Theory and Applications of Categories, Vol. 12, No. 4, 2004, pp. 195–224.

BAER INVARIANTS IN SEMI-ABELIAN CATEGORIES II: HOMOLOGY

T. EVERAERT AND T. VAN DER LINDEN

ABSTRACT. This article treats the problem of deriving the reflector of a semi-abelian category A onto a Birkhoff subcategory B of A. Basing ourselves on Carrasco, Cegarra and Grandjeán’s homology theory for crossed modules, we establish a connection between our theory of Baer invariants with a generalization—to semi-abelian categories— of Barr and Beck’s cotriple homology theory. This results in a semi-abelian version of Hopf’s formula and the Stallings-Stammbach sequence from group homology.

Contents

1 Introduction 195 2 Chain complexes 200 3 Simplicial objects 203 4 The Kan condition 206 5 Implications of the Kan condition 209 6 Cotriple homology and Hopf’s formula 215

1. Introduction

1.1. A semi-abelian category is a category with binary coproducts which is pointed, Barr exact and Bourn protomodular. In [18] we show the following.

1.2. Theorem. Let A be a pointed, Barr exact and Bourn protomodular category with enough projectives. Consider a short exact sequence

[figure: 0; K; A; f; B; 0]

in A. If V is a Birkhoff subfunctor of A, then an exact sequence

[figure: ΔV (A); ΔLf; ΔV (B); \frac{K}{V_{1}f}; U(A); Uf; U(B); 0]  (A)

The first author’s research is financed by a Ph.D. grant of the Institute of Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).

Received by the editors 2003-07-17 and, in revised form, 2004-02-10. Transmitted by Walter Tholen. Published on 2004-02-22. 2000 Mathematics Subject Classification: Primary 20J05 18G50 18C15; Secondary 18G30 18G35

18E25.

Key words and phrases: Baer invariant, semi-abelian category, cotriple homology. \begin{matrix}c \\ ◯ \end{matrix} T. Everaert and T. Van der Linden, 2004. Permission to copy for private use granted.

exists, which depends naturally on the given short exact sequence.

The aim of this article is to interpret sequence A as a generalization of the Stallings-Stammbach sequence from integral homology of groups [29], [30]. We do this by proving the following semi-abelian version of Hopf’s formula [21]:

H_{2}(X, U)_{𝔾} ≅ ΔV (X).  (B)

1.3. Let us start by explaining the right hand side of this formula, briefly recalling the main concepts from [18].

A presentation of an object A in a category A is a regular epimorphism (a coequalizer) p : A_{0} [figure]A. The category of presentations of objects of A—a morphism f = (f_{0}, f) : p

[figure]q being a commutative square

\xymatrix{A_{0} \ar@{|->}[d]_{p} \ar@{=>}[r]^{f_{0}} & B_{0} \ar@{|->}[d]^{q} \\ A \ar@{=>}[r]_{f} & B}

—is denoted by PrA. pr : PrA [figure]A denotes the forgetful functor which maps a presentation to the object presented, sending a morphism of presentations f = (f′, f) to f. Two morphisms of presentations f , g : p [figure]q are called isomorphic, notation f ≃ g, if pr f = pr g (or f = g). A functor B : PrA [figure]A is called a Baer invariant if f ≃ g implies that Bf = Bg.

If A has sufficiently many projective objects, then any Baer invariant B : PrA [figure]A induces a functor A [figure]A; for instance the functor ΔV : A [figure]A arises this way. Denoting W_{proj} the full subcategory of all projective objects of A (and i : W_{proj} [figure] PrA the canonical inclusion), we call choice of projective presentations in A any graph morphism c : A [figure]W_{proj} such that pr^{∘}i^{∘}c = 1_{A}. Then, for any choice of projective presentations c : A [figure]W_{proj} in A, B^{∘}i^{∘}c is a functor. Moreover, for any other choice of

projective presentations c′ : A  [figure]W_{proj}, a natural isomorphism B^{∘}i^{∘}c  [figure] C^{∘}i^{∘}c′ exists.

1.4. For the larger part of our theory of Baer invariants we need the category A to be semi-abelian. This means that A is pointed, Barr exact and Bourn protomodular with binary coproducts [25]. The reason we work in this context is that it is natural for the classical theorems of homological algebra—as the Snake Lemma, the 3 × 3 Lemma and Noether’s Isomorphism Theorems—to hold: see, e.g. Bourn [10] or Borceux and Bourn [5] for a new approach to these results, on which our paper [18] strongly depends.

A category A is regular [2] when it has finite limits and coequalizers of kernel pairs (i.e. the two projections k_{0}, k_{1} : R[f] [figure]A of the pullback of an arrow f : A [figure]B along itself), and when a pullback of a regular epimorphism along any morphism is again a regular epimorphism. In this case, every regular epimorphism is the coequalizer of its kernel pair, and every morphism f : A [figure]B has an image factorization f = Im f^{∘}p, unique up to isomorphism, where p : A [figure]I[f] is regular epi and the image Im f : I[f]

[figure]B of f is mono. Taking images is functorial. Moreover, in a regular category, regular epimorphism are stable under composition, and if a composition f^{∘}g is regular epi, then so is f. A regular category in which every equivalence relation is a kernel pair is called Barr exact.

A pointed category with pullbacks A is Bourn protomodular [7] as soon as the Split Short Five-Lemma holds. This means that for any commutative diagram

[figure: 0; K′; k′; u; A′; f′; v; B′; s′; w; 0; K; k; A; f; B; s]

such that f and f′ are split epimorphisms (with resp. splittings s and s′) and such that k = Ker f and k′ = Ker f′, u and w being isomorphisms implies that v is an isomorphism.

A sequence

\xymatrix{K \ar@{=>}[r]^{k} & A \ar@{=>}[r]^{f} & B}  (C)

in a pointed category is called short exact if k = Ker f and f = Coker k. We denote this situation

[figure: 0; K; k; A; f; B; 0.]

In a pointed, regular and protomodular category the exactness of sequence C is equivalent to demanding that k = Ker f and f is a regular epimorphism. Thus, a pointed, regular and protomodular category has all cokernels of kernels. A sequence of morphisms

\xymatrix{… \ar@{=>}[r] & A_{i+1} \ar@{=>}[r]^{f_{i+1}} & A_{i} \ar@{=>}[r]^{f_{i}} & A_{i−1} \ar@{=>}[r] & …}

in pointed, regular and protomodular category is called exact if, for any i, Im f_{i+1} = Ker f_{i}.

1.5. Let A be a semi-abelian category. A normal subfunctor V of 1_{A} (i.e. a kernel V [figure] 1_{A}) is called Birkhoff subfunctor of A if V preserves regular epimorphisms. Recall from Janelidze and Kelly [24] that a Birkhoff subcategory of A is a reflective subcategory B of A which is full and closed in A under subobjects and quotient objects. In [18], we show that Birkhoff subfunctors correspond bijectively to the Birkhoff subcategories of A: assuming that, for any A ∈ A, the sequence

[figure: 0; V (A); μ_{A}; A; η_{A}; U(A); 0]

is exact, V is a Birkhoff subfunctor if and only if U reflects A onto a Birkhoff subcategory. A Birkhoff subcategory of a semi-abelian category is semi-abelian.

Any Birkhoff subfunctor V of A induces a functor V_{1} : Pr A [figure]A, constructed as follows. Let p : A_{0} [figure]A be a presentation in A and (R[p], k_{0}, k_{1}) its kernel pair. Applying V, next taking the coequalizer of V k_{0} and V k_{1} and then the kernel of Coeq(V k_{0}, V k_{1}), one gets V_{1}p.

[figure: V (R[p]); V k_{0}; V k_{1}; 0; V_{1}p; V (A_{0}); Coeq (V k_{0},V k_{1}); Coeq[V k_{0}, V k_{1}]; 0]

Given a Birkhoff subfunctor V of A and a presentation p : A_{0} [figure]A, ΔV is the functor induced by the Baer invariant PrA [figure]A which maps p to

\frac{K[p] ∩ V (A_{0})}{V_{1}p}.

Consider the category Gp of groups and its Birkhoff subcategory Ab = Gp_{Ab} of abelian groups. The associated Birkhoff subfunctor V sends G to [G, G], the commutator subgroup of G. It is indeed well known that the abelianization of a group G, i.e. the reflection of G along the inclusion Ab [figure]Gp, is just G/[G, G]. For groups R ◁ F, [R, F] denotes the subgroup of F generated by the elements rfr^{−1}f^{−1}, with r ∈ R and f ∈ F. If p denotes the quotient F [figure]F/R, then V_{1}p = [R, F]. Now let

[figure: 0; R; F; p; G; 0]

be a presentation of a group G by a “group of generators” F and a “group of relations” R, i.e. a short exact sequence with F a free (or, equivalently, projective) group. Then it follows that

ΔV (G) = \frac{R ∩ [F, F]}{[R, F]}.

1.6. Hopf’s formula [21] is the isomorphism

H_{2}(G, ℤ) ≅ \frac{R ∩ [F, F]}{[R, F]};  (D)

here, H_{2}(G, ℤ) is the second integral homology group of G. It is clear that the left hand side of the formula B should, in a way, generalize this homology group.

That such a generalization is possible comes as no surprise, as Carrasco, Cegarra and Grandjeán (in their article [16]) already prove a generalized Hopf formula in the category CM of crossed modules. Furthermore, they obtain a five term exact sequence which generalizes the Stallings-Stammbach sequence [29], [30]. Their five term exact sequence becomes a particular case of ours; see Corollary 6.10.

Since the right hand side of B is defined relative to a Birkhoff subcategory B of a semiabelian category A, so must be the left-hand side. We restrict ourselves to the following situation: A is a semi-abelian category, monadic over Set; U : A [figure]B is a reflector onto a Birkhoff subcategory B of A; 𝔾 is the comonad on A, defined by the adjunction Set ⇀ A. In this situation, the formula B holds, for p : GX [figure]X the standard “𝔾-free” projective presentation of an object X.

