Theory and Applications of Categories, Vol. 23, No. 8, 2010, pp. 150–169.

THE THIRD COHOMOLOGY GROUP CLASSIFIES DOUBLE CENTRAL EXTENSIONS

Dedicated to Dominique Bourn on the occasion of his sixtieth birthday

DIANA RODELO AND TIM VAN DER LINDEN

Abstract. We characterise the double central extensions in a semi-abelian category in terms of commutator conditions. We prove that the third cohomology group H^{3}(Z, A) of an object Z with coefficients in an abelian object A classifies the double central extensions of Z by A.

Introduction

The second cohomology group H^{2}(Z, A) of a group Z with coefficients in an abelian group A is well-known to classify the central extensions of Z by A in the following manner (see for instance [29]). A central extension f of Z by A can be described as a short exact sequence

[figure: 0; A; kerf; X; f; Z; 0]

such that axa^{−1}x^{−1} = 1 for all a ∈ A and x ∈ X. Two extensions f: X → Z and f′: X′ → Z are equivalent if and only if there exists a group (iso)morphism x: X → X′ satisfying f′^{∘}x = f and x^{∘}kerf = kerf′. The induced equivalence classes, together with the classical Baer sum, form an abelian group Centr^{1}(Z, A), and this group is isomorphic to H^{2}(Z, A).

In [19], see also [7] and [10], this construction was extended categorically from the context of groups to semi-abelian categories [24]. This includes familiar results for, say, Lie algebras over a field, commutative algebras, non-unital rings, or (pre)crossed modules.

The aim of the present work is to prove a two-dimensional version of this result, at once in a categorical context: we show that the third cohomology group H^{3}(Z, A) of an object Z with coefficients in an abelian object A of a semi-abelian category A classifies the double central extensions in A of Z by A. Thus the connections between two interpretations of H^{3}(Z, A) are made explicit.

Research supported by Centro de Matemática da Universidade de Coimbra, by Fundação para a Ciência e a Tecnologia (under grant number SFRH/BPD/38797/2007) and by Vrije Universiteit Brussel.

Received by the editors 2009-01-14 and, in revised form, 2010-01-20.

Published on 2010-02-03 in the Bourn Festschrift.

2000 Mathematics Subject Classification: 18G50, 18G60, 20J, 55N.

Key words and phrases: cohomology, categorical Galois theory, semi-abelian category, higher central extension, Baer sum.

\begin{matrix}c \\ ◯ \end{matrix} Diana Rodelo and Tim Van der Linden, 2010. Permission to copy for private use granted.

THE THIRD COHOMOLOGY GROUP CLASSIFIES DOUBLE CENTRAL EXTENSIONS 151

On one hand, there is the direction approach to cohomology established by Bourn [5, 6, 7] and further investigated in collaboration with Rodelo [11, 31]; here the cohomology groups H^{n}A of an internal abelian group A are described through direction functors, in such a way that any short exact sequence of internal abelian groups induces a long exact cohomology sequence. This concept of direction may be understood as follows. It is well-known that in a Barr exact context, H^{1}A can be interpreted in terms of Atorsors. An A-torsor is a generalised affine space over A, an “abelian group without zero” where any choice of zero gives back A—its direction. Further borrowing intuition from affine geometry, H^{1}A is described in terms of autonomous Mal’tsev operations with given direction A. On level 2—the level which corresponds to the “third cohomology group” from the title—the direction functor theory is based on that of level 1: now H^{2}A is described in terms of internal groupoids with given direction A. By means of higher order internal groupoids, the theory is inductively extended to higher levels H^{n}A.

On the other hand, there is the approach to semi-abelian homology [2, 14] based on categorical Galois theory [3, 20] initiated by Janelidze [21, 22] and further worked out by Everaert, Gran and Van der Linden [12]. Here the basic situation is given by a semi-abelian category A and a Birkhoff subcategory B of A; the derived functors of the reflector I: A → B are computed in terms of higher Hopf formulae using the induced Galois structures of higher central extensions. In the specific case where B is the Birkhoff subcategory AbA determined by the abelian objects in A and I = ab is the abelianisation functor, one actually begins with the Galois structure

Γ = ([figure: A; ab; ⊥; AbA; ⊃], |ExtA|, |ExtAbA|).  (A)

The class of extensions |ExtA| (respectively |ExtAbA|) consists of the regular epimorphisms in A (in AbA) and forms the class of objects of the category ExtA (or ExtAbA) whose morphisms are commutative squares between extensions. The coverings with respect to this Galois structure Γ are exactly the central extensions in the sense of commutator theory: an extension f: X → Z is central if and only if [R[f], ∇_{X}] = Δ_{X}, i.e., the commutator of the kernel pair of f with the largest relation ∇_{X} on X is the smallest relation Δ_{X} on X. These central extensions, in turn, determine a reflective subcategory CExtA of ExtA; the reflector centr: ExtA → CExtA which sends f to the central extension centrf: X/[R[f], ∇_{X}] → Z is the centralisation functor. Thus we obtain the Galois structure

