists as follows: … ← 𝐻𝑛−1(ℐ) ← ℋ𝐵𝑛−1(ℐ) ← ℋℋ𝑛(ℐ) ← 𝐻𝑛(ℐ) ← ℋ𝐵𝑛(ℐ) ← ℋℋ𝑛+1(ℐ) ← …. The following definition covers the concept of ℋ-unitarity in 𝒜∞-algebras, focusing on the conditions under which a module ℛ is considered ℋ-unitary. 3.6. Definition Suppose that ℛ is an ℐ-bimodule since ℐ is an 𝒜∞-algebras. If every 𝒜∞-modules 𝒢 has an exact complex (ℛ ⊗ ℐ⨂∗, 𝜌∗)⊗ 𝒢, we can deduce that ℛ is ℋ-unitary. One states that ℐ is ℋ-unital when ℛ = ℐ, such ℛ is the left ℐ-module. It follows logically that ℛ⊗ ℐ is ℋ-unitary, if ℐ is ℋ-unital. In the following, we examine a theorem on quasi-isomorphisms between complexes in the context of 𝒜∞-algebras and bimodules, proving the results under specific assumptions about ℋ-unitarity. 3.7. Theorem Assume that the extension of 𝒜∞-algebras is given by 0 → ℐ → 𝒜 → ℬ → 0, and defines 𝒢 to be a 𝒜∞-modules and ℛ to be an 𝒜-bimodule. Then we can say that the following canonical inclusions: 𝑖: (ℛ ⊗ ℐ⨂∗, 𝜌∗)⊗ 𝒢 ↪ (ℛ ⊗𝒜⨂∗, 𝜌∗)⊗ 𝒢, (9) 𝑖′: (ℛ ⊗ ℐ⨂∗, 𝜌∗ ′)⊗ 𝒢 ↪ (ℛ ⊗𝒜⨂∗, 𝜌∗ ′)⊗ 𝒢 (10) are quasi-isomorphisms when the ℐ-bimodule ℛ is ℋ-unitary. Proof: By considering the filtration 𝐹0 ⊆ 𝐹0 ⊆ ⋯ of (ℛ ⊗𝒜⨂∗, 𝜌∗) and using [32], [33], assume that 𝒢 is an 𝒜∞-modules, such that: 𝐹ℓ ≔ ℛ 𝜌1 ← ℛ ⊗𝒜 𝜌2 ← ℛ⊗𝒜⊗2 𝜌3 ← … 𝜌ℓ ← ℛ ⊗𝒜⊗𝑝 𝜌ℓ+1 ← ℛ ⊗ ℐ ⊗𝒜⊗ℓ 𝜌ℓ+2 ← ℛ ⊗ ℐ⊗2 ⊗𝒜⊗ℓ 𝜌ℓ+3 ← … For all ℓ ≥ 0, we have: (𝐹ℓ+1⊗ 𝒢 𝐹ℓ ⊗𝒢) ∗ = (ℛ ⊗ ℐ⨂ ∗−ℓ−1, 𝜌∗ ′)⊗ ℬ⊗𝒜⊗ℓ ⊗𝒢, (11) According to theory, this is exact. Taking into consideration the homology of long exact sequence for all 𝑛 ≥ 0 connected to 0 → 𝐹𝑛 ⊗𝒢 → 𝐹𝑛+1 ⊗𝒢 → 𝐹𝑛+1⊗𝒢 𝐹𝑛⊗𝒢 → 0 (12) As can be seen, the canonical map 𝐹0 → 𝐹ℓ represents quasi-isomorphism for every ℓ, 𝑖 is also a quasi- isomorphism consequently. In a similar demonstration, the same applies to 𝑖′. Remark: Note that the theorem (3.7) given above may also be proved in the case when ℐ is a right ideal of 𝒜 instead of a two-sided ideal. 9484 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate The following corollary provides further insight into quasi-isomorphisms in the context of extensions of 𝒜∞-algebras, focusing on modules and the conditions for ℋ-unitarity. 3.8. Corollary Suppose that 0 → ℐ → 𝒜 → ℬ → 0 is an extension of the 𝒜∞-algebras where ℐ ⊂ 𝒜 ⊂ ℬ and 𝒢 is a 𝑘-module and use [34]. The canonical arrows: 𝜋: (ℬ ⊗𝒜⊗∗, 𝜌∗)⊗ 𝒢 → (ℬ ⊗ℬ⊗∗, 𝜌∗)⊗ 𝒢, 𝜋′: (ℬ ⊗𝒜⊗∗, 𝜌′∗)⊗ 𝒢 → (ℬ ⊗ℬ⊗∗, 𝜌′∗)⊗ 𝒢, are quasi-isomorphisms when ℐ is an ℋ-unital. Proof: We must take into consideration for all ℓ ≥ 0, the quotient complex �̃�ℓ for (ℬ ⊗𝒜⊗∗, 𝜌∗) provided by: �̃�ℓ ≔ ℬ 𝜌1 ← ℬ ⊗ℬ 𝜌2 ← ℬ ⊗ℬ⊗2 𝜌3 ← … 𝜌ℓ ← ℬ ⊗ℬ⊗ℓ 𝜌ℓ+1 ← ℬ ⊗ℬ⊗ℓ ⊗𝒜 𝜌ℓ+2 ← ℬ ⊗ ℬ⊗ℓ ⊗𝒜⊗2 𝜌ℓ+3 ← … . to demonstrate whether 𝜋 is a quasi-isomorphism, consider that the canonical projections 𝜋ℓ: �̃�ℓ ⊗𝒢 → �̃�ℓ+1⊗𝒢. Given that ℬ(ℓ) = ℬ ⊗ℬ⊗ℓ ⊗ ℐ, the straightforward calculation reveals that: 𝐾𝑒𝑟(𝜋ℓ) = (ℬ(ℓ)⊗𝒜⊗∗−ℓ−1 , 𝜌∗) ⊗ 𝒢. Therefore, according to Theorem (3.7), 𝐾𝑒𝑟(𝜋ℓ) is quasi-isomorphic to: (ℬ(ℓ)⊗ ℐ⊗∗−ℓ−1 , 𝜌∗) ⊗ 𝒢 = (ℬ(ℓ)⊗ ℐ⊗∗−ℓ−1 , 𝜌∗ ′) ⊗ 𝒢, (13) which is exact by assumption. For 𝜋′, a similar proof applies. Now, we conclude with a theorem establishing the equivalence of various conditions related to ℋ- unitarity, excision, and homology requirements for 𝒜∞-algebras, providing a comprehensive view of their interrelations. 