Carrasco, Cegarra and Grandjeán define their homology of crossed modules by deriving the functor ab : CM [figure]CM_{Ab}, which sends a crossed module (T, G, ∂) to its abelianization (T, G, ∂)_{ab}—an object of the abelian category CM_{Ab}. More precisely, using the monadicity of the forgetful functor U : CM [figure]Set, they obtain a comonad 𝔾 on CM. This, for any crossed module (T, G, ∂), yields a canonical simplicial object 𝔾(T, G, ∂) in CM. The n-th homology object (an abelian crossed module) of a crossed module (T, G, ∂) is then defined as H_{n−1}C𝔾(T, G, ∂)_{ab}, the (n−1)-th homology object of the unnormalized chain complex associated with the simplicial object 𝔾(T, G, ∂)_{ab}. This is an application of Barr and Beck’s cotriple homology theory [1], which gives a way of deriving any functor U : A [figure]B from an arbitrary category A equipped with a comonad 𝔾 to an abelian category B.

To prove our Hopf formula, we use methods similar to those of Carrasco, Cegarra and Grandjeán—in fact, our proof of Theorem 6.9 is a modification of their [16, Theorem 12]. Therefore, a semi-abelian notion of homology must be introduced. This is done in Section 6. In Sections 2 through 5 the necessary theory is developed.

1.7. Since chain complexes are crucial in any homology theory, in Section 2, we consider them in a semi-abelian context. More precisely, in categories that are pointed, regular and protomodular. A morphism in such a category is proper [10] when its image is a kernel. We call a chain complex proper whenever all its differentials are. As in the abelian case, the n-th homology object of a proper chain complex C with differentials d_{n} is said to be H_{n}C = Cok[C_{n+1} [figure]K[d_{n}]]. We prove that this equals the dual K_{n}C = K[Cok[d_{n+1}]

[figure]C_{n−1}]. Moreover, any short exact sequence of proper chain complexes gives rise to a long exact sequence of homology objects.

In Section 3 we extend the homology theory of Section 2 to simplicial objects. Therefore, we consider the Moore functor N : SA [figure]ChA. Suppose that A is a pointed category with pullbacks. Let us write ∂_{i} for the face operators of a simplicial object A in A. The normalized chain complex N(A) of A is the chain complex with N_{0}A = A_{0},

N_{n}A = ⋂_{i=0}^{n−1} K[∂_{i} : A_{n} [figure]A_{n−1}]

and differentials d_{n} = ∂_{n}^{∘} ⋂_{i} Ker ∂_{i} : N_{n}A [figure]N_{n−1}A, for n ≥ 1, and A_{n} = 0, for n < 0. When A is pointed, exact and protomodular, we prove that the Moore functor maps simplicial objects to proper chain complexes. This allows us to define the n-th homology object of A as H_{n}A = H_{n}N(A). Furthermore, we show that, if ϵ : A [figure]A_{−1} is a contractible augmented simplicial object, then H_{0}A = A_{−1} and, for n ≥ 1, H_{n}A = 0.

Recall from [15] and [14] that a finitely complete category is called Mal’cev, if every reflexive relation is an equivalence relation. A regular category is Mal’cev if and only if the composition of equivalence relations is commutative. A finitely complete protomodular category is always Mal’cev [8].

The validity of our generalized Hopf formula depends strongly on the fact that the Moore functor N : SA [figure]ChA is exact. This essentially amounts to the fact that, in a regular Mal’cev category, any regular epimorphism of simplicial objects is a Kan fibration. We prove this in Section 4, generalizing Carboni, Kelly and Pedicchio’s result [14] that in a regular Mal’cev category, every simplicial object is Kan. The exactness of N is shown in Section 5. We get that any short exact sequence of simplicial objects induces a long exact sequence of homology objects. In Section 5 we moreover prove Dominique Bourn’s conjecture that, for n ≥ 1, the homology H_{n}A of a simplicial object A in a semi-abelian category A is an abelian object of A.

Finally, in Section 6, we generalize Barr and Beck’s notion of cotriple homology [1] to the situation where B is a pointed, regular and protomodular category. Their definition is modified to the following. Let A be a category, 𝔾 a comonad on A and U : A [figure]B a functor. The n-th homology object H_{n}(X, U)_{𝔾} of X with coefficients in U relative to the cotriple 𝔾 is the object H_{n−1}NU(𝔾X), the (n − 1)-th homology object of the normalized chain complex associated with the simplicial object U(𝔾X) of B. We show that H_{1}(X, U)_{𝔾} = U(X), give a proof of formula B and obtain a version of the Stallings-Stammbach sequence.

1.8. For the basic theory of semi-abelian categories we refer to the Borceux’s survey [3] and Borceux and Bourn’s book [5]. For general category theory we used Borceux [4] and Mac Lane [27]. Weibel’s book [31] provides an excellent introduction to homological algebra. For the theory of model categories the reader is referred to Quillen [28] and Hovey [22].

Acknowledgements. Thanks to Marino Gran for starting this work, by pointing us to the subject of Baer invariants in semi-abelian categories; to George Janelidze and Rudger Kieboom for fruitful discussion and infinite support; to Francis Borceux and Dominique Bourn for kindly making available a preprint of their book [5]; to Dominique Bourn for suggesting us Theorem 5.5; to the referee for helpful comments and suggestions.

2. Chain complexes

2.1. Notation. Given a morphism f : A [figure]B in A, (if it exists) its kernel is denoted by Ker f : K[f] [figure]A, its image by Im f : I[f] [figure]B and its cokernel by Coker f : B [figure]Cok[f]. In a diagram, the forms [figure: A; B] , [figure: A; B] and \xymatrix{A \ar@{|->}[r] & B} signify that the arrow is, respectively, a monomorphism, a normal monomorphism and a regular epimorphism.

Recall that a chain complex C is a collection of morphisms (d_{n} : C_{n} [figure]C_{n−1})_{n∈ℤ} such that d_{n}^{∘}d_{n+1} = 0, for all n ∈ ℤ. Although usually considered in an abelian context, chain complexes of course make sense in any pointed category A. Obtaining a good notion of homology objects H_{n}C of a chain complex C, however, demands a stronger assumption on A.

When A is an abelian category, H_{n}C is K[d_{n}]/I[d_{n+1}] (see, for example, [31]). Since this is just Cok[C_{n+1} [figure]K[d_{n}]], it seems reasonable to define H_{n}C this way, supposed that the considered kernels and cokernels exist in A. Yet, one could also suggest the dual K_{n}C = K[Cok[d_{n+1}] [figure]C_{n−1}], since, in the abelian case, this equals H_{n}C. Let A be a pointed, regular and protomodular category. Recall that a morphism is proper if its image is a kernel. We call a chain complex C in A proper whenever all its differentials are. We will prove in Proposition 2.3 that for any proper complex C, the homology objects H_{n}C and K_{n}C are isomorphic. Furthermore, as the Snake Lemma holds in A—see Bourn [10, Theorem 14]—we get Proposition 2.4: any short exact sequence of proper chain complexes induces a long exact sequence of homology objects.

Let ChA be the category of chain complexes in A, morphisms being commutative ladders, and let PChA be the full subcategory of proper chain complexes. For a complex C ∈ PChA and n ∈ ℤ, let H_{n}C be its n-th homology object, and K_{n}C its dual, as defined above. Note that H_{n}C and K_{n}C exist, as A has all cokernels of kernels. Further remark that, like A, the category ChA is pointed, regular and protomodular. This is not the case for PChA, since PChA e.g. need not have kernels. By an exact sequence of proper chain complexes, we mean an exact sequence in ChA such that the objects are proper chain complexes.

We need the following

2.2. Lemma. [10, 5] Let A be a pointed, regular and protomodular category. Consider the following commutative diagram, where k = Ker f, f′ is regular epi and the left hand square a pullback:

[figure: K′; k′; u; A′; f′; v; B′; w; 0; K; k; A; f; B.]

If k′ = Ker f′, then w is a monomorphism.

2.3. Proposition. Let A be a pointed, regular and protomodular category. For any n ∈ ℤ, H_{n} and K_{n} are naturally isomorphic functors PChA [figure]A.

Proof. Consider the commutative diagram of solid arrows

[figure: C_{n+1}; d′_{n+1}; d_{n+1}; C_{n}; d_{n}; Coker d_{n+1}; C_{n−1}; K[d_{n}]; Ker d_{n}; Coker d′_{n+1}; p_{n}; Cok[d_{n+1}]; d′′_{n}; H_{n}C; j_{n}; (λ_{n})_{C}; K_{n}C.; Ker d′′_{n}]  (E)

Note that all cokernels exist, because d_{n+1} and d′_{n+1} are proper. Since H_{n}C is a cokernel and K_{n}C is a kernel, unique morphisms j_{n} and p_{n} exist that keep the diagram commutative. A natural transformation λ : H_{n} [figure] K_{n} is defined by the resulting unique (λ_{n})_{C}. To prove it an isomorphism, first consider the following diagram with exact rows.

[figure: 0; K[d_{n}]; p_{n}; C_{n}; d_{n}; Coker d_{n+1}; C_{n−1}; 0; K_{n}C; Ker d′′_{n}; Cok[d_{n+1}]; d′′_{n}; C_{n−1}]

1_{C_{n−1}} being a monomorphism, the left hand square is a pullback, and p_{n} is a regular epimorphism.