Γ_{1} = ([figure: ExtA; centr; ⊥; CExtA; ⊃], |Ext^{2}A|, |ExtCExtA|).  (B)

The class |Ext^{2}A| consists of double extensions in A, which are defined as commutative squares

\xymatrix{X \ar@{=>}[r]^{c} \ar@{=>}[d]_{d} & C \ar@{=>}[d]^{g} \\ D \ar@{=>}[r]_{f} & Z}

such that the maps c, d, f, g and the comparison map (d, c): X → D ×_{Z} C to the pullback of f with g are regular epimorphisms. The elements of |ExtCExtA| are such double extensions, but with the extra condition that both d and g are central. The coverings with respect to the Galois structure Γ_{1} are used in the computation of the third homology functor H_{3}(−, ab): A → AbA (see [12]) and form the main subject of the present paper— they are the “double central extensions” from the title.

We start by recalling the main properties of the Galois structure Γ_{1} in Section 1. In Section 2 we characterise the Γ_{1}-coverings in terms of commutators (as Janelidze does in the category of groups [21] and Gran and Rossi do in the context of Mal’tsev varieties [18]) and in terms of internal pregroupoids in the sense of [27]. Section 3 recalls the definition of the third cohomology group in semi-abelian categories from [31]. We obtain a natural notion of direction for double extensions and show in Section 4 that the set Centr^{2}(Z, A) of equivalence classes of double central extensions of an object Z by an abelian object A carries a canonical abelian group structure. In Section 5 we conclude the paper with the isomorphism H^{3}(Z, A) ≅ Centr^{2}(Z, A) between the third cohomology group of an object Z with coefficients in an abelian object A and the group Centr^{2}(Z, A).

We conjecture that this result may be generalised to higher degrees, so that also for n > 2 the (n + 1)-st cohomology group H^{n+1}(Z, A) of Z with coefficients in A classifies the n-fold central extensions of Z by A. This will be the subject of future work.

Acknowledgements. Many thanks to Tomas Everaert and George Janelidze for important comments and suggestions on the text. Thanks also to Tomas for the present, more elegant, proof of Theorem 2.8.

1. Preliminaries

1.1. Semi-abelian categories. The basic context where we shall be working is that of semi-abelian categories [24]. Some examples are the categories Gp of all groups, Rng of non-unital rings, Lie_{𝕂} of Lie algebras over a field 𝕂, XMod of crossed modules, and Loop of loops. We briefly recall the main definitions.

A category is semi-abelian when it is pointed, Barr exact and Bourn protomodular and has binary coproducts. A category is pointed when it has a zero object 0, a terminal object which is also initial. A Barr exact category is regular—finitely complete with pullback-stable regular epimorphisms and coequalisers of kernel pairs—and such that every internal equivalence relation is a kernel pair [1]. When a category is pointed and regular, Bourn protomodularity can be defined via the regular Short Five Lemma [4]: given any commutative diagram of regular epimorphisms with their kernels

[figure: K[f]; kerf; k; X; x; f; Z; z; K[f′]; kerf′; X′; f′; Z′]

the morphisms k and z being isomorphisms implies that x is an isomorphism.

Any semi-abelian category A is a Mal’tsev category, i.e., it is finitely complete and in A every reflexive relation is an equivalence relation.

1.2. Internal pregroupoids. The concept of internal pregroupoid due to Kock [27] (called herdoid in [26]; see also [25]) generalises internal groupoids in the following manner: in a pregroupoid, the domain and codomain of a map may live in different objects, and no identities need to exist.

1.3. Definition. Let A be a finitely complete category. A pregroupoid (X, d, c, p) in A is a span

[figure: X; d; c; D; C]  (C)

with a partial ternary operation p on X satisfying

1. p(α, β, γ) is defined if and only if c(α) = c(β) and d(γ) = d(β);

2. dp(α, β, γ) = d(α) and cp(α, β, γ) = c(γ) if p(α, β, γ) is defined;

3. p(α, α, γ) = γ if p(α, α, γ) is defined, and p(α, γ, γ) = α if p(α, γ, γ) is defined;

4. p(α, β, p(γ, δ, ϵ)) = p(p(α, β, γ), δ, ϵ) if either side is defined.

We denote the category of (pre)groupoids in A by (Pre)GdA.

An “element” α of X should be interpreted as a map α: d(α) → c(α); its domain d(α) is an element of D, while its codomain c(α) is an element of C. The operation p sends a composable triple (α, β, γ)

\xymatrix{d(α) \ar@{=>}[r]^{α} \ar@{=>}[dr]_{δ} & c(α) \\ d(γ) \ar@{=>}[r]_{γ} \ar@{=>}[ur]^{β} & c(γ)}

to the dotted diagonal δ = p(α, β, γ): d(α) → c(γ).

