@ Appl. Gen. Topol. 22, no. 1 (2021), 31-46doi:10.4995/agt.2021.13148 © AGT, UPV, 2021 Further remarks on group-2-groupoids Sedat Temel Department of Mathematics, Recep Tayyip Erdogan University, Turkey (sedat.temel@erdogan.edu.tr) Communicated by E. Minguzzi Abstract The aim of this paper is to obtain a group-2-groupoid as a 2-groupoid object in the category of groups and also as a special kind of an inter- nal category in the category of group-groupoids. Corresponding group- 2-groupoids, we obtain some categorical structures related to crossed modules and group-groupoids and prove categorical equivalences be- tween them. These results enable us to obtain 2-dimensional notions of group-groupoids. 2010 MSC: 20L05; 18D05; 18D35; 20J15. Keywords: crossed module; group-groupoid; 2-groupoid. 1. Introduction There are several 2-dimensional notions of groupoids such as double groupoids, 2-groupoids, and crossed modules over groupoids. The purpose of this pa- per is to obtain 2-dimensional notions of group-groupoids which are internal groupoids in the category of groups and widely used under the name of 2- groups. The term ”categorification”, which was first used by Louis Crane [13] in the context of mathematical physics, is the process of replacing set-theoretic theo- rems by category-theoretic concepts. The aim of categorification is to develop a richer case of existing mathematics by replacing sets with categories, functions with functors and equations between functions with natural isomorphisms be- tween functors. In this approach, the categorified version of a group is called a group-groupoid [2, 5]. Group-groupoids, which are also known as G-groupoids [6] or 2-groups [4], are internal categories (hence internal groupoids) in the Received 17 February 2020 – Accepted 07 November 2020 http://dx.doi.org/10.4995/agt.2021.13148 S. Temel category Gp of groups [22, 23]. Equivalently, group-groupoids can be thought as group objects in the category Cat of small categories [6, 23]. Another useful viewpoint of group-groupoids is to think them as crossed modules over groups. Crossed modules which can be viewed as 2-dimensional groups [7] are widely used in homotopy theory [8], homological algebra [16], and algebraic K-theory [21]. The well-known categorical equivalence between crossed modules and group-groupoids is proved by Brown and Spencer [6]. This equivalence is introduced in [4] by obtaining a group-groupoid as a 2-category with a unique object. Crossed modules, and their higher dimensional analogues, provide algebraic models for homotopy n-types; the group-2-groupoids of this paper in principle provide algebraic models for certain homotopy 3-types. In the previous paper [1], the notions of a group-2-groupoid were introduced and compared with a corresponding structure related to crossed modules over groups. On the other hand, the main objective of this paper is to obtain the structure of a group-2-groupoid as a 2-groupoid object in the category of groups and also as a special kind of internal category in the category of group- groupoids. In section 4, we present the notion of crossed modules over group- groupoids and prove that there is a categorical equivalence between group- 2-groupoids and crossed modules over group-groupoids using the categorical equivalence between 2-groupoids and crossed modules over groupoids given in [17]. In section 5, we show that group-2-groupoids are categorically equivalent to special kind of internal categories in the category of crossed modules. 