Considering the image factorizations of d′_{n+1} and d_{n+1}, there is a morphism i such that the diagram with exact rows

[figure: C_{n+1}; I[d_{n+1}]; C_{n+1}; C_{n+1}; C_{n+1}; I[d′_{n+1}]; I[d′_{n+1}]; I[d_{n+1}]; I[d_{n+1}]; C_{n}; Ker d_{n}; I[d′_{n+1}]; I[d_{n+1}]; i; I[d′_{n+1}]; K[d_{n}]; Im d′_{n+1}; K[d_{n}]; C_{n}; Im d_{n+1}; C_{n}; Cok[d_{n+1}]; Coker d_{n+1}; K[d_{n}]; C_{n}; K[d_{n}]; H_{n}C; Coker d′_{n+1}; H_{n}C; Cok[d_{n+1}]; j_{n}; Cok[d_{n+1}]; 0; H_{n}C; Cok[d_{n+1}]; H_{n}C; 0; 0; 0; C_{n+1}; K[d_{n}]; d′_{n+1}; C_{n+1}; C_{n}; d_{n+1}]

commutes. Clearly, it is both a monomorphism and a regular epimorphism, thus an isomorphism. Because d_{n+1} is proper, Im d_{n+1} is a kernel. We get that also Im d′_{n+1} is a kernel. Now the middle square is a pullback, so Lemma 2.2 implies that j_{n} is a monomorphism.

Accordingly, since E commutes, (λ_{n})_{C} is both regular epi and mono, hence it is an isomorphism.

For n ∈ ℤ, let \overline{d}_{n} denote the unique map such that the diagram

[figure: C_{n}; Coker d_{n+1}; d_{n}; C_{n−1}; Cok[d_{n+1}]; _{\overline{d}_{n}}; K[d_{n−1}]; Ker d_{n−1}]

commutes. Since d_{n} is proper, so is \overline{d}_{n}.

The following is a straightforward generalization of the abelian case—see, for instance, Theorem 1.3.1 in Weibel [31].

2.4. Proposition. Let A be a pointed, regular and protomodular category. Any short exact sequence of proper chain compexes

[figure: 0; C′′; C′; C; 0]

gives rise to a long exact sequence of homology objects

\xymatrix{… \ar@{=>}[r] & H_{n+1}C \ar@{=>}[r]^{δ_{n+1}} & H_{n}C′′ \ar@{=>}[r] & H_{n}C′ \ar@{=>}[r] & H_{n}C \ar@{=>}[r]^{δ_{n}} & H_{n−1}C′′ \ar@{=>}[r] & …}  (F)

which depends naturally on the given short exact sequence.

Proof. Since the d_{n} and \overline{d}_{n} are proper, mimicking the abelian proof—using the Snake Lemma twice—we get an exact sequence

\xymatrix{K_{n}C′′ \ar@{=>}[r] & K_{n}C′ \ar@{=>}[r] & K_{n}C \ar@{=>}[r] & H_{n−1}C′′ \ar@{=>}[r] & H_{n−1}C′ \ar@{=>}[r] & H_{n−1}C}

for every n ∈ ℤ. By Proposition 2.3 we can paste these together to F. The naturality follows from the naturality of the Snake Lemma.

Note that Bourn’s version of the Snake Lemma [10, Theorem 14] states only the existence and the exactness of the sequence. However, it is quite clear from the construction of the connecting morphism that the sequence is, moreover, natural.

3. Simplicial objects

In this section we extend the homology theory of Section 2 to simplicial objects. We start by considering the Moore functor N : SA [figure]ChA, which maps a simplicial object A in a pointed category with pullbacks A to the normalized chain complex N(A). We prove that, when A is, moreover, exact and protomodular, N(A) is always proper, and then define the n-th homology object of A as H_{n}A = H_{n}N(A). Furthermore, we show that, if ϵ : A [figure]A_{−1} is a contractible augmented simplicial object, then H_{0}A = A_{−1} and, for n ≥ 1, H_{n}A = 0.

When working with simplicial objects in a category A, we will use the notations of [31]. The simplicial category Δ has, as objects, finite ordinals [n] = {0, …, n}, for n ∈ ℕ and, as morphisms, monotone functions. The category SA of simplicial objects and simplicial maps of A is the functor category Fun(Δ^{op}, A). Thus a simplicial object A : Δ^{op} [figure]A corresponds to the following data: a sequence of objects (A_{n})_{n∈ℕ}, face operators ∂_{i} : A_{n}

[figure]A_{n−1} and degeneracy operators σ_{i} : A_{n} [figure]A_{n+1}, for i ∈ [n] and n ∈ ℕ, subject to the simplicial identities

∂_{i}^{∘}∂_{j} = ∂_{j−1}^{∘}∂_{i} if i < j
σ_{i}^{∘}σ_{j} = σ_{j+1}^{∘}σ_{i} if i ≤ j

∂_{i}^{∘}σ_{j} = \begin{cases}σ_{j−1}^{∘}∂_{i} & if i < j \\ 1 & if i = j or i = j + 1 \\ σ_{j}^{∘}∂_{i−1} & if i > j + 1. \end{cases}

An augmented simplicial object ϵ : A [figure]A_{−1} consists of a simplicial object A and a map ϵ : A_{0} [figure]A_{−1} with ϵ^{∘}∂_{0} = ϵ^{∘}∂_{1}. It is contractible if there exist maps f_{n} : A_{n}

[figure]A_{n+1}, n ≥ −1, with ϵ^{∘}f_{−1} = 1_{A_{−1}}, ∂_{0}^{∘}f_{0} = f_{−1}^{∘}ϵ, ∂_{n+1}^{∘}f_{n} = 1_{A_{n}} and ∂_{i}^{∘}f_{n} = f_{n−1}^{∘}∂_{i}, for 0 ≤ i ≤ n and n ∈ ℕ.

3.1. Definition. Let A be a simplicial object in a pointed category A with pullbacks. The normalized, or Moore, chain complex N(A) is the chain complex with N_{0}A = A_{0},

N_{n}A = ⋂_{i=0}^{n−1} K[∂_{i} : A_{n} [figure]A_{n−1}]

and differential d_{n} = ∂_{n}^{∘} ⋂_{i} Ker ∂_{i} : N_{n}A [figure]N_{n−1}A, for n ≥ 1, and A_{n} = 0, for n < 0. This gives rise to a functor N : SA [figure]ChA.

3.2. Remark. Note that, in the above definition, ∂_{n}^{∘} ⋂_{i} Ker ∂_{i} : N_{n}A [figure]A_{n−1} may indeed be considered as an arrow d_{n} : N_{n}A [figure]N_{n−1}A: the map clearly factors over ⋂_{i} Ker ∂_{i} : N_{n−1}A [figure]A_{n−1}.

3.3. Remark. Obviously, for n ≥ 1, the object of n-cycles Z_{n}A = K[d_{n}] of a simplicial object A of A is equal to ⋂_{i=0}^{n} K[∂_{i} : A_{n} [figure]A_{n−1}].

3.4. Remark. The functor N : SA [figure]ChA preserves limits. Indeed, limits in SA and ChA are computed degreewise, and taking kernels and intersections (pulling back), as occurs in the construction of N, commutes with taking arbitrary limits in A. In Section 5 we shall prove that N, moreover, preserves regular epimorphism, hence is exact.

In a protomodular category A, an intrinsic notion of normal monomorphism exists (see Bourn [9]). We will, however, not introduce this notion here. It will be sufficient to note that, if A is moreover exact, the normal monomorphisms are just the kernels. To prove Theorem 3.6, we need the following

3.5. Lemma. [Non-Effective Trace of the 3 × 3 Lemma [11, Theorem 4.1]] Consider, in a regular and protomodular category, a commutative square with horizontal regular epimorphisms

\xymatrix{A′ \ar@{=>}[d]_{v} \ar@{|->}[r]^{f′} & B′ \ar@{=>}[d]^{w} \\ A \ar@{|->}[r]_{f} & B.}

If w is a monomorphism and v a normal monomorphism, then w is a normal monomorphism.

3.6. Theorem. Let A be a pointed, exact and protomodular category and A a simplicial object in A. Then N(A) is a proper chain complex of A.

Proof. Any differential d_{n}, when viewed as an arrow to A_{n−1}, is a composition of a normal monomorphism (an intersection of kernels), and a regular epimorphism (a split epimorphism, by the simplicial identities). Hence, the Non-Effective Trace of the 3 × 3 Lemma 3.5 implies that d_{n} : N_{n}A [figure]A_{n−1} is proper. This clearly remains true when, following Remark 3.2, we consider d_{n} as an arrow to N_{n−1}A.

3.7. Definition. Suppose A is pointed, exact and protomodular. The object H_{n}A = H_{n}N(A) will be called the n-th homology object of A, and the resulting functor H_{n} : SA

[figure]A the n-th homology functor, for n ∈ ℕ.

In order to compute, in Proposition 3.11, the homology of a contractible augmented simplicial object, we first make the following, purely categorical, observations. We start by recalling a result due to Dominique Bourn.

3.8. Lemma. [7, Proposition 14] If, in a protomodular category, a square with vertical regular epimorphisms

\xymatrix{A \ar@{=>}[r] \ar@{=>}[d] & B \ar@{|->}[d] \\ C \ar@{=>}[r] & D}

is a pullback, it is also a pushout.

A fork is a diagram such as G below where e^{∘}∂_{0} = e^{∘}∂_{1}.

3.9. Proposition. [(cf. 10, Corollary 6)] In a pointed and protomodular category A, let

\xymatrix{A \ar@{=>}[r]^{∂_{1}} \ar@{=>}[r]_{∂_{0}} & B \ar@{=>}[r]^{e} & C}  (G)

be a fork and t : B [figure]A a map with ∂_{0}^{∘}t = ∂_{1}^{∘}t = 1_{B}. Then the following are equivalent:

1. e is a coequalizer of ∂_{0} and ∂_{1};

2. the square

\xymatrix{A \ar@{=>}[d]_{∂_{0}} \ar@{=>}[r]^{∂_{1}} & B \ar@{=>}[d]^{e} \\ B \ar@{=>}[r]_{e} & C}

is a pushout;

3. e is a cokernel of ∂_{1}^{∘}Ker ∂_{0}.