When A is a Mal’tsev category, conditions (2) and (4) of Definition 1.3 are automatically satisfied, see Proposition 2.6.11 in [2] or Proposition 4.1 in [8]. Moreover, if it exists, a pregroupoid structure p on a span (X, d, c) is necessarily unique. In this case we shall say that the span (X, d, c) is a pregroupoid and drop the structure p from the notation. In case (X, d, c) is the underlying span of a reflexive graph, the given splitting i: C = D → X of d and c provides identities for the composition, so that the pregroupoid becomes a groupoid, and the composite p(α, β, γ) may be viewed as γ^{∘}β^{−1}^{∘}α.

1.4. The commutator of equivalence relations. Let R = (R, r_{0}, r_{1}) and S = (S, s_{0}, s_{1}) be equivalence relations on an object X of an exact Mal’tsev category A. Let R ×_{X} S denote the pullback of r_{1} and s_{0}.

[figure: R ×_{X} S; p_{R}; p_{S}; S; s_{0}; i_{S}; R; r_{1}; i_{R}; X]

The object R ×_{X} S “consists of” triples (α, β, γ) where αRβ and βSγ. We say that R and S centralise each other when there exists a connector between R and S: a morphism p: R ×_{X} S → X which satisfies p(α, α, γ) = γ and p(α, γ, γ) = α [8]; see also [2, Definition 2.6.1]. As explained in the introduction of [8], an internal pregroupoid structure p on a span (X, d, c) is the same thing as a connector between the kernel pairs R[c] and R[d] of c and d; so a connector between equivalence relations R and S is nothing but a pregroupoid structure on the induced span of coequalisers X/R ← X → X/S.

When A is a semi-abelian category, the commutator of R and S [30], denoted by [R, S], is the universal equivalence relation on X which, when divided out, makes them centralise each other. More precisely, [R, S] is the kernel pair R[ψ] of the map ψ in the diagram

[figure: R; i_{R}; r_{0}; R ×_{X} S; T; X; ψ; S; i_{S}; s_{1}]

where the dotted arrows denote the colimit of the outer square [2, Section 2.8]. The direct images ψR and ψS of R and S along the regular epimorphism ψ centralise each other; hence R and S centralise each other if and only if [R, S] = Δ_{X} [8, Proposition 4.2].

An equivalence relation R on an object X is central when R and ∇_{X} centralise each other—when [R, ∇_{X}] = Δ_{X}. An object X is abelian when [∇_{X}, ∇_{X}] = Δ_{X}; this happens when X admits a (unique) internal abelian group structure.

1.5. Central extensions. The definition of central extensions for groups was extended to varieties of Ω-groups by Fröhlich and Lue [16, 28] and then to the categorical context by Janelidze and Kelly [23]. They have also been defined via commutator theory; as explained in [17], the two approaches are equivalent. We sketch the categorical definition.

Let A be a semi-abelian category. For any object X of A we may take the kernel of the X-component of the unit of the adjunction in A to obtain a short exact sequence

[figure: 0; ⟨X⟩; μ_{X}; X; η_{X}; abX; 0.]

Thus we acquire a functor ⟨−⟩: A → A and a natural transformation μ: ⟨−⟩ ⇒ 1_{A}.

1.6. Lemma. The functors ab and ⟨−⟩ preserve pullbacks of regular epimorphisms along split epimorphisms.

Proof. It is shown in [17] that the functor ab has this property. Since kernels commute with pullbacks, it follows that the functor ⟨−⟩ has the same property.

An extension f: X → Z is central (with respect to the Galois structure Γ in diagram A) if and only if either one of the projections p_{0} or p_{1} of its kernel pair (R[f], p_{0}, p_{1}) is a trivial extension, i.e., a pullback of an extension in AbA. It follows that f is central if and only if the right hand side square in the diagram

[figure: 0; ⟨R[f]⟩; ⟨p_{0}⟩; μ_{R[f]}; R[f]; p_{0}; η_{R[f]}; abR[f]; abp_{0}; 0; 0; ⟨X⟩; μ_{X}; X; η_{X}; abX; 0]

is a pullback or, equivalently, ⟨p_{0}⟩ is an isomorphism. Hence the kernel of ⟨p_{0}⟩, which is denoted by ⟨f⟩_{1}, is zero if and only if f is central. The object ⟨f⟩_{1} may be considered as a normal subobject of X through the composite μ_{X}^{∘}⟨p_{1}⟩^{∘}kerp_{0}, see Lemma 4.12 in [13]. Now the centralisation functor centr: ExtA → CExtA from the Galois structure Γ_{1} in diagram B takes the extension f: X → Z and maps it to the quotient centrf: X/⟨f⟩_{1} → Z of f: X → Z by the extension ⟨f⟩_{1} → 0.

Given an object Z and an abelian object A, a central extension of Z by A is a central extension f: X → Z with kernel K[f] = A. The group of isomorphism classes of central extensions of Z by A is denoted Centr^{1}(Z, A). Recall the following result from [19].

1.7. Proposition. If A is a semi-abelian category and Z is an object of A then the functor Centr^{1}(Z, −): AbA → Ab preserves finite products.