2. Preliminaries Let C be a finitely complete category and D0, D1 are objects of the ambient category C. An internal category D = (D0, D1, s, t, ε, m) in C consists of an object D0 in C called the object of objects and an object D1 in C called the object of arrows (i.e. morphisms), together with morphisms s, t: D1 → D0, ε: D0 → D1 in C called the source, the target and the identity maps, respec- tively, D1 s // t // D0 εtt such that sε = tε = 1D0 and a morphism m: D1 ×D0 D1 → D1 of C called the composition map (usually expressed as m(f, g) = g ◦ f) where D1 ×D0 D1 is the pullback of s, t such that εs(f) ◦ f = f = f ◦ εs(f) [22]. An internal groupoid in C is an internal category with a morphism η : D1 → D1, η(f) = f in C called inverse such that f ◦ f = 1s(f), f ◦ f = 1t(f). We write C(x, y) for all morphisms from x to y where x, y ∈ C0. If C(x, y) = ∅ for all x, y ∈ C0 such that x 6= y, then C is called totally disconnected cate- gory. We introduce the definition of a 2-category as given in [4]. A 2-category C = (C0, C1, C2) consists of a set of objects C0, a set of 1-morphisms C1, and © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 32 Further remarks on group-2-groupoids a set of 2-morphisms C2 as follows: x f (( g 66 ✤✤ ✤✤ �� α y with maps s: C1 → C0, s(f) = x, sh : C2 → C0, sh(α) = x, sv : C2 → C1, sv(α) = f, t: C1 → C0, t(f) = y, th : C2 → C0, th(α) = y, tv : C2 → C1, tv(α) = g, called the source and the target maps, respectively, the com- position of 1-morphisms as in an ordinary category, the associative horizontal composition of 2-morphisms ◦h : C2 ×C0 C2 → C2 as x f (( g 66 ✤✤ ✤✤ �� α y f1 (( g1 66 ✤✤ ✤✤ �� δ z = x f1◦f (( g1◦g 66 ✤✤ ✤✤ �� δ◦hα z , where C2 ×C0 C2 = {(α, δ) ∈ C2 × C2|sh(δ) = th(α)} and the associative verti- cal composition of 2-morphisms ◦v : C2 ×C1 C2 → C2 as x f !! ✤✤ ✤✤ �� α == h ✤✤ ✤✤ �� β g // y = x f (( h 66 ✤✤ ✤✤ �� β◦vα y where C2 ×C1 C2 = {(α, β) ∈ C2 × C2|sv(β) = tv(α)} such that satisfying the following interchange rule: (θ ◦v δ) ◦h (β ◦v α) = (θ ◦h β) ◦v (δ ◦h α) whenever one side makes sense, and the identity maps ε: C0 → C1, ε(x) = 1x, εh : C0 → C2, εh(x) = 11x such that α ◦h 11x = α = 11y ◦h α and εv : C1 → C2, εv(f) = 1f such that α ◦v 1f = α = 1g ◦v α. Therefore, the construction of a 2-category C = (C0, C1, C2) contains compatible category structures C1 = (C0, C1, s, t, ε, ◦), C2 = (C0, C2, sh, th, εh, ◦h), and C3 = (C1, C2, sv, tv, εv, ◦v) such that the following diagram commutes. C2 sv // tv // th ��✸ ✸✸ ✸✸ ✸✸ ✸ ✸✸ ✸✸ ✸ sh ��✸ ✸ ✸✸ ✸ ✸✸ ✸✸ ✸✸ ✸ ✸ C1 εv ss C0 �� t ☞☞☞ ☞☞☞ ☞☞ ☞☞☞ ☞☞ �� s ☞ ☞☞ ☞☞☞ ☞☞ ☞☞☞ ☞☞ εh TT ε JJ Let C and C′ be 2-categories. A 2-functor is a map F : C → C′ sending each object of C to an object of C′, each 1-morphism of C to 1-morphism of C′ and © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 33 S. Temel 2-morphism of C to 2-morphism of C′ as follows: x f (( g 66 ✤✤ ✤✤ �� α y 7→ F(x) F (f) ** F (g) 44 ✤✤ ✤✤ �� F (α) F(y) such that F(f1 ◦ f) = F(f1) ◦ F(f), F(δ ◦h α) = F(δ) ◦h F(α), F(β ◦v α) = F(β) ◦v F(α), F(11x) = 1F (1x) = 11F (x), F(1f ) = 1F (f). Hence 2-categories form a category which is denoted by 2Cat [24]. A strict 2-groupoid is a 2-category all of whose 1-morphisms are invertible and in which all 2-morphisms are invertible horizontally and vertically. x f (( g 66 ✤✤ ✤✤ �� α y f̄ '' ḡ 77 ✤✤ ✤✤ �� ᾱh x = x 1x (( 1x 66 ✤✤ ✤✤ �� 11x x , x f �� ✤✤ ✤✤ �� α >> f ✤✤ ✤✤ �� ᾱ v g // y = x f (( f 66 ✤✤ ✤✤ �� 1f y Let G, G′ be 2-groupoids. A morphism of 2-groupoids is a 2-functor F : G → G′ which preserves the 2-groupoid structures. Thus, 2-groupoids and their mor- phisms form a category which is denoted by 2Gpd [24]. A group-groupoid is an internal category in Gp [22]. Also, a group-groupoid can be obtained as a group object in the category Cat of small categories (or in Gpd). A morphism of group-groupoids is a morphism of groupoids which pre- serves group structures. Hence we can define the category of group-groupoids, which is denoted by 2Gp or GpGd. For further details about group-groupoids, see [24, 6, 4]. By a crossed module as defined by Whitehead, it is meant a pair M, N of groups together with an action •: N × M → M of groups and a morphism ∂ : M → N of groups such that ∂(n•m) = n∂(m)n−1 and ∂(m)•m′ = mm′m−1 [28, 29]. Let K = (M, N, ∂, •), K′ = (M′, N′, ∂′, •′) be crossed modules and λ1 : N → N′, λ2 : M → M ′ be morphisms of groups. If λ1, λ2 satisfies the conditions λ1∂ = ∂ ′λ2 and λ2(n • m) = λ1(n) • ′ λ2(m), then 〈λ2, λ1〉: K → K ′ is called morphism of crossed modules [6]. Hence crossed modules and their morphisms form a category which we denote by Cm. The following theorem was proved by Brown and Spencer in [6]: Theorem 2.1. The category of group-groupoids and the category of crossed modules are equivalent. Let G = (X, G) and H = (X, H) be groupoids over the same object set X such that H is totally disconnected. We recall from [8, 17, 11] that an action © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 34 Further remarks on group-2-groupoids of G on H is a partially defined map •: G × H → H, (g, h) 7→ g • h such that the following conditions satisfies [AG 1] g • h is defined iff t(h) = s(g), and t(g • h) = t(g), [AG 2] (g2 ◦ g1) • h = g2 • (g1 • h), [AG 3] g•(h2◦h1) = (g•h2)◦(g•h1), for h1, h2 ∈ H(x, x) and g ∈ G(x, y), [AG 4] 1x • h = h, for h ∈ H(x, x). From this conditions, it can be easily obtain that g•1x = 1y, for g ∈ G(x, y). Using this action of G on H, we can obtain a groupoid which is called semi- direct product of G and H denoted by G ⋉ H. Let x g // y h // y are morphisms of G and H, respectively, then (g, h) is a morphism as follows x (g,h) // y where the structure maps are defined by s(g, h) = s(g), t(g, h) = t(g), ε(x) = (1x, 1x). If x g // y h // y g1 // z h1 // z then the composition of morphisms is defined by (g1, h1) ◦ (g, h) = (g1 ◦ g, h1 ◦ (g1 • h)). The notion of crossed modules over groupoids is introduced by Brown- Higgins [9, 10] and Brown-Icen [11]. Let G = (X, G) and H = (X, H) be groupoids over the same object set X such that H is totally disconnected. A crossed module K = (H, G, ∂, •) over groupoids consists of a morphism ∂ = (1, ∂): H → G of groupoids which is identity on objects together with an action •: G × H → H of groupoids which satisfies ∂(g • h) = g ◦ ∂(h) ◦ g and ∂(h) • h1 = h ◦ h1 ◦ h, for h, h1 ∈ H(x, x) and g ∈ G(x, y). Let K = (H, G, ∂, •) and K′ = (H′, G′, ∂′, •′) be crossed modules over groupoids. A morphism of crossed modules over groupoids is a mapping λ = 〈λ2, λ1, λ0〉: K → K′ which satisfies λ2∂ = ∂ ′λ1 and λ1(g•h) = λ2(g)• ′λ1(h) where (λ0, λ1): H → H′ and (λ0, λ2): G → G ′ are morphisms of groupoids. Hence the category of crossed modules over groupoids can be defined which we denoted by Cmg. The following result was proved by Icen in [17]. Since we need some details in section 4, we give a sketch proof in terms of our notations. Theorem 2.2. The categories of 2-groupoids and of crossed module over groupoids are equivalent. Proof. For any 2-groupoid G = (G0, G1, G2), we know that B = (G0, G1) is a groupoid. Let A(x) = {α ∈ G2|sv(α) = ε(x)}, for x ∈ G0 and A = {A(x)}x∈G0. Then A = (G0, A) is a totally disconnected groupoid. Now we define a functor © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 35 S. Temel γ : 2Gpd → Cmg as an equivalence of categories such that γ(G) = (A, B, ∂) is a crossed module over groupoids with ∂ : A → B, ∂(α) = tv(α) and an action of groupoids such that f • α = 1f ◦h α ◦h 1f . y 1y (( ∂(f•α) 66 ✤✤ ✤✤ �� f•α y = y f (( f 66 ✤✤ ✤✤ �� 1f x 1x (( ∂(α) 66 ✤✤ ✤✤ �� α x f (( f 66 ✤✤ ✤✤ �� 1f y Clearly ∂(f • α) = f ◦ ∂(α) ◦ f and ∂(α) • α1 = α ◦h α1 ◦h α h, for f ∈ G1(x, y) and α, α1 ∈ A(x). Let F = (F0, F1, F2) be a morphism of 2-groupoids. Then γ(F) = 〈F2 ∣ ∣ A , F1, F0〉 is a morphism of crossed modules over groupoids. Now we define a functor θ : Cmg → 2Gpd which is an equivalence of cat- egories. Let K = (A, B, ∂) be a crossed module over groupoids A = (X, A) and B = (X, B). Then 2-groupoid θ(K) = (X, B, B ⋉ A) is a 2-groupoid which is constructed as in the following way. The set of 2-morphisms is the semi-direct product B ⋉ A = {(b, a)|b ∈ B, a ∈ A, s(a) = t(a) = t(b)}. If x b // y a // y , then (b, a) is a 2-morphism as follows: x b (( ∂(a)◦b 66 ✤✤ ✤✤ �� (b,a) y where the horizontal composition of 2-morphisms is defined by (b1, a1) ◦h (b, a) = (b1 ◦ b, a1 ◦ (b1 • a)) when y b1 // z a1 // z and