Proof. The equivalence of 1. and 2. is obvious. In the diagram

[figure: K[∂_{0}]; Ker ∂_{0}; A; ∂_{0}; ∂_{1}; B; e; 0; B; e; C,]

the left hand side square is a pushout: indeed, Lemma 3.8 applies, since it is a pullback along the split, hence regular, epimorphism ∂_{0}. Consequently, the outer rectangle is a pushout if and only if the right square is, which means that 2. and 3. are equivalent.

3.10. Corollary. If A is a pointed, exact and protomodular category and A a simplicial object of A (with face operators ∂_{0}, ∂_{1} : A_{1} [figure]A_{0}), then H_{0}A = Coeq[∂_{0}, ∂_{1}].

A fork is split if there are two more arrows s : C [figure]B and t : B [figure]A such that e^{∘}s = 1_{C}, ∂_{1}^{∘}t = 1_{B} and ∂_{0}^{∘}t = s^{∘}e. Every split fork is a coequalizer diagram.

3.11. Proposition. If ϵ : A [figure]A_{−1} is a contractible augmented simplicial object in a pointed, exact and protomodular category A, then H_{0}A = A_{−1} and, for n ≥ 1, H_{n}A = 0.

Proof. The contractibility of ϵ : A [figure]A_{−1} implies that the fork

\xymatrix{A_{1} \ar@{=>}[r]^{∂_{1}} \ar@{=>}[r]_{∂_{0}} & A_{0} \ar@{=>}[r]^{ϵ} & A_{−1}}
is split (by the arrows f_{−1} : A_{−1}  [figure]A_{0} and f_{0} : A_{0}  [figure]A_{1}). We get that it is a

coequalizer diagram. The first equality now follows from Corollary 3.10.

In order to prove the other equalities, first recall from Remark 3.3 that, for n ≥ 1,

Z_{n}A = K[d_{n} : N_{n}A  [figure]N_{n−1}A] = ⋂_{i=0}^{n} K[∂_{n} : A_{n}  [figure]A_{n−1}].
We are to show that the image of d_{n+1} : N_{n+1}A  [figure]N_{n}A is Ker d_{n} : K[d_{n}]  [figure]N_{n}A.

But, for any i ≤ n, the left hand downward-pointing arrow in the diagram with exact rows

[figure: 0; K[∂_{i}]; Ker ∂_{i}; A_{n+1}; ∂_{n+1}; ∂_{i}; A_{n}; ∂_{n}; 0; 0; K[∂_{i}]; Ker ∂_{i}; A_{n}; ∂_{i}; f_{n}; A_{n−1}; f_{n−1}; 0]

is a split epimorphism, because both its upward and downward pointing squares commute. It follows that the intersection N_{n+1}A = ⋂_{i≤n} K[∂_{i}] [figure]⋂_{i≤n} K[∂_{i}] = K[d_{n}] is a split, hence a regular, epimorphism, and Im d_{n+1} = Ker d_{n}.

4. The Kan condition

The results in this section will allow us to prove, in Section 5, two important facts concerning the functors H_{n} : PChA [figure]A and N : SA [figure]PChA: Theorem 5.5 and Proposition 5.6. We recall the result from Carboni, Kelly and Pedicchio [14] that every simplicial object in a regular Mal’cev category is Kan. We add that every regular epimorphism between simplicial objects of a regular Mal’cev category is a Kan fibration.

Kan complexes and Kan fibrations are very important in the homotopy theory of simplicial sets (or simplicial objects in a variety over Set). In their article [14], Carboni, Kelly and Pedicchio extend the notion of Kan complex to an arbitrary category A. When A is regular, their definition amounts to the one stated in Definition 4.1. (They consider only horns with n ≥ 2; indeed, any simplicial object fulfils the Kan condition for n = 1.) In the same spirit, we propose an extension of the notion of Kan fibration to a regular category A.

4.1. Definition. Consider a simplicial object K in a regular category A. For n ≥ 1 and k ∈ [n], a family

x = (x_{i} : X [figure]K_{n−1})_{i∈[n],i≠k}

is called an (n, k)-horn of K if it satisfies ∂_{i}^{∘}x_{j} = ∂_{j−1}^{∘}x_{i}, for i < j and i, j ≠ k.

We say that K is Kan if, for every (n, k)-horn x = (x_{i} : X [figure]K_{n−1}) of K, there is a regular epimorphism p : Y [figure]X and a map y : Y [figure]K_{n} such that ∂_{i}^{∘}y = x_{i}^{∘}p for i ≠ k.

A map f : A [figure]B of simplicial objects is said to be a Kan fibration if, for every

(n, k)-horn x = (x_{i} : X  [figure]A_{n−1}) of A and every b : X  [figure]B_{n} with ∂_{i}^{∘}b = f_{n−1}^{∘}x_{i} for
all i ≠ k, there is a regular epimorphism p : Y  [figure]X and a map a : Y  [figure]A_{n} such

that f_{n}^{∘}a = b^{∘}p and ∂_{i}^{∘}a = x_{i}^{∘}p for all i ≠ k.

Obviously, a simplicial object K is Kan if and only if the unique map K [figure]∗ from K to a terminal object ∗ of SA is a Kan fibration.

In case A is Set, these notions have an equivalent formulation in which X and Y are both equal to a terminal object ∗. These equivalent formulations are the classical definitions of Kan simplicial set and Kan fibration—see, for instance, Weibel [31]. There is also the following connection between Definition 4.1 and the set-theoretic notions.

4.2. Proposition. Let A be a regular category and Υ : A [figure]Set a functor which preserves regular epimorphisms and has a left adjoint Φ : Set [figure]A. Then

1. for any Kan simplicial object K of A, the simplicial set Υ(K) is Kan;

2. for any Kan fibration f : A [figure]B of simplicial objects in A, the simplicial map Υf : Υ(A) [figure]Υ(B) is a Kan fibration of simplicial sets.

Proof. Let ζ : 1_{Set} [figure] Υ^{∘}Φ denote the unit of the adjunction and

φ_{X,A} : hom_{A}(Φ(X), A) ≅ hom_{Set}(X, Υ(A))

the canonical isomorphism, natural in X ∈ Set^{op} and A ∈ A. Suppose that f : A [figure]B is a Kan fibration; consider an (n, k)-horn x = (x_{i} : X [figure]Υ(A)_{n−1})_{i∈[n],i≠k} of Υ(A) and a map b : X [figure]Υ(B_{n}) with Υ∂_{i}^{∘}b = Υf_{n−1}^{∘}x_{i} for all i ≠ k. We give a proof of 2.

Note that the collection (φ^{−1}(x_{i}) : Φ(X) [figure]A_{n−1})_{i∈[n],i≠k} is an (n, k)-horn of A such that ∂_{i}^{∘}φ^{−1}(b) = f_{n−1}^{∘}φ^{−1}(x_{i}): indeed, by the naturality of φ^{−1},

∂_{i}^{∘}φ^{−1}(b) = φ^{−1}(Υ∂_{i}^{∘}b) = φ^{−1}(Υf_{n−1}^{∘}x_{i}) = f_{n−1}^{∘}φ^{−1}(x_{i}).

Since f is Kan, we get a regular epimorphism p_{1} : Y_{1} [figure]Φ(X) and a map a_{1} : Y_{1}

[figure]A_{n} such that f_{n}^{∘}a_{1} = φ^{−1}(b)^{∘}p_{1} and ∂_{i}^{∘}a_{1} = φ^{−1}(x_{i})^{∘}p_{1}, for i ≠ k. Now consider the pullback square

[figure: Y; p; z; X; ζ_{Φ(X)}; Υ(Y_{1}); Υp_{1}; ΥΦ(X).]

Because Set is regular, p is a regular epimorphism. Put a = Υa_{1}^{∘}z : Y [figure]Υ(A_{n}); then p and a are the required maps:

Υf_{n}^{∘}a = Υf_{n}^{∘}Υa_{1}^{∘}z
= Υ(f_{n}^{∘}a_{1})^{∘}z
= Υ(φ^{−1}(b)^{∘}p_{1})^{∘}z
= Υφ^{−1}(b)^{∘}Υp_{1}^{∘}z
= Υφ^{−1}(b)^{∘}ζ_{Φ(X)}^{∘}p
= φφ^{−1}(b)^{∘}p
= b^{∘}p

and, similarly, Υ∂_{i}^{∘}a = Υx_{i}^{∘}p, for all i ≠ k.

We recall some basic definitions and observations from [14]. In a category with finite

limits, a relation R : A  [figure]B from A to B is a subobject (d_{0}, d_{1}) : R  [figure]A × B. If a
map (f, g) : X  [figure]A × B factorizes through (d_{0}, d_{1}), then the map h : X  [figure]R with

(f, g) = (d_{0}, d_{1})^{∘}h is necessarily unique; we will denote the situation by g(R)f. SR : A

[figure]C denotes the composition of a relation R : A  [figure]B with a relation S : B  [figure]C.

4.3. Proposition. [14, Proposition 2.1] Let A be a regular category.

1. A map b : X  [figure]B factorizes through the image of a map f : A  [figure]B if and
only if there is a regular epimorphism p : Y  [figure]X and a map a : Y  [figure]A with

b^{∘}p = f^{∘}a;

2. given relations R : A  [figure]B and S : B  [figure]C and maps a : X  [figure]A and c : X

[figure]C, c(SR)a if and only if there is a regular epimorphism p : Y [figure]X and a map b : Y [figure]B with b(R)a^{∘}p and c^{∘}p(S)b.

Recall from [15] and [14] that a finitely complete category is called Mal’cev, if every reflexive relation is an equivalence relation. It follows from 2. that in a regular category, the composition of relations is associative. A regular category is Mal’cev if and only if the composition of equivalence relations is commutative [14], i.e. when for any two equivalence relations S and R on an object X the equality SR = RS holds. Every finitely complete protomodular (hence, a fortiori, every semi-abelian) category is Mal’cev [8]. In a regular Mal’cev category, the equivalence relations on a given object X constitute a lattice, the join of two equivalence relations being their composition.