Proof. We shall only repeat the main point of the construction behind [19, Proposition 6.1]. Let a: A → B be a morphism of abelian objects in A and f: X → Z a central extension of Z by A. Let A ⊕ B denote the biproduct of A with B in AbA. The functor Centr^{1}(Z, −) maps the equivalence class of f to the equivalence class of the central extension f′ in the diagram with exact rows

[figure: 0; A ⊕ B; kerf×1_{B}; [a,1_{B}]; X × B; f∘pr_{X}; Z; 0; 0; B; X′; f′; Z; 0.]  (D)

The extension f′ is central as a quotient of the central extension f^{∘}pr_{X}.

1.8. Double central extensions. Let A be a semi-abelian category. Recall that a double extension (of an object Z) in A is a commutative square

\xymatrix{X \ar@{=>}[r]^{c} \ar@{=>}[d]_{d} & C \ar@{=>}[d]^{g} \\ D \ar@{=>}[r]_{f} & Z}  (E)

such that all its maps and the comparison map (d, c): X → D ×_{Z} C to the pullback of f with g are regular epimorphisms.

By definition, a double extension is central when it is a covering with respect to the Galois structure Γ_{1}. Hence the double extension E, considered as a map (c, f): d → g in the category ExtA, is central if and only if the first projection

[figure: R[c]; p_{0}; R[(c,f)]; X; d; R[f]; p_{0}; D; R[c]; p_{0}; X; R[c]/⟨R[(c, f)]⟩_{1}; X/⟨d⟩_{1}]

of its kernel pair—the left hand side square—is a trivial extension with respect to Γ_{1}. (Alternatively, one could use the square of second projections.) This means that the comparison map to its reflection into CExtA—the right hand side square—is a pullback. For this to happen, the natural map ⟨R[(c, f)]⟩_{1} → ⟨d⟩_{1} must be an isomorphism. This, in turn, is equivalent to the square

\xymatrix{⟨R[d]□R[c]⟩ \ar@{|->}[r]^{⟨p_{0}⟩} \ar@{|->}[d]_{⟨p_{0}⟩} & ⟨R[d]⟩ \ar@{|->}[d]^{⟨p_{0}⟩} \\ ⟨R[c]⟩ \ar@{|->}[r]_{⟨p_{0}⟩} & ⟨X⟩}

being a pullback, because ⟨R[(c, f)]⟩_{1} and ⟨d⟩_{1} are the kernels of the vertical maps above. Here (R[d]□R[c], p_{0}, p_{1}) denotes the kernel pair of R[(c, f)]; it consists of all quadruples (α, β, γ, δ) ∈ X^{4} in the configuration

(\begin{matrix}α & c & β \\ d & & d \\ δ & c & γ \end{matrix}) ,

called a diamond in [25]: d(α) = d(δ), c(α) = c(β), c(γ) = c(δ) and d(γ) = d(β).

2. Characterisation of double central extensions in terms of commutators

In this section we characterise the coverings with respect to the Galois structure Γ_{1} in terms of internal pregroupoids. This characterisation turns out to be equivalent to the conditions given by Janelidze in [21] and Gran and Rossi in [18]—and thus we prove a categorical version of the next result.

2.1. Proposition. [18, 21] Let A be a Mal’tsev variety. A double extension E in A is central if and only if [R[d], R[c]] = Δ_{X} = [R[d] ∩ R[c], ∇_{X}].

Double extensions may be characterised in terms of spans in a slice category as follows.

2.2. Definition. A span (X, d, c) in a regular category A

1. has global support when !_{D}: D → 1 and !_{C}: C → 1 are regular epimorphisms;

2. is aspherical when also (d, c): X → D × C is a regular epimorphism.

2.3. Proposition. Let A be a semi-abelian category. A commutative square E in A is a double extension if and only if (X, d, c) is an aspherical span in A ↓ Z.

2.4. Definition. Suppose that A is regular. A pregroupoid (X, d, c, p) has global support or is aspherical whenever the span (X, d, c) has global support or is aspherical. This definition applies in the obvious way to internal groupoids.

Because of Proposition 2.3, which exhibits the close connection between double extensions in A and spans in a slice category A ↓ Z, we are also mostly interested in pregroupoids in slice categories. For A Mal’tsev, asking that a span (X, d, c) is a pregroupoid in A ↓ Z amounts to asking that (X, d, c) is a pregroupoid in A. When A is semi-abelian, this happens precisely when the first equality [R[d], R[c]] = Δ_{X} of Proposition 2.1 holds.

2.5. Definition. Suppose that A is semi-abelian and let Z be an object of A. An aspherical (pre)groupoid (X, d, c) in A ↓ Z is central when (d, c): X → D ×_{Z} C is a central extension in A.

Since R[d] ∩ R[c] = R[(d, c): X → D ×_{Z} C], this makes the centrality of the aspherical pregroupoid (X, d, c) equivalent to the second equality [R[d] ∩ R[c], ∇_{X}] = Δ_{X} of Proposition 2.1. And thus we proved:

2.6. Proposition. Let A be a semi-abelian category. A double extension E in A satisfies

[R[d], R[c]] = Δ_{X} = [R[d] ∩ R[c], ∇_{X}]  (F)

if and only if the span (X, d, c) is a central pregroupoid in the slice category A ↓ Z.