the vertical composition of 2-morphisms is defined by ( ∂(a) ◦ b, a2 ) ◦v (b, a) = (b, a2 ◦ a) when y a2 // y . The source and the target maps are defined by sh(b, a) = s(b), sv(b, a) = b, th(b, a) = t(b), tv(b, a) = ∂(a) ◦ b, respectively, the identity maps are defined by εh(x) = (1x, 1x), εv(b) = (b, 1y), and the inversion maps are defined by (b, a) v = (∂(a) ◦ b, a), (b, a) h = (b, b • a). Let λ = 〈λ2, λ1, λ0〉 be a morphism of crossed modules over groupoids. Then θ(λ) = (λ0, λ2, λ2 × λ1) is a morphism of 2-groupoids. A natural equivalence S : θγ → 12Gpd is defined via the map SG : θγ(G) → G which is defined to be identity on objects and on 1-morphisms, on 2-morphisms is defined by α 7→ (f, α ◦h 1f). Clearly SG is an isomorphism and preserves compositions. Now, given a crossed module K = (A, B, ∂, •) over groupoids, we define a natural equivalence T : 1Cmg → γθ by a map TK : K → γθ(K) which is defined to be identity on objects and on B, while on A is defined by a 7→ (s(a), a). � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 36 Further remarks on group-2-groupoids 3. Group-2-groupoids In [1], a group-2-groupoid is defined as a group object in 2Cat using similar methods given in [6, 23]. In other words, a group-2-groupoid G is a small 2-groupoid equipped with the following 2-functors satisfying group axioms, written out as commutative diagrams (1) µ: G × G → G called product, (2) inv : G → G called inverse and (3) id: {∗} → G (where {∗} is a singleton) called unit or identity. Then, the product of x a '' b 77 ✤✤ ✤✤ �� α y and x ′ a ′ (( b ′ 77 ✤✤ ✤✤ �� α ′ y′ is written by x · x′ a·a′ ** b·b′ 44 ✤✤ ✤✤ �� α·α ′ y · y′ , the inverse of x a '' b 77 ✤✤ ✤✤ �� α y is x −1 a −1 )) b −1 55 ✤✤ ✤✤ �� α−1 y −1 where id{∗} = e 1e '' 1e 77 ✤✤ ✤✤ �� 11e e . The condition 1 above gives us the following interchange rules (a1 ◦ a) · (a ′ 1 ◦ a ′) = (a1 · a ′ 1) ◦ (a · a ′), (δ ◦h α) · (δ ′ ◦h α ′) = (δ · δ′) ◦h (αα ′), (β ◦v α) · (β ′ ◦v α ′) = (β · β′) ◦v (α · α ′) whenever compositions are defined. We can obtain from the condition 2 that (a1 ◦ a) −1 = a−11 ◦ a −1, (δ ◦h α) −1 = δ−1 ◦h α −1, (β ◦v α) −1 = β−1 ◦v α −1, 1−1x = 1x−1, 1 −1 1x = 11 x−1 and 1−1a = 1a−1. Moreover, the structure of a group-2-groupoid G = (G0, G1, G2) contains compatible group-groupoids G = (G0, G1), G ′ = (G0, G2) and G ′′ = (G1, G2) [1]. Equivalently we shall describe a group-2-groupoid as a 2-groupoid object in the category Gp of groups. Let C0, C1 and C2 be objects of a finitely complete category C. If C1 = (C0, C1, s, t, ε, ◦), C2 = (C0, C2, sh, th, εh, ◦h), and C3 = (C1, C2, sv, tv, εv, ◦v) are internal categories in C such that the following diagram commutes whenever the usual interchange rule satisfies between ◦h and ◦v, then (C0, C1, C2) is called an internal 2-category in C. C2 sv // tv // th ��✸ ✸✸ ✸✸ ✸✸ ✸✸ ✸ ✸✸ ✸ sh ��✸ ✸ ✸✸ ✸✸ ✸ ✸✸ ✸✸ ✸✸ C1 εv ss C0 �� t ☞ ☞☞ ☞☞☞ ☞☞☞ ☞☞ ☞☞ �� s ☞ ☞☞☞ ☞☞ ☞☞☞ ☞☞ ☞☞ εh TT ε JJ © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 37 S. Temel Proposition 3.1. A 2-category object in Gp is a group-2-groupoid. Proof. Let G = (G0, G1, G2) is a 2-category object in Gp and µ0, µ1, µ2 be multiplications of groups G0, G1, G2, respectively. Then, we can define a multiplication µ: G × G → G as a 2-functor such that µ = µ0 on objects, µ = µ1 on 1-morphisms and µ = µ2 on 2-morphisms. Similarly, we can define 2-functors id: 1 → G (where 1 is the terminal object of 2Cat, i.e. the one- object discrete 2-category) which picks out an identity object, an identity 1- morphism and an identity 2-morphism and inv : G → G picks out inverses for multiplications. Since a = 1s(a)a −11t(a) from [6] and α v = 1sv(α)α −11tv(α), αh = 11sh(α) α−111th(α) from [1], G is a 2-groupoid. Then, G is a group object in 2Cat and so G is a group-2-groupoid. � Example 3.2. Every group-groupoid can be thought as a group-2-groupoid in which all 2-morphisms are identities as follows: x a '' a 77 ✤✤ ✤✤ �� 1a y · x ′ a ′ (( a ′ 77 ✤✤ ✤✤ �� 1 ′ a y ′ = x · x′ a·a′ ** a·a′ 44 ✤✤ ✤✤ �� 1a·a′ y · y ′ It is mentioned that a group-groupoid is a 2-category with a single object [4]. Then, we shall need a different viewpoint on group-groupoids as a special kind of group-2-groupoids: Proposition 3.3. A group-2-groupoid with a single object is a group-groupoid in which both groups are necessarily abelian. Proof. In this approach, the composition of 1-morphisms and the horizontal composition of 2-morphisms are defined by multiplications of groups as follows: ⋆ a (( b 66 ✤✤ ✤✤ �� α ⋆ a ′ (( b ′ 66 ✤✤ ✤✤ �� α ′ ⋆ = ⋆ a ′∗a (( b ′∗b 66 ✤✤ ✤✤ �� α ′∗α ⋆ It is proved in [23] that a′ ∗ a = a′ · a = a · a′. Using similar way, we get α′ ∗ α = (α′ · 1e) ∗ (1e · α) = (α ′ ∗ 1e) · (1e ∗ α) = α ′ · α and α ′ · α = (1e ∗ α ′) · (α ∗ 1e) = (1e · α) ∗ (α ′ · 1e) = α · α ′ . � A third way to understand group-2-groupoids is to view them as double group-groupoids which are defined in [26] (see also [27]). Recall that a double category is a category object internal to Cat. Hence the structure of a double category contains four different but compatible category structures as partially © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 38 Further remarks on group-2-groupoids shown in the following diagram D2 s // t // t �� s �� DV1 t �� s �� ε tt DH1 s // t // ε II D0 ε UU ε kk where DH1 and D V 1 are called horizontal and vertical edge categories, respec- tively, and D2 is called the set of squares. For further details, see [12, 14, 15, 20]. The structure of a 2-category may be regarded as a double category in which all vertical morphisms are identities (or D2 and D H 1 have the same objects) [12, 20]. Therefore, a group-2-groupoid is a special kind of an internal category in the category GpGd of group-groupoids. 4. Crossed modules over group-groupoids In this section, we work on crossed modules over groupoids by replacing such groupoids with group-groupoids. Using the natural equivalence between crossed modules over groupoids and 2-groupoids given in [17], we will prove that there is a categorical equivalence between group-2-groupoids and crossed modules over group-groupoids. Definition 4.1. Let G = (X, G) and H = (X, H) are group-groupoids over the same object set, H be totally disconnected and K = (H, G, ∂) be a crossed module over G and H such that ∂ is a homomorphism of group-groupoids and the following interchange rule holds: (g • h) · (g′ • h′) = (g · g′) • (h · h′) where g, g′ ∈ G, h, h′ ∈ H. Then K is called a crossed module over group- groupoids. A morphism of crossed modules over group-groupoids is a morphism of crossed modules of groupoids which preserves group structures. Then, we can construct the category of crossed modules over group-groupoids which we de- note by Cmg*. Theorem 4.2. The categories Cmg* and Gp2Gd are equivalent. Proof. The idea of the proof is to show that the functor of 2.2 restricts to an equivalence of categories. Let A = (X, A) and B = (X, B) are group-groupoids and K = (A, B, ∂) is a crossed module over A and B. Then θ(K) = (X, B, B⋉A) is a group-2-groupoid via the process of the proof 2.2. The group multiplication of 2-morphisms in θ(K) is defined by (b, a) · (b′, a′) = (b · b′, a · a′). © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 39 S. Temel We draw such pairs as x b (( ∂(a)◦b 66 ✤✤ ✤✤ �� (b,a) y · x ′ b ′ )) ∂(a′)◦b′ 55 ✤✤ ✤✤ �� (b ′ ,a ′) y′ = x · x′ b·b′ ,, ∂(a·a′)◦(b·b′) 22 ✤✤ ✤✤ �� (b·b ′ ,a·a′) y · y′ Now we will verify that compositions and the group multiplication satisfy the interchange rule. [ (b1, a1) ◦h (b, a) ] · [ (b′1, a ′ 1) ◦h (b ′, a′) ] = [ (b1 ◦ b, a1 ◦ (b1 • a)) ] · [ (b′1 ◦ b ′, a′1 ◦ (b ′ 1 • a ′)) ] = ( (b1 ◦ b) · (b ′ 1 ◦ b ′), (a1 ◦ (b1 • a) · (a ′ 1 ◦ (b ′ 1 • a ′)) ) = ( (b1 · b ′ 1) ◦ (b · b ′), (a1 · a ′ 1) ◦ ( (b1 • a) · (b ′ 1 • a ′) ) ) = ( (b1 · b ′ 1) ◦ (b · b ′), (a1 · a ′ 1) ◦ ( (b1 · b ′ 1) • (a · a ′) ) ) = (b1 · b ′ 1, a1 · a ′ 1) ◦h (b · b ′, a · a′) = [ (b1, a1) · (b ′ 1, a ′ 1) ] ◦h [ (b, a) · (b′, a′) ] and [ (∂(a) ◦ b, a2) ◦v (b, a) ] · [ (∂(a′) ◦ b′, a′2) ◦v (b ′, a′) ] = (b, a2 ◦ a) · (b ′, a′2 ◦ a ′) = (b · b′, (a2 · a ′ 2) ◦ (a · a ′)) = [ ∂(a · a′) ◦ (b · b′), a2 · a ′ 2 ] ◦v (b · b ′, a · a′) = [ (∂(a) ◦ b, a2) · (∂(a ′) ◦ b′, a′2) ] ◦v [ (b, a) · (b′, a′) ] whenever all above compositions are defined. Now let G = (G0, G1, G2) be a group-2-groupoid. Then γ(G) is a crossed module over groupoids internal to Gp. We will verify that the interchange law holds: (f•α)·(f′•α′) = (1f ◦hα◦h1f )·(1f′◦hα ′◦h1f′) = 1f·f′◦h(α·α ′)◦h1f·f′ = (f·f ′)•(α·α′) Now we will show that SG preserves the group multiplication: SG(α · α ′) = (f · f′, (α · α′) ◦h 1f·f′) = ( f · f′ , (α · α′) ◦h (1f · 1f′) ) = ( f · f′ , (α ◦h 1f ) · (α ′ ◦h 1f′) ) = (f, α ◦h 1f ) · (f ′, α′ ◦h 1f′) = SG(α) · SG(α ′) Other details are straightforward and so are omitted. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 40 Further remarks on group-2-groupoids 5. Group-2-groupoids as internal categories in Cm A group-2-groupoid can be also thought as a special case of an internal category in the category Cm of crossed modules (see, e.g., [25] and [26] for more details about internal categories in Cm). This idea comes from that the structure of a group-2-groupoid contains three compatible group-groupoid structures. Given a group-2-groupoid, we can extract crossed modules as fol- lows: G2 sv // tv // th ��✷ ✷ ✷✷ ✷ ✷✷ ✷ ✷✷ ✷✷ ✷ ✷ sh ��✷ ✷ ✷ ✷✷ ✷✷ ✷ ✷✷ ✷✷ ✷ ✷ G1 εv ss Ker(sh) sv // tv // th|Ker(sh) ��✼ ✼✼ ✼✼ ✼✼ ✼✼ ✼✼ ✼✼ ✼✼ Ker(s) t|Ker(s) ��✟✟ ✟✟ ✟✟ ✟✟ ✟✟ ✟✟ ✟✟ ✟ εv qq 7→ G0 �� t ☞ ☞☞ ☞ ☞☞ ☞☞ ☞ ☞☞ ☞☞ ☞ �� s ☞ ☞☞ ☞☞ ☞ ☞☞ ☞ ☞☞ ☞☞ ☞ εh TT ε JJ G0 Then, we obtain an internal groupoid in Cm (Ker(sh), G0) s // t // (Ker(s), G0) ǫpp where the structure maps are defined by s = 〈sv, 1〉, t = 〈tv, 1〉, ǫ = 〈εv, 1〉 as morphisms of crossed modules. Here s, t, ǫ are equivariant maps, since sv(x • α) = x • sv(α), tv(x • α) = x • tv(α) and εv(x • f) = x • εv(f), for all x ∈ G0 and α ∈ Ker(sh). The actions of G0 on Ker(sh) and on Ker(s) are drawn in the following diagram: e x•f ++ x•g 33 ✤✤ ✤✤ �� x•α xyx −1 := x 1x (( 1x 66 ✤✤ ✤✤ �� 11x x · e f (( g 66 ✤✤ ✤✤ �� α y · x −1 1−1x ** 1−1x 44 ✤✤ ✤✤ �� 1 −1 1x x−1 We denote the category of such internal groupoids in Cm by IGCm. We know from [25, 26] that internal categories in the category Cm of crossed mod- ules are naturally equivalent to crossed squares which in turn should be viewed as a ”crossed module of crossed modules”. Hence an object of the category IGCm can be viewed as a special kind of crossed square. Let G = (G0, G1, X, ∂0, ∂1) be an object of IGCm. Then, the following diagram is commutative. G1 s // t // ∂1 ❆ ❆❆ ❆❆ ❆❆ ❆ G0 ∂0~~⑥⑥ ⑥⑥ ⑥⑥ ⑥⑥ ε ss X © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 41 S. Temel Let G = (G0, G1, X, ∂0, ∂1), G ′ = (G′0, G ′ 1, X ′, ∂′0, ∂ ′ 1) be objects of IGCm. If (λ1, λ2) is an endomorphism of the group-groupoid G = (G0, G1), and 〈λ1, λ0〉, 〈λ2, λ0〉 are morphisms of crossed modules (G0, X, ∂0), (G1, X, ∂1), respectively, then λ = (λ2, λ1, λ0) is called a morphism of IGCm. Lemma 5.1. Let G = (G0, G1, X, ∂0, ∂1) be an object of IGCm. Then x • (β ◦ α) = (x • β) ◦ (x • α) for x ∈ X, α, β ∈ G1 where s(β) = t(α). Proof. Let a α // b β // c . We know from [6] that β ◦α = β ·1−1 b ·α. Then, we get x • (β ◦ α) = x • (β · 1−1 b · α) = (x • β) · (x • 1−1 b ) · (x • α) = (x • β) · (x • 1b) −1 · (x • α) = (x • β) · 1−1 (x•b) · (x • α) = (x • β) ◦ (x • α) � Example 5.2. Every crossed module K = (M, N, ∂) over groups is an object of IGCm with the discrete groupoid of M where n • 1m = 1n•m and ∂1(1m) = ∂(m). Theorem 5.3. There is an equivalence between IGCm and Gp2Gd. Proof. A functor γ : Gp2Gd → IGCm is defined in the following way. Let H = (H0, H1, H2) be a group-2-groupoid. Then γ(H) = (G0, G1, X, ∂0, ∂1) is an object of IGCm where G0 = Ker(s), G1 = Ker(sh), X = H0, ∂0 = t ∣ ∣ Ker(s) and ∂1 = th ∣ ∣ Ker(sh) H2 sv // tv // th ��✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ sh ��✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ H1 εv rr G1 s ′ // t ′ // ∂1 ��✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ G0 ∂0 ��✍✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ε ′ ss H : 7→ γ(H) : H0 �� t ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ �� s ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ εh TT ε JJ X with actions x•f = 1x·f ·1 −1 x and x•α = 11x ·α·1 −1 1x , for x ∈ X, f ∈ G0, α ∈ G1. Now we will verify that s′, t′, ε′ are equivariant maps. s ′(x•α) = s′(11x ·α·1 −1 1x ) = sv(11x)·sv(α)·sv(1 −1 1x ) = 1x ·sv(α)·1 −1 x = x•s ′(α), t′(x • α) = t′(11x · α · 1 −1 1x ) = tv(11x) · tv(α) · tv(1 −1 1x ) = 1x · tv(α) · 1 −1 x = x • t ′(α) © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 42 Further remarks on group-2-groupoids and ε′(x•f) = ε′(1x·f ·1x −1) = εv(1x)·εv(f)·εv(1x −1) = 11x ·εv(f)·1 −1 1x = x•ε′(f). Let F = (F0, F1, F2) be a morphism of group-2-groupoids. Then γ(F) = (F2|Ker(sh), F1|Ker(s), F0) is a morphism of IGCm. Next, we define a functor θ : IGCm → Gp2Gd is an equivalence of cat- egories. Given an object G = (G0, G1, X, ∂0, ∂1) of IGCm, we can obtain a group-2-groupoid θ(G) = H = (H0, H1, H2) where H0 = X, H1 = X⋉G0, H2 = X ⋉ G1 as in the following way. Let a α // b be a morphism of G. Then pairs x (x,a) // ∂0(a) · x and x (x,b) // ∂0(b) · x are obtained as morphisms of the group-groupoid (H0, H1), and a pair x (x,α) // ∂1(α) · x is obtained as a morphism of the group-groupoid (H0, H2). Since ∂1(α) · x = ∂0s(α) · x = ∂0(a) · x, ∂1(α) · x = ∂0t(α) · x = ∂0(b) · x, then (x, α) can be considered as a 2-morphism as follows: x (x,a) ,, (x,b) 22 ✤✤ ✤✤ �� (x,α) ∂1(α) · x Let a α // b β // c . Then, the vertical composition of (x, α) and (x, β) is defined by (x, β) ◦v (x, α) = (x, β ◦ α) where the source and the target maps are defined by sv(x, α) = (x, s(α)) and tv(x, α) = (x, t(α)), respectively, and the identity map is defined by εv(x, a) = (x, 1a). Given morphisms a α // b and a1 α1 // b1 , we obtain pairs (x, α), (∂1(α) · x, α1) and we define their horizontal composite by (∂1(α) · x, α1) ◦h (x, α) = (x, α1 · α) where the source and the target maps are defined by sh(x, α) = x, th(x, α) = ∂1(α) · x, respectively, and the identity map is defined by εh(x) = (x, 1e). Clearly the vertical composition and the horizontal composition satisfy the usual interchange rule. The product of (x, α) and (x′, α′) is written by (x, α) · (x′, α′) = (x · x′, α · (x • α′)) for a α // b and a′ α ′ // b′ . If λ = (λ2, λ1, λ0) is a morphism of G, then θ(λ) = (λ0, λ0 × λ1, λ0 × λ2) is morphism of θ(G). A natural equivalence S : 1Gp2Gd → θγ is defined with a map SG : G → θγ(G) which is defined such that to be the identity on objects, SG(f) = © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 43 S. Temel (x, f ·1−1x ) and SG(α) = (x, α·1 −1 1x ) for f ∈ G1, α ∈ G2 where x = s(f) = sh(α). Clearly SG is an isomorphism and preserves the group operations and compo- sitions as follows: SG(α) · SG(α ′) = (x, α · 1−11x ) · (x ′ , α ′ · 1−11x′ ) = ( x · x′, α · 1−11x · (x • (α ′ · 1−11x′ )) ) = ( x · x′, α · 1−11x · 11x · α ′ · 1−11x′ · 1−11x ) = ( x · x′, α · α′ · 1−11xx′ ) = SG(α · α ′) where s(α) = x, s(α′) = x′, SG(δ◦hα) = SG(δ·1 −1 1y ·α) = (x, δ·1−11y ·α·1 −1 1x ) = (y, δ·1−11y )◦h(x, α·1 −1 1x ) = SG(δ)◦hSG(α) where t(α) = s(δ) = y and SG(β) ◦v SG(α) = (x, β · 1 −1 1x ) ◦v (x, α · 1 −1 1x ) = ( x, (β · 1−11x ) ◦v (α · 1 −1 1x ) ) = ( x, (β ◦v α) · (1 −1 1x ◦v 1 −1 1x ) ) = ( x, (β ◦v α) · 1 −1 1x ) = SG(β ◦v α) where sv(β) = tv(α). To define a natural equivalence T : 1IGCm → γθ, a map TG is defined such that to be identity on X, TG(a) = (e, a) for a ∈ G0 and TG(α) = (e, α) for α ∈ G1. Obviously TG is an isomorphism and preserves the composition and the group multiplication as follows: TG(β ◦ α) = (e, β ◦ α) = (e, β) ◦ (e, α) = TG(β) ◦ TG(α) TG(α) · TG(α ′) = (e, α) · (e, α′) = (e, α · (e • α′)) = (e, α · α′) = TG(α · α ′). Other details are straightforward and so are omitted. � Acknowledgements. We would like to thank the referee for his/her useful remarks and suggestions which help us to improve the paper. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 44 Further remarks on group-2-groupoids References [1] N. Alemdar and S. Temel, Group-2-groupoids and 2G-crossed modules, Hacet. J. Math. Stat. 48, no. 5 (2019), 1388–1397. [2] J. C. Baez and A. D. Lauda, Higher Dimensional Algebra V: 2-Groups, Theory Appl. Categ. 12, no. 14 (2004), 423–491. [3] J. Bènabou, Introduction to bicategories, In: Reports of the Midwest Category Seminar. Lecture Notes in Mathematics 47 (1967), 1–77. [4] J. C. Baez, A. Baratin, L. Freidel and D. K. Wise, Infinite-Dimensional Representations of 2-Groups, Mem. Amer. Math. Soc. 219, 1032 (2012). [5] J. C. Baez and D. Stevenson, The classifying space of a topological 2-group, Abel Sym- posia, vol 4. Springer, Berlin, Heidelberg. [6] R. Brown and C. B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Indagat. Math. (Proceedings) 79, no. 4 (1976), 296–302. [7] R. Brown, Higher dimensional group theory, Low Dimensional Topology, Cambridge University Press, 1982. [8] R. Brown, P. J. Higgins and R. Sivera, Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids, European Mathematical Society Tracts in Mathematics 15, 2011. [9] R. Brown and P. J. Higgins, Crossed complexes and non-abelian extensions, Category Theory. Lecture Notes in Mathematics, 962, Springer, Berlin, Heidelberg 1982. [10] R. Brown and P. J. Higgins, Tensor Products and Homotopies for ω−groupoids and crossed complexes, J. Pure and Appl. Algebra 47 (1987), 1–33. [11] R. Brown and I. Icen, Homotopies and automorphisms of crossed module over groupoids, Appl. Categ. Structures, 11 (2003) 185-206. [12] R. Brown and G. H. Mosa, Double categories, 2-Categories, thin structures and connec- tions, Theory Appl. Categ. 5, no. 7 (1999), 163–175. [13] L. Crane, Categorical physics (arXiv:hep-th/9301061v1). [14] C. Ehresmann, Catègories doubles et catègories structurès, Comptes rendus de l’Acadèmie des Sciences 256 (1963), 1198–1201. [15] C. Ehresmann, Catègories structurès, Annales Scientifiques de l’Ècole Normale Supèrieure 80 (1963), 349–425. [16] J. Huebschmann, Crossed n-fold extensions of groups and cohomology, Comment. Math. Helvetici 55 (1980), 302–314. [17] I. Icen, The equivalence of 2-groupoids and crossed modules, Commun. Fac. Sci. Univ. Ank. Series A1 49 (2000), 39–48. [18] G. Janelidze, L. Marki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002), 367–386. [19] K. H. Kamps and T. Porter, 2-Groupoid enrichments in homotopy theory and algebra, K-Theory 25 (2002), 373-409. [20] G. M. Kelly and R. Street, Review of the Elements of 2-Categories, Category Seminar. Lecture Notes in Mathematics, vol 420. (1974) Springer, Berlin, Heidelberg. [21] J.-L. Loday, Cohomologie et groupe de Steinberg relatifs, J. Algebra 54 (1978), 178–202. [22] S. Maclane, Categories for the Working Mathematician, Graduate Text in Mathematics, (1971) Springer-Verlag New York. [23] M. Forrester-Barker, Group objects and internal categories (arXiv:math/0212065). [24] B. Noohi, Notes on 2-groupoids, 2-groups and crossed modules, Homology Homotopy Appl. 9, no. 1 (2007), 75–106. [25] T. Sahan and J. J. Mohammed, Categories internal to crossed modules, Sakarya Uni- versity Journal of Science 23, no. 4 (2019), 519–531. [26] S. Temel, T. Şahan and O. Mucuk, Crossed modules, double group-groupoids and crossed squares, Filomat 34, no. 6 (2020), 1755-1769. [27] S. Temel, Internal categories in crossed semimodules and Schreier internal categories, Math. Sci. Appl. E-Notes 8, no. 2 (2020), 86–95. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 45 S. Temel [28] J. H. C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55 (1949), 453–496. [29] J. H. C. Whitehead, Note on a previous paper entitled “On adding relations to homotopy group”, Ann. Math. 47 (1946), 806–810. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 46