Carboni, Kelly and Pedicchio extend the classical result of Moore that any simplicial group is Kan: in [14], it is proven that any simplicial object in a regular Mal’cev category is Kan. This property is even seen to characterize the Mal’cev categories among the regular ones. We will add that any regular epimorphism between simplicial objects in a regular Mal’cev category is a Kan fibration.

4.4. Proposition. [14, Theorem 4.2] Let A be a regular Mal’cev category. Then

1. every simplicial object K of A is Kan;

2. if f : A [figure]B is a regular epimorphism between simplicial objects of A, then it is a Kan fibration.

Proof. In our proof of 2. we will repeatedly use Proposition 4.3. For n ≥ 1 and k ∈ [n], let

x = (x_{i} : X [figure]A_{n−1})_{i∈[n],i≠k}

be an (n, k)-horn of A and let b : X [figure]B_{n} be a map with ∂_{i}^{∘}b = f_{n−1}^{∘}x_{i}, for i ∈ [n] and i ≠ k. Because f_{n} is regular epi, there is a regular epimorphism p_{1} : Y_{1} [figure]X and a map c : Y_{1} [figure]A_{n} with f_{n}^{∘}c = b^{∘}p_{1}. For i ∈ [n] and i ≠ k, put c_{i} = ∂_{i}^{∘}c : Y_{1} [figure]A_{n−1}. Let R[f] denote the kernel relation of f. Now

f_{n−1}^{∘}c_{i} = f_{n−1}^{∘}∂_{i}^{∘}c = ∂_{i}^{∘}f_{n}^{∘}c = ∂_{i}^{∘}b^{∘}p_{1} = f_{n−1}^{∘}x_{i}^{∘}p_{1},

and, consequently, c_{i}(R[f_{n−1}])x_{i}^{∘}p_{1}. By the simplicial identities, this defines an (n, k)-horn

((c_{i}, x_{i}^{∘}p_{1}) : Y_{1} [figure]A_{n−1})_{i∈[n],i≠k}

of R[f]. This simplicial object being Kan yields a regular epimorphism p_{2} : Y_{2} [figure]Y_{1} and a (d, e) : Y_{2} [figure]R[f]_{n} such that ∂_{i}^{∘}d = c_{i}^{∘}p_{2} and ∂_{i}^{∘}e = x_{i}^{∘}p_{1}^{∘}p_{2}, for i ∈ [n] and i ≠ k, and f_{n}^{∘}d = f_{n}^{∘}e. Thus, c^{∘}p_{2}(D)d, where D is the equivalence relation ⋀_{i∈[n],i≠k} D_{i} and D_{i} is the kernel relation of ∂_{i} : A_{n} [figure]A_{n−1}. It follows that c^{∘}p_{2}(DR[f_{n}])e. By the Mal’cev property, DR[f_{n}] is equal to R[f_{n}]D. This, in turn, implies that there exists a regular epimorphism p_{3} : Y [figure]Y_{2} and a map a : Y [figure]A_{n} such that a(D)e^{∘}p_{3} and c^{∘}p_{2}^{∘}p_{3}(R[f_{n}])a. The required maps are now a and p = p_{1}^{∘}p_{2}^{∘}p_{3} : Y [figure]X: indeed, f_{n}^{∘}a = f_{n}^{∘}c^{∘}p_{2}^{∘}p_{3} = b^{∘}p and ∂_{i}^{∘}a = ∂_{i}^{∘}e^{∘}p_{3} = x_{i}^{∘}p.

5. Implications of the Kan condition

In this section we consider two important implications of Proposition 4.4. First we show that for a simplicial object A of A, being Kan implies that H_{n}A is abelian (n ≥ 1). Next we prove that N : SA [figure]PChA is an exact functor. This is an implication of the fact that every regular epimorphism between simplicial objects is a Kan fibration. Finally, using the exactness of N, we prove that every short exact sequence of simplicial objects induces a long exact homology sequence (Corollary 5.7). We obtain it as an immediate consequence of Proposition 2.4.

Let A be a Kan simplicial object in a regular category A and n ≥ 1. Recall from Remark 3.3 the notation Z_{n}A = K[d_{n}] = ⋂_{i=0}^{n} K[∂_{i}]. We write z_{n} : Z_{n}A [figure]A_{n} for the inclusion ⋂_{i=0}^{n} Ker ∂_{i}. Basing ourselves on Weibel [31], we say that morphisms x : X

[figure]Z_{n}A and x′ : X′ [figure]Z_{n}A are homotopic, and write x ∼ x′, if there is an arrow

y : Y  [figure]A_{n+1} (called a homotopy from x to x′) and regular epimorphisms p : Y  [figure]X

and p′ : Y [figure]X′ such that

∂_{i}^{∘}y = \begin{cases}0, & if i < n; \\ z_{n}^{∘}x^{∘}p, & if i = n; \\ z_{n}^{∘}x′^{∘}p′, & if i = n + 1. \end{cases}

5.1. Proposition. Let A be a Kan simplicial object in a regular category A. For n ≥ 1, ∼ defines an equivalence relation on the class of arrows x : X [figure]Z_{n}A in A with codomain Z_{n}A.

Proof. Due to to fact that A is Kan, the proof of 8.3.1 in Weibel [31] may be copied: one just considers arrows with codomain A_{n} instead of elements of A_{n} and reads “0” instead of “∗”.

In Borceux and Bourn [5] and Borceux [3] a notion of commutator is introduced which generalizes a definition by Huq [23]. It is based on a construction due to Bourn [6]. Other notions of commutator exist; we use this one because it allows us to take the commutator of subobjects which are not necessarily kernels.

5.2. Definition. [6, Proposition 1.9, 3, Definition 6.4] In a semi-abelian category A, let f : A [figure]C and g : B [figure]C be two morphisms with the same codomain. Then the arrow ψ, obtained by taking the colimit of the diagram of solid arrows

[figure: A × B; D; D; C,; A; A × B; (1_{A},0); A; D; A; C,; f; A × B; D; φ; A × B; B; (0,1_{B}); D; C,; ψ; D; B; C,; B; g]  (H)

is a regular epimorphism. The kernel of ψ is denoted [f, g] and called the commutator of f and g. An object A of A is called abelian if [1_{A}, 1_{A}] = 0.

Note that the above colimit exists, as a semi-abelian category has all finite limits. Recall, furthermore, that, in a semi-abelian category, all regular epimorphisms are cokernels, hence D = C/[f, g].

Proposition 9 of [9] states that an object is abelian if and only if it can be provided with the structure of an internal abelian group. The full subcategory A_{Ab} of all abelian objects is a Birkhoff subcategory of A [3, Theorem 7.1 and 7.2]. The component at an object A of A of the unit of the adjunction is given by ψ : A [figure]D = A/[1_{A}, 1_{A}].

5.3. Notation. In a regular category A, consider a monomorphism m : A_{1} [figure]A and a regular epimorphism p : A [figure]B. Taking the image factorization

[figure: A_{1}; m; pA_{1}; p(m); A; p; B]

of p^{∘}m yields a monomorphism p(m) : pA_{1} [figure]B called the direct image of m along p.

In our proof of Theorem 5.5 we need the following properties of the commutator [f, g].

5.4. Proposition. In a semi-abelian category A, let f : A  [figure]C and g : B  [figure]C

be two morphisms with the same codomain.

1. If f or g is 0 then [f, g] = 0.

2. For any regular epimorphism p : C [figure]C′, p[f, g] = [p^{∘}f, p^{∘}g]. This means that there exists a regular epimorphism \overline{p} such that the square

[figure: [f, g]; \overline{p}; [p^{∘}f, p^{∘}g]; C; p; C′]

commutes.

3. If k : A [figure]C is a kernel, then [k, k] factors over A, and A/[k, k] is an abelian object of A.

Proof. 1. is obvious. The rest of the proof is based on Huq [23, Proposition 4.1.4]. As in Definition 5.2, let ψ : C [figure]D and φ : A × B [figure]D, resp. ψ′ : C′ [figure]D′ and φ′ : A × B [figure]D′, denote the couniversal arrows obtained from the construction of [f, g] and [p^{∘}f, p^{∘}g]. Then

[figure: A × B; D′; D′; C,; A; A × B; (1_{A},0); A; D′; ψ′∘p∘f; A; C,; f; A × B; D′; φ′; A × B; B; (0,1_{B}); D′; C,; ψ′∘p; D′; B; ψ′∘p∘g; C,; B; g]

is a cocone on the diagram of solid arrows H. The couniversal property of colimits yields a unique map d : D [figure]D′. In the commutative diagram of solid arrows

[figure: 0; D; d; C; ψ; p; [f, g]; \overline{p}; Ker ψ; 0; 0; D′; C′; ψ′; [p^{∘}f, p^{∘}g]; Ker ψ′; 0,]

there exits a unique map \overline{p} : [f, g] [figure][p^{∘}f, p^{∘}g] such that the right hand square commutes.

For 2. we must show that if p is a regular epimorphism, then so is \overline{p}. To do so, we prove that k = Ker ψ′^{∘}Im \overline{p} is a kernel of ψ′. By the Non-Effective Trace of the 3 × 3 Lemma 3.5, k is a kernel; hence, it is sufficient that ψ′ be a cokernel of k.

Let z : C′ [figure]Z be a map such that z^{∘}k = 0. Then z^{∘}p^{∘}Ker ψ = 0, which yields a map y : D [figure]Z with y^{∘}ψ = z^{∘}p. We get the following cocone.

[figure: A × B; Z; Z; C′; A; A × B; (1_{A},0); A; Z; z∘p∘f; A; C′; p∘f; A × B; Z; y∘φ; A × B; B; (0,1_{B}); Z; C′; z; Z; B; z∘p∘g; C′; B; p∘g]

Thus we acquire an arrow x : D′ [figure]Z such that x^{∘}ψ′ = z.