2.7. Proposition. In a semi-abelian category, condition F is preserved and reflected by pullbacks of double extensions along double extensions.

Proof. The proof given in Section 4 of [18] in the context of Mal’tsev varieties is still valid in the present situation.

We obtain the next result, which is further worked out in [15] in the more general context of exact Mal’tsev categories.

2.8. Theorem. Consider a double extension E in a semi-abelian category A. The following are equivalent:

1. E is a double central extension;

2. (X, d, c) is a central pregroupoid in A ↓ Z;

3. [R[d], R[c]] = Δ_{X} = [R[d] ∩ R[c], ∇_{X}].

Proof. By Proposition 2.6 we already know that (2) and (3) are equivalent. To see that (1) implies (3), suppose that E is a double central extension. Then either one of the projections of its kernel pair is trivial with respect to Γ_{1}, meaning that it is a pullback of a double extension between central extensions (i.e., a morphism of the category CExtA). This latter double extension satisfies the condition corresponding to F; hence applying Proposition 2.7 twice shows that (3) holds.

Now we prove that (2) implies (1). The pregroupoid structure of (X, d, c) is a connector p: R[c] ×_{X} R[d] → X. As explained in Subsection 1.8, we are to show that the outer square in the diagram

[figure: ⟨R[d]□R[c]⟩; ⟨π⟩; ⟨p_{0}⟩; ⟨p_{0}⟩; ⟨R[d]⟩; ⟨p_{0}⟩; ⟨R[c] ×_{X} R[d]⟩; ⟨R[c]⟩; ⟨p_{0}⟩; ⟨X⟩]

is a pullback. Here π: R[d]□R[c] → R[c] ×_{X} R[d] is defined by

(\begin{matrix}α & c & β \\ d & & d \\ δ & c & γ \end{matrix}) ↦ (α, β, γ).

By Lemma 1.6 we know that the inner quadrangle is a pullback, hence it suffices that ⟨π⟩ is an isomorphism. The left hand side square

\xymatrix{R[d]□R[c] \ar@{=>}[r]^{π} \ar@{=>}[d]_{q} & R[c] ×_{X} R[d] \ar@{=>}[d]^{p} & ⟨R[d]□R[c]⟩ \ar@{=>}[r]^{⟨π⟩} \ar@{=>}[d]_{⟨q⟩} & ⟨R[c] ×_{X} R[d]⟩ \ar@{=>}[d]^{⟨p⟩} \\ R[d] ∩ R[c] \ar@{=>}[r]_{p_{0}} & X & ⟨R[d] ∩ R[c]⟩ \ar@{=>}[r]_{⟨p_{0}⟩} & ⟨X⟩,}

where q is defined by

(\begin{matrix}α & c & β \\ d & & d \\ δ & c & γ \end{matrix}) ↦ (p(α, β, γ), δ),

is a pullback. Since p_{0} is a split epimorphism we may again use Lemma 1.6 to show that also the right hand side square above is a pullback. It follows that ⟨π⟩ is an isomorphism if and only if ⟨p_{0}⟩ is an isomorphism, so that the internal pregroupoid (X, d, c) is central if and only if E is a double central extension.

3. The third cohomology group

In this section we translate the description of the second order direction functor and its associated cohomology groups, developed in [31] for Barr exact categories, to the context of semi-abelian categories. A similar translation was made in [31] for Moore categories (i.e., strongly protomodular semi-abelian categories) where the connection with n-fold crossed extensions is explored. In that context, Z-modules are simpler since, due to strong protomodularity, they are just split exact sequences whose kernel is an abelian object (compare with the definition of Z-module given below). We shall, however, not need this simplification; in fact it turns out that we may focus on trivial modules. Note that what we call the third cohomology group here is actually the second cohomology group in [31]; the dimension shift is there for historical reasons, in order to comply with the “non-abelian” numbering used in classical cohomology of groups. From now on, A will denote a semi-abelian category and Z a fixed object of A.

An aspherical (abelian) groupoid in A ↓ Z consists of a commutative diagram

[figure: X; d; c; f∘d=f∘c; Y; f; i; Z]  (G)

such that the top line is a groupoid in A, and both f and (d, c): X → R[f] are regular epimorphisms. Such an internal groupoid has an underlying double extension

\xymatrix{X \ar@{|->}[r]^{c} \ar@{|->}[d]_{d} & Y \ar@{|->}[d]^{f} \\ Y \ar@{|->}[r]_{f} & Z.}  (H)

We denote by AsphGd(A ↓ Z) the category of aspherical groupoids in A ↓ Z.

The category Mod_{Z}A of Z-modules is the category Ab(A ↓ Z) of abelian groups in A ↓ Z. So, a Z-module gives us a split exact sequence

[figure: 0; A; kerp; P; p; Z; s; 0]

where A is an abelian object and p is a split epimorphism (equipped with an additional structure making it an abelian group in A ↓ Z). Using the equivalence between split epimorphisms and internal actions [9], we can replace P with a semi-direct product Z ⋉ (A, ξ). For simplicity, we denote a Z-module just by its induced Z-algebra (A, ξ).