3.9. Theorem Let the assumption that ℐ is a 𝒜∞-algebras, and then the next propositions are equivalent: (1) The 𝒜∞-algebras ℐ remains ℋ-unital. (2) The 𝒜∞-algebras ℐ fulfills the 𝐻-homology excision. (3) The 𝒜∞-algebras ℐ fulfills the excision requirement of bar homology. (4) The 𝒜∞-algebras ℐ fulfills the excision requirement of simplicial homology. Proof: The proposition (1) is equivalent to (2) when we let 0 → ℐ → 𝒜 → ℬ → 0 be an 𝒜∞- algebras as a pure extension and 𝒢 be a 𝑘-module, and the canonical projection given by: 𝜋: (𝒜 ⊗𝒜⨂∗, 𝜌∗) ⊗ 𝒢 → (ℬ ⊗ℬ⨂∗, 𝜌∗) ⊗ 𝒢. Suppose that the commutation diagram of short exact sequences is as follows: 0 0 → → (ℐ ⊗𝒜⨂ ∗, 𝜌∗) ⊗ 𝒢 |𝑗 𝑘𝑒𝑟 (𝜋) → → (𝒜 ⊗𝒜⨂ ∗, 𝜌∗) ⊗ 𝒢 | = (𝒜 ⊗𝒜⨂ ∗, 𝜌∗) ⊗ 𝒢 → 𝜋 → (ℬ ⊗𝒜⨂ ∗, 𝜌∗) ⊗ 𝒢 |𝜋1 (ℬ ⊗𝒜⨂∗ , 𝜌∗) ⊗ 𝒢 → → 0 0 According to Corollary (3.8), 𝜋1 indicates a quasi-isomorphism. As a result, 𝑗 is as well. We continue the proof using the Theorem (3.7). The proposition (1) is equivalent to (3): Comparable to (1) ⟹ (2). The proposition (2) and (3) is equivalent to (4): The long exact sequence: ⋯ ← 𝐻𝑛−1(ℐ) ← ℋ𝐵𝑛−1(ℐ) ← ℋℋ𝑛(ℐ) ← 𝐻𝑛(ℐ) ← ℋ𝐵𝑛(ℐ) ← ℋℋ𝑛+1(ℐ) ← ⋯ gives this simple consequence. The proposition (2) is equivalent to (1): Assume that 𝑘-algebra 𝒜 = ℐ ⊕ 𝒢 with 𝑘-module 𝒢 in addition to the canonical projection: 𝜋: (𝒜 ⊗𝒜⨂ ∗, 𝜌∗) → (𝒢 ⊗ 𝒢⨂∗, 𝜌∗), 9485 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 9472-9486, 2024 DOI: 10.55214/25768484.v8i6.4026 © 2024 by the authors; licensee Learning Gate where the product is provided by (𝑢, 𝑣)(𝑢′, 𝑣′) = (𝑢𝑢′, 0). Given that 𝑘𝑒𝑟 (𝜋) is a direct summation of the complex 𝒢 ⊗ (ℐ ⊗ ℐ⨂ ∗−1, 𝜌∗ ′) ⊕ (ℐ ⊗ ℐ⨂∗, 𝜌∗ ′), ℐ satisfy the excision to 𝐻-homology. In such a case, 𝒢 ⊗ (ℐ ⊗ ℐ⨂ ∗−1, 𝜌∗ ′) is exact. The proposition (3) is equivalent to (1): Comparable to (2) ⟹ (1). The proposition (4) is equivalent to (1): Allow 𝒜 and 𝒢 are being in (2) ⟹ (1). Let the canonical projection given by �̅�: 𝒞∗∗(𝒜) → 𝒞∗∗(𝒢) and 𝛽 be the sub-complex of 𝑘𝑒𝑟 (𝜋) produced by the components (𝒶0⊗⋯⊗𝒶𝑛, 𝒶 ′ 0⊗⋯⊗𝒶′𝑛−1) that include some 𝒶𝑖 with some 𝒶′n in 𝒢. So 𝛽 is exact such that 𝑘𝑒𝑟(�̅�) = 𝒞∗∗(ℐ) ⊕ 𝛽 and ℐ fulfills excision instead of simplicial homology. Consider the case when ℐ cannot be ℋ-unital. Suppose 𝑥 ∈ 𝒢 ⊗ ℐ⨂𝑛 represents a cycle that does not represent a boundary for 𝜌𝑛 ′ , it is obvious that (0, 𝑁(𝑥)) in is a cycle of degree 𝑛 + 1 that is not a boundary, which is in direct opposition to the exactness of 𝛽. 3.7. Concluding Remarks The core focus was on 𝒜∞-algebras, defined by Stasheff identities that ensure homotopy associativity. We examined various examples and their applications, highlighting their versatility. Morphisms between 𝒜∞-algebras were also analyzed, including strict morphisms and quasi- isomorphisms. 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