[1_{A}, 1_{A}] is a subobject of [k, k]: take p = k and f = g = 1_{A} in the discussion above. Hence, using that A/[1_{A}, 1_{A}] is abelian and that A_{Ab} is closed under quotients (by definition of a Birkhoff subcategory), the first statement of 3. implies the second one. This first statement follows from the fact that

[figure: A × A; Cok[k]; Cok[k]; C,; A; A × A; (1_{A},0); A; Cok[k]; 0; A; C,; k; A × A; Cok[k]; 0; A × A; A; (0,1_{A}); Cok[k]; C,; Coker k; Cok[k]; A; 0; C,; A; k]

is a cocone. Thus a map may be found such that the right hand square in

[figure: 0; [k, k]; i; C; ψ; \frac{C}{[k,k]}; 0; 0; A; k; C; Coker k; Cok[k]; 0]

commutes, whence the map i.

The following was suggested to us by Dominique Bourn. A more conceptual proof of this theorem may be found in his forthcoming paper [12]. It is well-known to hold in case A is the category Gp of groups.

5.5. Theorem. Let A be a simplicial object in a semi-abelian category A. For all n ≥ 1, H_{n}A is an abelian object of A.

Proof. Let n ≥ 1. Consider the subobjects k_{n} : K_{n} = [z_{n}, z_{n}] [figure]A_{n} of A_{n} and

s_{n} : S_{n} = [σ_{n−1}^{∘}z_{n}, σ_{n}^{∘}z_{n}] [figure]A_{n+1}

of A_{n+1}. By the second statement of Proposition 5.4,

∂_{i}S_{n} = [∂_{i}^{∘}σ_{n−1}^{∘}z_{n}, ∂_{i}^{∘}σ_{n}^{∘}z_{n}],

for 0 ≤ i ≤ n+1. Hence, by the simplicial identities and the first statement of Proposition 5.4, ∂_{i}S_{n} = 0, for i ≠ n, and ∂_{n}S_{n} = [z_{n}, z_{n}] = K_{n}. This last equality means that there exists a regular epimorphism filling the square

[figure: S_{n}; s_{n}; K_{n}; k_{n}; A_{n+1}; ∂_{n}; A_{n}.]

By the third statement of Proposition 5.4, there is a map l_{n} : K_{n} [figure]Z_{n}A such that z_{n}^{∘}l_{n} = k_{n}. It follows that l_{n} ∼ 0. Now, by Proposition 4.4, A is Kan; hence, Proposition 5.1 implies that 0 ∼ l_{n}. Thus there exists a morphism x : X [figure]A_{n+1} and a regular epimorphism p : X [figure]K_{n} such that ∂_{n+1}^{∘}x = z_{n}^{∘}l_{n}^{∘}p, i.e. the outer rectangle in

[figure: X; x; p; q; A_{n+1}; ∂_{n+1}; (I); N_{n+1}A; ⋂_{i=1}^{n} Ker ∂_{i}; d′_{n+1}; K_{n}; l_{n}; Z_{n}A; z_{n}; A_{n}]

commutes, and such that, moreover, ∂_{i}^{∘}x = 0, for i ≠ n + 1. It follows that an arrow q exists such that, in the diagram above, the triangle commutes. As z_{n} is a monomorphism, we get the commutativity of the trapezium (I). Now, in the diagram with exact rows

[figure: X; (I); q; p; K_{n}; l_{n}; Z_{n}A; \frac{Z_{n}A}{K_{n}}; r; 0; N_{n+1}A; d′_{n+1}; Z_{n}A; H_{n}A; 0,]

an arrow r exists such that the right hand square commutes. Indeed, as p is an epimorphism, Coker (l_{n}^{∘}p) = Coker l_{n}. This map r is a regular epimorphism. Since, by Proposition 5.4, Z_{n}A/K_{n} = Z_{n}A/[z_{n}, z_{n}] is an abelian object, and since a quotient of an abelian object of A is abelian, so is the object H_{n}A.

Recall from Remark 3.4 that N preserves kernels. If the category A is regular Mal’cev, Proposition 4.4 implies that N preserves regular epimorphisms. If, moreover, A is protomodular, all regular epimorphisms are cokernels, and we obtain

5.6. Proposition. If A is a pointed, regular and protomodular category, then the functor N : SA [figure]PChA is exact.

Proof. We prove that N preserves regular epimorphisms. Therefore, let f : A [figure]B be a regular epimorphism between simplicial objects of A. Then N_{0}f = f_{0} is regular epi, as well as N_{n}f = 0 : 0 [figure]0, for n < 0. For n ≥ 1, let b : X [figure]N_{n}B be a map. We are to show—see Proposition 4.3—that there is a regular epimorphism p : Y [figure]X and a map a : Y [figure]N_{n}A with b^{∘}p = N_{n}f^{∘}a. Now (x_{i} = 0 : X [figure]A_{n−1})_{i∈[n−1]} is an (n, n)-horn of A with ∂_{i}^{∘} ⋂_{i∈[n−1]} Ker ∂_{i}^{∘}b = 0 = f_{n−1}^{∘}x_{i}, for i ∈ [n − 1]. By Proposition 4.4, f is a Kan fibration, which implies that there is a regular epimorphism p : Y [figure]X and a map a′ : Y [figure]A_{n} such that f_{n}^{∘}a′ = ⋂_{i∈[n−1]} Ker ∂_{i}^{∘}b^{∘}p and ∂_{i}^{∘}a′ = x_{i}^{∘}p. Then the unique factorization a : Y [figure]N_{n}A of a′ over ⋂_{i∈[n−1]} Ker ∂_{i} : N_{n}A [figure]A_{n} is the required map.

Proposition 2.4 may now immediately be extended to simplicial objects.

5.7. Corollary. Let A be a pointed, exact and protomodular category. Any short exact sequence of simplicial objects

[figure: 0; A′′; A′; A; 0]

gives rise to a long exact sequence of homology objects

\xymatrix{… \ar@{=>}[r] & H_{n+1}A \ar@{=>}[r]^{δ_{n+1}} & H_{n}A′′ \ar@{=>}[r] & H_{n}A′ \ar@{=>}[r] & H_{n}A \ar@{=>}[r]^{δ_{n}} & H_{n−1}A′′ \ar@{=>}[r] & … \\ … \ar@{=>}[r] & H_{1}A \ar@{=>}[r]_{δ_{1}} & H_{0}A′′ \ar@{=>}[r] & H_{0}A′ \ar@{|->}[r] & H_{0}A \ar@{=>}[r]_{δ_{0}} & 0}

which depends naturally on the given short exact sequence of simplicial objects. If A is semi-abelian then, except for the lowest three terms H_{0}A′′, H_{0}A′ and H_{0}A, all of its terms are abelian objects of A.

Although we shall not use it, we think it worth mentioning that this result can be formulated in terms of homological δ-functors. By a (universal) homological δ-functor between pointed, exact and protomodular categories A and B we mean a collection of functors (T_{n} : A [figure]B)_{n∈ℕ} which preserve binary products, together with a collection of connecting morphisms (δ_{n})_{n∈ℕ} as in [31, Definition 2.1.1 and 2.1.4].

5.8. Notation. For any simplicial object A in A, let us denote by A^{−} the simplicial object defined by A_{n}^{−} = A_{n+1}, ∂_{i}^{−} = ∂_{i} : A_{n+1} [figure]A_{n}, and σ_{i}^{−} = σ_{i} : A_{n+1} [figure]A_{n+2}, for i ∈ [n], n ∈ ℕ. This is the simplicial object obtained from A by leaving out A_{0} and, for n ∈ ℕ, all ∂_{n} : A_{n} [figure]A_{n−1} and σ_{n} : A_{n} [figure]A_{n+1}. Observe that ∂ = (∂_{n+1})_{n} defines a simplicial morphism from A^{−} to A. Furthermore, remark that the augmented simplicial

object ∂_{0} : A^{−}  [figure]A_{0} is contractible—for n ≥ −1, put f_{n} = σ_{n+1} : A_{n+1}  [figure]A_{n+2}.

5.9. Proposition. Let A be a pointed, exact and protomodular category. The sequence of functors (H_{n} : SA [figure]A)_{n∈ℕ}, together with the connecting morphisms (δ_{n})_{n∈ℕ}, form a universal homological δ-functor.

Proof. To prove that a functor H_{n} : SA [figure]A preserves binary products, it suffices that, for proper complexes C and C′ in A, H_{n}(C ×C′) = H_{n}C ×H_{n}C′. One shows this by using that for f and f′ proper, Coker (f × f′) = Coker f × Coker f′. This follows from the fact that in any regular category, a product of two regular epimorphisms is regular epi. The universality is proven by modifying the proof of Theorem 2.4.7 in [31], replacing, for a simplicial object A, the projective object P by A^{−}.

6. Cotriple homology and Hopf’s formula

The aim of this section is to prove our generalized Hopf formula (Theorem 6.9) and our version of the Stallings-Stammbach sequence (Corollary 6.10).

We recall from Barr and Beck [1] (or Weibel [31]) the definition of cotriple homology, and slightly generalize it to categories that are pointed, exact and protomodular. Let A be an arbitrary category. A comonad 𝔾 on A will be denoted by

𝔾 = (G : A  [figure]A, ϵ : G  [figure] 1_{A}, δ : G  [figure] G^{2}).

For any object X of A, recall that the axioms of comonad state that ϵ_{GX}^{∘}δ_{X} = Gϵ_{X}^{∘}δ_{X} = 1_{GX} and δ_{GX}^{∘}δ_{X} = Gδ_{X}^{∘}δ_{X}. Putting

∂_{i} = G^{i}ϵ_{G^{n−i}X} : G^{n+1}X  [figure]G^{n}X  and σ_{i} = G^{i}δ_{G^{n−i}X} : G^{n+1}X  [figure]G^{n+2}X,

for 0 ≤ i ≤ n, makes the sequence (G^{n+1}X)_{n∈ℕ} a simplicial object 𝔾X of A. This induces a functor A [figure]SA, which, when confusion is unlikely, will be denoted by 𝔾.