In the context of semi-abelian categories, the direction functor from [31, Definition 3.7] determines a functor d_{Z}: AsphGd(A ↓ Z) → Mod_{Z}A which maps an aspherical internal groupoid G to the Z-module d_{Z}(G) = (A, ξ) defined by the downward pullback/upward pushout

\xymatrix{R[(d, c)] \ar@{|->}[r] \ar@{|->}[d]^{p_{0}} & Z ⋉ (A, ξ) \ar@{|->}[d]^{p} \\ X \ar@{=>}[u]^{(1_{X},1_{X})} \ar@{|->}[r]_{f∘d} & Z. \ar@{=>}[u]^{s}}  (I)

More precisely, the pair (p, s): Z ⋉ (A, ξ) ⇄ Z arises as a pushout of (1_{X}, 1_{X}) along f^{∘}d but, using the properties of G, one may show that the square of downward arrows in I is a pullback [6]. Thus we see that A = K[p] = K[p_{0}] = K[(d, c)] = K[d] ∩ K[c].

3.1. Remark. Suppose we have a symmetric monoidal category (C, ⊗, E) such that the property

∀C ∈ C, ∃\overline{C} ∈ C: C ⊗ \overline{C} ∼ E  (J)

holds, where ∼ means “is connected to (by a zigzag)”. Then it is easy to check that the monoidal structure of C induces an abelian group structure on the set π_{0}C of its connected components (equivalence classes with respect to ∼). The addition is defined by \{C_{1}\} + \{C_{2}\} = \{C_{1} ⊗ C_{2}\}, the zero is \{E\} and −\{C\} = \{\overline{C}\}.

It is shown in [6] that the fibres of d_{Z} are symmetric monoidal categories with property J. The tensor product is called the Baer sum since it gives the Baer sum of (2-fold) extensions in the classical examples. So, for any Z-module (A, ξ), π_{0}d_{Z}^{−1}(A, ξ) is an abelian group.

3.2. Definition. [31] Let (A, ξ) be a Z-module. The third cohomology group

H^{3}(Z, (A, ξ))

of Z with coefficients in (A, ξ) is the abelian group π_{0}d_{Z}^{−1}(A, ξ) of equivalence classes of aspherical internal groupoids in A ↓ Z with direction (A, ξ). This defines an additive functor

H^{3}(Z, −): Mod_{Z}A → Ab.

We are especially interested in the case of trivial Z-modules (A, τ), i.e., abelian objects A with the trivial Z-action τ. In this situation we write H^{3}(Z, A) for H^{3}(Z, (A, τ)). The functor H^{3}(Z, −) restricts to an additive functor AbA → Ab.

3.3. Proposition. The direction of an aspherical groupoid G in A ↓ Z is a trivial Z-module (A, τ) in A if and only if G is a central groupoid.

Proof. Let us first suppose that d_{Z}(G) = (A, τ). Then, d_{Z}(G), defined by

(p, s): Z ⋉ (A, τ) ⇄ Z

in diagram I, is the product projection with its canonical inclusion

(pr_{Z}, (1_{Z}, 0)): Z × A ⇄ Z.

It follows that the pullback (p_{0}, (1_{X}, 1_{X})): R[(d, c)] ⇄ X is also a product projection with its canonical inclusion, namely (pr_{X}, (1_{X}, 0)): X × A ⇄ X. In particular, the splitting (1_{X}, 1_{X}) is a normal monomorphism in A, which by Theorem 5.2 in [8] (see also Corollary 6.1.8 in [2]) means that R[(d, c)] = R[d]∩R[c] is central. Hence [R[d]∩R[c], ∇_{X}] = Δ_{X} and the groupoid is central.

Conversely, suppose that G is a central groupoid in A ↓ Z. By the same arguments as above we see that (p_{0}, (1_{X}, 1_{X})) and hence (p, s) are product projections with their canonical inclusions. It follows that A has a trivial Z-action τ.

3.4. Corollary. Let G be an aspherical groupoid in A ↓ Z and let H be the corresponding double extension. Then H is a double central extension if and only if d_{Z}(G) is a trivial Z-module (A, τ) in A.

Thus we see that the direction of a central internal groupoid G is just the intersection A = K[d] ∩ K[c] of the kernels of d and c; indeed, this object A is always abelian as the kernel of the central extension (d, c). In view of this fact we can extend the concept of direction to double central extensions.

4. The group of equivalence classes of double central extensions

4.1. Definition. The direction of a double central extension E is the abelian object K[d] ∩ K[c]. This defines a functor D_{Z}: CExt_{Z}^{2}A → AbA, where CExt_{Z}^{2}A denotes the category of double central extensions of the object Z of A.

The fibre D_{Z}^{−1}A of this functor over an abelian object A is the category of double central extensions of Z by A. Two double central extensions of Z by A which are connected by a zigzag in D_{Z}^{−1}A are called equivalent. The equivalence classes form the set Centr^{2}(Z, A) = π_{0}D_{Z}^{−1}A of connected components of this category.

4.2. Remark. Depending on the context it might not be clear whether Centr^{2}(Z, A) is indeed a set (rather than a proper class) but in any case Theorem 5.3 implies that Centr^{2}(Z, A) is only as large as is H^{3}(Z, A).

4.3. Remark. The double central extension E induces a 3 × 3 diagram

[figure: A; K[d]; K[g]; K[c]; X; c; d; C; g; K[f]; D; f; Z]

and the object A in this diagram is the direction of E.

We now show that Centr^{2}(Z, A) carries a canonical abelian group structure.

4.4. Proposition. Let A be a semi-abelian category and let Z be an object of A. Mapping an abelian object A of A to the set Centr^{2}(Z, A) of equivalence classes of double central extensions of Z by A gives a finite product-preserving functor

Centr^{2}(Z, −): AbA → Set.

Proof. Let a: A → B be a morphism of abelian objects in A and E a double central extension of Z by A. Then (d, c): X → D ×_{Z} C is a central extension of D ×_{Z} C by A, and the construction of Proposition 1.7 yields a central extension (d′, c′) of D ×_{Z} C by B. The morphism Centr^{2}(Z, a) now maps the equivalence class of E to the class of the right hand side square below. Indeed, since the left hand side square

[figure: X × B; c∘pr_{X}; d∘pr_{X}; (d,c)∘pr_{X}; C; g; D ×_{Z} C; _{pr_{D}}; D; f; Z; B; 0; 0; 0; X′; c′; d′; (d′,c′); C; g; D ×_{Z} C; D; f; Z]

—which arises from the regular epimorphism in the top sequence in D—is a double central extension as the product of E with the middle double central extension, so is its right hand side quotient. The functoriality of Centr^{2}(Z, −) now follows from the functoriality of Centr^{1}(Z, −).

It is clear that Centr^{2}(Z, −) preserves the terminal object: any double central extension with direction 0 is connected to

Z  Z Z  Z.

To show that Centr^{2}(Z, −) also preserves binary products, we must provide an inverse to the map

(Centr^{2}(Z, pr_{A}), Centr^{2}(Z, pr_{B})): Centr^{2}(Z, A × B) → Centr^{2}(Z, A) × Centr^{2}(Z, B).

This inverse is given by the product in the category CExt_{Z}^{2}A of double central extensions of Z. Let indeed the two squares

\xymatrix{X \ar@{|->}[r]^{c} \ar@{|->}[dd]_{d} & C \ar@{|->}[dd]^{g} &  & X′ \ar@{|->}[r]^{c′} \ar@{|->}[dd]_{d′} & C′ \ar@{|->}[dd]^{g′} \\  &  & and \\ D \ar@{|->}[r]_{f} & Z &  & D′ \ar@{|->}[r]_{f′} & Z}

be double central extensions of Z by A and B, respectively. Then their product in CExt_{Z}^{2}A is the square

\xymatrix{X ×_{Z} X′ \ar@{|->}[r]^{c×_{Z}c′} \ar@{|->}[d]_{d×_{Z}d′} & C ×_{Z} C′ \ar@{|->}[d]^{g∘pr_{C}} \\ D ×_{Z} D′ \ar@{|->}[r]_{f∘pr_{D}} & Z.}

In fact, this square represents a pregroupoid in A ↓ Z as a product of two such pregroupoids, and the comparison map (d ×_{Z} d′, c ×_{Z} c′) to the pullback is a central extension as a pullback of the central extension (d, c) × (d′, c′). Finally, the direction of this double central extension is the kernel of (d ×_{Z} d′, c ×_{Z} c′), which is nothing but A × B.

4.5. Corollary. The functor Centr^{2}(Z, −) uniquely factors through the forgetful functor Ab → Set to yield a functor Centr^{2}(Z, −): AbA → Ab.

Proof. Any abelian object of A carries a canonical internal abelian group structure; we just showed that the functor Centr^{2}(Z, −) preserves such structures. See also Remark 5.5.

5. H^{3}(Z, A) and Centr^{2}(Z, A) are isomorphic

5.1. Proposition. (Cf. Section 3 in [27]) Let A be a finitely complete category. The forgetful embedding GdA ↪ PreGdA has a right adjoint gd: PreGdA → GdA. Moreover, when A is semi-abelian, Z is an object of A and A is an abelian object of A, this adjunction restricts to the fibres of the direction functors d_{Z} and D_{Z}

[figure: d_{Z}^{−1}(A, τ); ⊂; ⊥; D_{Z}^{−1}A.; gd]  (K)

Proof. Given an internal pregroupoid (X, d, c), the induced internal groupoid gd(X, d, c) has as underlying reflexive graph