6.1. Definition. Let 𝔾 be a comonad on a category A. Let B be a pointed, exact and protomodular category and U : A [figure]B a functor. We say that the object

H_{n}(X, U)_{𝔾} = H_{n−1}NU(𝔾X)

is the n-th homology object of X with coefficients in U relative to the cotriple 𝔾. This defines a functor H_{n}(·, U)_{𝔾} : A [figure]B, for any n ∈ ℕ_{0}.

We now make the following assumptions:

1. U is the reflector of a semi-abelian category A onto a Birkhoff subcategory B (which is itself semi-abelian—see [18]);

2. A is monadic over Set;

3. 𝔾 = (G : A  [figure]A, ϵ : G  [figure] 1_{A}, δ : G  [figure] G^{2}) is the resulting comonad on A.
Let Υ : A  [figure]Set and Φ : Set  [figure]A denote the respective right and left adjoint
functors and ϵ : Φ^{∘}Υ  [figure] 1_{A} and ζ : 1_{Set}  [figure] Υ^{∘}Φ the counit and unit. Then G = Φ^{∘}Υ, ϵ

is just the counit and δ is the natural transformation defined by δ_{X} = Φζ_{Υ(X)}, for X ∈ A.

6.2. Remark. The requirement that A be monadic over Set implies that A is complete, cocomplete and exact (see e.g. Borceux [4, Theorem II.4.3.5]); hence, if A is, moreover, pointed and protomodular, it is semi-abelian.

Reciprocally, any variety of algebras over Set is monadic—see e.g. Cohn [17], Borceux [4] or Mac Lane [27]—and thus semi-abelian varieties form an example of the situation considered. A characterisation of such varieties of algebras over Set is given by Bourn and Janelidze in their paper [13]. More generally, in [19], Gran and Rosický characterize semi-abelian categories, monadic over Set.

6.3. Remark. Note that, by Beck’s Theorem, the monadicity of Υ implies that for any object X of A, the diagram

\xymatrix{G^{2}X \ar@{=>}[r]^{Gϵ_{X}} \ar@{=>}[r]_{ϵ_{GX}} & GX \ar@{=>}[r]^{ϵ_{X}} & X}  (I)

is a coequalizer (see, for instance, the proof of Theorem VI.7.1 in [27], or Lemma II.4.3.3 in [4]).

6.4. Remark. Because the forgetful functor Υ : A [figure]Set preserves regular epimorphisms (again Borceux [4, Theorem II.4.3.5]) and because, in Set, every object is projective, A is easily seen to have enough projectives. In particular, any GX is projective. Accordingly, for any object X of A, the map ϵ_{X} : GX [figure]X is a projective presentation, the “𝔾-free” presentation of X.

Recalling Corollary 4.7 in [18], we get that ΔV (GX) = 0.

The following characterization of H_{1}(X, U)_{𝔾} is an immediate consequence of Proposition 3.9.

6.5. Proposition. For any object X of A, H_{1}(X, U)_{𝔾} ≅ U(X).

Proof. On one hand, H_{1}(X, U)_{𝔾} = H_{0}NU(𝔾X) is a cokernel of UGϵ_{X}^{∘}Ker Uϵ_{GX}. On the other hand, I is a coequalizer diagram, and U preserves coequalizers. Since, moreover, the map Uδ_{X} : UGX [figure]UG^{2}X is a splitting for both UGϵ_{X} and Uϵ_{GX}, Proposition 3.9 applies, and U(X) is a cokernel of UGϵ_{X}^{∘}Ker Uϵ_{GX} as well.

In the proof of Theorem 6.9, we will need some basic facts concerning the Quillen model structure (SSet, fib, cof, we) on the category of simplicial sets and simplicial maps. For a complete description of this model category and its properties we refer the reader to Quillen [28] and Hovey [22]. We shall, however, only use the following. fib, cof and we are classes of morphisms of SSet, respectively called fibrations, cofibrations and weak equivalences, subject to certain axioms. Any simplicial set X is cofibrant, meaning that the unique map ∅ [figure]X from the initial object to X is in cof. Dually, a simplicial set X is fibrant, meaning that the unique map X [figure]∗ from X to the terminal object is in fib, if and only if it is Kan. More generally, a simplicial map is a fibration precisely when it is a Kan fibration. Any contractible augmented simplicial set ϵ : A [figure]A_{−1} gives rise to a weak equivalence \overline{ϵ} : A [figure]\overline{A_{−1}}. Here \overline{A_{−1}} denotes the constant functor Δ^{op}

[figure]Set mapping every object of Δ^{op} to A_{−1} and every morphism to 1_{A_{−1}}. Conversely, if \overline{ϵ} : A [figure]\overline{A_{−1}} is a weak equivalence, then ϵ : A [figure]A_{−1} is contractible. The category of simplicial objects in the category Set_{∗} of pointed sets and basepoint-preserving maps has a model structure, induced by the one on SSet, as follows: a pointed simplicial map is a fibration, cofibration or weak equivalence if and only if it is in SSet. Finally, we need the following

6.6. Lemma. [28, Proposition I.3.5] Let (C, fib, cof, we) be a pointed model category. If, in the diagram

[figure: K[p]; γ; Ker p; E; β; p; B; α; K[p′]; Ker p′; E′; p′; B′]

of C, every object is both fibrant and cofibrant, p and p′ are fibrations, and α and β are weak equivalences, then the induced map γ is a weak equivalence.

Keeping in mind that a functor category Fun(C, A) has the limits and colimits of A, computed pointwise, the following follows immediately from the definitions.

6.7. Lemma. Let C be a small category, A a semi-abelian category and U : A [figure]B a reflector onto a Birkhoff subcategory B of A. Then

1. the functor category Fun(C, A) is semi-abelian;

2. Fun(C, PrA) = PrFun(C, A);

3. Fun(C, B) is a Birkhoff subcategory of Fun(C, A); 4. the functor Fun(C, U) = U^{∘}(·) : Fun(C, A) [figure]Fun(C, B) is its reflector;

5. V_{1}^{Fun(C,B)} = Fun(C, V_{1}^{B}) : Fun(C, PrA) [figure]Fun(C, A), where V^{Fun(C,B)} and V^{B} are the Birkhoff subfunctors associated with Fun(C, B) and B, respectively.

These properties hold, in particular, when C is Δ^{op}, i.e. when all functor categories are categories of simplicial objects and simplicial maps.

6.8. Remark. By the way of obtaining the sequence A from the given short exact sequence—applying the Snake Lemma, see [18]—we get that the square in

[figure: 0; V_{1}f; K; \frac{K}{V_{1}f}; 0; A; η_{A}; U(A)]

is commutative.

6.9. Theorem. For any object X of A, H_{2}(X, U)_{𝔾} ≅ ΔV (X).

Proof. By Remark 6.4, the tail of the exact sequence A, induced by the short exact sequence

0  [figure]K[ϵ_{X}] [figure: Ker ϵ_{X}]GX [figure: ϵ_{X}]X  [figure]0,

becomes

[figure: 0; ΔV (X); \frac{K[ϵ_{X}]}{V_{1}ϵ_{X}}; ψ; UGX.]

Keeping in mind Lemma 6.7, note that U𝔾ϵ_{X} : U𝔾GX [figure]U𝔾X is a presentation in SA. By Remark 6.4, ΔV 𝔾X is zero: it can be computed pointwise and every (𝔾X)_{n} is projective. Hence, Theorem 1.2 induces the exact sequence

0  [figure]\frac{K[𝔾ϵ_{X}]}{V_{1}𝔾ϵ_{X}} [figure: Ker U𝔾ϵ_{X}]U𝔾GX [figure: U𝔾ϵ_{X}]U𝔾X  [figure]0

of simplicial objects in B. Now Corollary 5.7 implies that

[figure: 0; H_{2}(X, U)_{𝔾}; H_{0} \frac{K[𝔾ϵ_{X}]}{V_{1}𝔾ϵ_{X}}; φ; UGX]  (J)

is an exact sequence in B. Indeed, recalling the notation from 5.8, as 𝔾GX = (𝔾X)^{−}, ϵ_{GX} : 𝔾GX [figure]GX is a contractible augmented simplicial object. Proposition 3.11 implies that H_{1}U𝔾GX = 0 and H_{0}U𝔾GX = UGX.

Accordingly, ΔV (X) is a kernel of ψ and H_{2}(X, U)_{𝔾} is a kernel of φ. We prove that ψ and φ are equal.

Consider the following diagram, where the κ_{i} are the face operators of K[𝔾ϵ_{X}] and κ is defined as K(ϵ_{GX}), as pictured in diagram L below.

[figure: 0; V_{1}G^{2}ϵ_{X}; V_{1}(Gϵ_{GX},Gϵ_{X}); V_{1}(ϵ_{G^{2}X},ϵ_{GX}); K[G^{2}ϵ_{X}]; κ_{1}; κ_{0}; \frac{K[G^{2}ϵ_{X}]}{V_{1}G^{2}ϵ_{X}}; \frac{κ_{1}}{V_{1}(Gϵ_{GX} ,Gϵ_{X})}; \frac{κ_{0}}{V_{1}(ϵ_{G^{2}X} ,ϵ_{GX})}; 0; 0; V_{1}Gϵ_{X}; V_{1}(ϵ_{GX},ϵ_{X}); K[Gϵ_{X}]; p; (I); κ; \frac{K[Gϵ_{X}]}{V_{1}Gϵ_{X}}; \frac{κ}{V_{1}(ϵ_{GX} ,ϵ_{X})}; 0; 0; V_{1}ϵ_{X}; K[ϵ_{X}]; q; \frac{K[ϵ_{X}]}{V_{1}ϵ_{X}}; 0]  (K)

We claim that, if the middle fork in K is a coequalizer diagram, then φ equals ψ. Indeed, by Remark 6.3 and Proposition 3.9, the square

\xymatrix{G^{2}X \ar@{|->}[r]^{Gϵ_{X}} \ar@{|->}[d]_{ϵ_{GX}} & GX \ar@{|->}[d]^{ϵ_{X}} \\ GX \ar@{|->}[r]_{ϵ_{X}} & X}

is a pushout; Proposition 5.1 and 5.2 of [18] now imply that V_{1}(ϵ_{GX}, ϵ_{X}) is a regular epimorphism. We get that (I) is a pushout. Hence, if the middle fork is a coequalizer diagram, so is the right fork, and then Corollary 3.10 implies that

\frac{K[ϵ_{X}]}{V_{1}ϵ_{X}} = H_{0} \frac{K[𝔾ϵ_{X}]}{V_{1}𝔾ϵ_{X}}.

Consequently, using the functoriality of H_{0}, we get that φ is the unique morphism such that the right hand square (III) in the diagram

[figure: K[Gϵ_{X}]; (II); p; Ker Gϵ_{X}; \frac{K[Gϵ_{X}]}{V_{1}Gϵ_{X}}; (III); \frac{κ}{V_{1}(ϵ_{GX} ,ϵ_{X})}; Ker UGϵ_{X}; \frac{K[ϵ_{X}]}{V_{1}ϵ_{X}}; φ; G^{2}X; η_{G^{2}X}; UG^{2}X; Uϵ_{GX}; UGX]

commutes. By Remark 6.8, also the square (II) is commutative. Further note—again using Remark 6.8—that ψ is unique in making

[figure: K[ϵ_{X}]; Ker ϵ_{X}; q; \frac{K[ϵ_{X}]}{V_{1}ϵ_{X}}; ψ; GX; η_{GX}; UGX]

commute. Now

φ^{∘} \frac{κ}{V_{1}(ϵ_{GX},ϵ_{X})}^{∘}p = Uϵ_{GX}^{∘}η_{G^{2}X}^{∘}Ker Gϵ_{X}
= η_{GX}^{∘}ϵ_{GX}^{∘}Ker Gϵ_{X}
= η_{GX}^{∘}Ker ϵ_{X}^{∘}κ
= ψ^{∘}q^{∘}κ
= ψ^{∘} \frac{κ}{V_{1}(ϵ_{GX},ϵ_{X})}^{∘}p,

where the second equality follows from the naturality of η, and the third one holds by definition of κ. This proves our claim that φ = ψ.

To see that the middle fork in diagram K is indeed a coequalizer diagram, note that it is defined by the exactness of the rows in the diagram

[figure: 0; K[𝔾ϵ_{X}]; κ; 𝔾GX; 𝔾ϵ_{X}; ϵ_{GX}; 𝔾X; ϵ_{X}; 0; 0; K[ϵ_{X}]; GX; ϵ_{X}; X; 0.]  (L)

(Read it as a diagram in A.) Since A is a pointed category, we get a diagram of pointed simplicial sets

[figure: 0; ΥK[𝔾ϵ_{X}]; Υ\overline{κ}; Υ𝔾GX; Υ𝔾ϵ_{X}; Υ\overline{ϵ_{GX}}; Υ𝔾X; Υ\overline{ϵ_{X}}; 0; Υ\overline{K[ϵ_{X}]}; Υ\overline{GX}; Υϵ_{X}; Υ\overline{X}.]  (M)

Indeed, as a right adjoint functor, Υ preserves limits, so Υ(0) is a terminal object ∗ of Set. Moreover, the rows of Diagram M are exact. Considered as unpointed augmented simplicial sets, Υϵ_{X} : Υ𝔾X [figure]ΥX and Υϵ_{GX} : Υ𝔾GX [figure]ΥGX are contractible: e.g. for the first one, a contraction is defined by f_{n} = ζ_{ΥG^{n+1}X}, where ζ : 1_{A} [figure] Φ^{∘}Υ is the unit of the adjunction Set ⇀ A. (This is Proposition 5.3 of Barr and Beck [1].) It follows that Υ\overline{ϵ_{X}} and Υ\overline{ϵ_{GX}} are weak equivalences of SSet and, consequently, also of SSet_{∗}. Furthermore, Proposition 4.4 and Proposition 4.2 imply that Υ𝔾ϵ_{X} and Υϵ_{X} are fibrations between fibrant objects. Now Lemma 6.6 applies, and Υ\overline{κ} is a weak equivalence. But this means that Υκ : ΥK[𝔾ϵ_{X}] [figure]ΥK[ϵ_{X}] is a contractible augmented simplicial object, and then Beck’s Theorem implies (see, again, the proof of Theorem VI.7.1 in Mac Lane [27] or Lemma II.4.3.3 in Borceux [4]) that the middle fork in K is a coequalizer.

Combining Theorems 1.2 and 6.9 yields the following generalization of the Stallings-Stammbach sequence.

6.10. Corollary. Let

[figure: 0; K; A; f; B; 0]

be a short exact sequence in A. There exists an exact sequence

H_{2}(A, U)_{𝔾} [figure: H_{2}(f,U)_{𝔾}]H_{2}(B, U)_{𝔾}  [figure]\frac{K}{V_{1}f}  [figure]H_{1}(A, U)_{𝔾} [figure: H_{1}(f,U)_{𝔾}]H_{1}(B, U)_{𝔾}  [figure]0

in B which depends naturally on the given short exact sequence.

6.11. Example. [Crossed modules] Recall that a crossed module (T, G, ∂) is a group homomorphism ∂ : T [figure]G together with an action of G on T (mapping a couple (g, t) ∈ G × T to ^{g}t ∈ T) satisfying

1. ∂(^{g}t) = g∂tg^{−1}, for all g ∈ G, t ∈ T;

2. ^{∂t}s = tst^{−1}, for all s, t ∈ T.

A morphism of crossed modules (f, ϕ) : (T, G, ∂) [figure](T′, G′, ∂′) is a pair of group

homomorphisms f : T  [figure]T′, ϕ : G  [figure]G′ with

1. ∂′^{∘}f = ϕ^{∘}∂;

2. f(^{g}t) =^{ϕ(g)} f(t), for all g ∈ G, t ∈ T.

It is well known that CM is equivalent to a variety of Ω-groups, namely to the variety of 1-categorical groups (see Loday [26]). Hence, it is semi-abelian [25]. Moreover, under this equivalence, a crossed module (T, G, ∂) corresponds to the semidirect product G ⋉ T equipped with the two appropriate homomorphisms. G ⋉ T and G × T have the same underlying set; thus the forgetful functor U : CM [figure]Set sends a crossed module (T, G, ∂) to the product T ×G of its underlying sets. This determines a comonad 𝔾 on CM. We get the cotriple homology of crossed modules described by Carrasco, Cegarra and Grandjeán in [16] as a particular case of Definition 6.1 by putting U the usual abelianization functor ab : CM [figure]CM_{Ab} (as defined for Ω-groups, see Higgins [20]).

In [16], Carrasco, Cegarra and Grandjeán give an explicit proof that

[(T, G, ∂), (T, G, ∂)] = V (T, G, ∂)

equals ([T, G], [G, G], ∂). For any crossed module (T, G, ∂), and any two normal subgroups K ◁ G, S ◁ T, let [K, S] denote the (normal) subgroup of T generated by the elements (^{k}s)s^{−1}, for k ∈ K, s ∈ S. It may be shown that, for (N, R, ∂) ◁ (Q, F, ∂),

[(N, R, ∂), (Q, F, ∂)] = V_{1}((Q, F, ∂) [figure]\frac{(Q,F,∂)}{(N,R,∂)})

is equal to ([R, Q][F, N], [R, F], ∂), but the proof involves somewhat fussy calculations, so will be omitted.

We recall the definition from [16]. Let (T, G, ∂) be a crossed module and n ≥ 1. The n-th homology object H_{n}(T, G, ∂) of (T, G, ∂) is

H_{n−1}C𝔾(T, G, ∂)_{ab}.

Here C𝔾(T, G, ∂)_{ab} is the unnormalized chain complex associated with 𝔾(T, G, ∂)_{ab}, defined by (C𝔾(T, G, ∂)_{ab})_{n} = (𝔾(T, G, ∂)_{ab})_{n} and

d_{n} = ∂_{0}^{ab} − ∂_{1}^{ab} + … + (−1)^{n}∂_{n}^{ab}.

𝔾(T, G, ∂)_{ab} denotes the simplicial abelian crossed module ab^{∘}𝔾(T, G, ∂). The ∂_{i}^{ab} are its face operators.

C𝔾(T, G, ∂)_{ab} need not be the same as N𝔾(T, G, ∂)_{ab}. However, their homology objects are equal, since 𝔾(T, G, ∂)_{ab} is a simplicial object in the abelian category CM_{Ab} (see, for instance, Weibel [31, Theorem 8.3.8]). Hence H_{n}(T, G, ∂) = H_{n}((T, G, ∂), ab)_{𝔾}.

The Hopf formula obtained in [16, Theorem 13 and above] is a particularization of our Theorem 6.9. In this particular case, the exact sequence of Corollary 6.10 becomes the exact sequence of [16, Theorem 12 (i)].

6.12. Example. [Groups] It is shown in [16, Theorem 10] that cotriple homology of crossed modules encompasses classical group homology; hence, so does our theory. If

[figure: 0; R; F; G; 0]

is a presentation of a group G by generators and relations, U : Gp [figure]Gp_{Ab} = Ab is the abelianization functor and 𝔾 the “free group on a set”-monad, the sequence in Corollary 6.10 becomes the Stallings-Stammbach sequence in integral homology of groups [29], [30]. The isomorphism in Theorem 6.9 is nothing but Hopf’s formula [21]

H_{2}(G, U)_{𝔾} ≅ \frac{R ∩ [F, F]}{[R, F]}.

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