\xymatrix{R[c] ×_{X} R[d] \ar@{=>}[r]^{dom} \ar@{=>}[r]_{cod} & X, \ar@{=>}[l]|{id}}

where dom and cod are the first and third projections and id is the diagonal. This reflexive graph is a groupoid, in which the composition maps a pair (αR[c]βR[d]γ, γR[c]δR[d]ϵ) to the triple (α, p(δ, γ, β), ϵ), where p is the pregroupoid structure of (X, d, c). The (X, d, c)- component of the counit of the adjunction is defined by the map

(p, d, c): (R[c] ×_{X} R[d], dom, cod) → (X, d, c)

in PreGdA; and given an internal groupoid

[figure: X; d; c; v; Y; i]

with inversion map v, the associated unit component is

\xymatrix{X \ar@{=>}[r]^{d} \ar@{=>}[r]_{c} \ar@{=>}[d]_{(i∘d,v,i∘c)} & Y, \ar@{=>}[l]|{i} \ar@{=>}[d]^{i} \\ R[c] ×_{X} R[d] \ar@{=>}[r]^{dom} \ar@{=>}[r]_{cod} & X. \ar@{=>}[l]|{id}}  (L)

One easily checks that the triangular identities hold.

Corollary 3.4 implies that the embedding GdA ↪ PreGdA restricts to the fibres of d_{Z} and D_{Z}. Now suppose that E ∈ D_{Z}^{−1}A; then (X, d, c) is a central pregroupoid in A ↓ Z by Theorem 2.8, and A = K[(d, c)]. Using that the square

\xymatrix{R[c] ×_{X} R[d] \ar@{=>}[r]^{p} \ar@{=>}[d]_{(dom,cod)} & X \ar@{=>}[d]^{(d,c)} \\ X ×_{Z} X \ar@{=>}[r]_{d×_{Z}c} & D ×_{Z} C}  (M)

is a pullback, we see that (dom, cod) is a central extension and A = K[(dom, cod)]. Hence the groupoid gd(X, d, c) in A ↓ Z is central, which by Proposition 3.3 means that it has direction (A, τ), that is, it is in the fibre d_{Z}^{−1}(A, τ)—so the functor gd also restricts to the fibres of the direction functors d_{Z} and D_{Z}.

To see that these restrictions are still adjoint to each other, it suffices to prove that the components of the unit and the counit are in the fibre of 1_{(A,τ)} (respectively 1_{A}). This is the case, because both the square M and the similar square corresponding to L are pullbacks.

5.2. Remark. Consider an adjunction

[figure: C; F; ⊥; D.; G]

1. The functors F and G induce functions φ: π_{0}C → π_{0}D, defined by φ{C} = {F C}, and γ: π_{0}D → π_{0}C, defined by γ{D} = {GD}, respectively.

2. F being left adjoint to G implies that φ^{−1} = γ, i.e., π_{0}C ≅ π_{0}D. In fact, (φ^{∘}γ){D} = {F GD} = {D}, for any object D of D, since F GD is connected to D by the Dcomponent of the counit of the adjunction; thus φ^{∘}γ = 1_{π_{0}D}. Similarly γ^{∘}φ = 1_{π_{0}C}, using the unit of the adjunction instead.

Now suppose that the category C carries a symmetric monoidal structure (C, ⊗, E) as in Remark 3.1.

3. π_{0}C is an abelian group.

4. π_{0}D is an abelian group with addition given by {D_{1}} + {D_{2}} = {F ( GD_{1} ⊗ GD_{2} )}, zero \{F E\} and −\{D\} = \{F ( \overline{GD})\}.

5. The function φ is a group isomorphism with inverse γ.

5.3. Theorem. In a semi-abelian category, the third cohomology group H^{3}(Z, A) of an object Z with coefficients in an abelian object A is isomorphic to the group Centr^{2}(Z, A) of equivalence classes of double central extensions of Z by A.

Proof. By the unicity in Corollary 4.5, to show that H^{3}(Z, −) and Centr^{2}(Z, −) are isomorphic as functors AbA → Ab, it suffices to give a bijection between the underlying sets H^{3}(Z, A) and Centr^{2}(Z, A), natural in A. Through Remark 5.2, the adjunction K from Proposition 5.1 induces the needed isomorphisms φ: H^{3}(Z, A) → Centr^{2}(Z, A) and γ: Centr^{2}(Z, A) → H^{3}(Z, A).

5.4. Remark. The isomorphisms of Theorem 5.3 may also be obtained as follows. We have φ: H^{3}(Z, A) → Centr^{2}(Z, A): {G} ↦ {H} and

γ: Centr^{2}(Z, A) → H^{3}(Z, A): {E} ↦ {gd(E)} ,

where

gd(E) = [figure: R[c] ×_{X} R[d]; dom; cod; X; f∘d=g∘c; id; Z]

such that φ^{∘}γ = 1_{Centr^{2}(Z,A)}, because for any double central extension E of Z by A, (φ^{∘}γ){E} is equal to {E} through (p, d, c), the E-component of the counit of the adjunction K

[figure: R[c] ×_{X} R[d]; dom; cod; p; X; g∘c; c; X; d; c; C; g; X; f∘d; d; Z; D; f; Z;]

and γ^{∘}φ = 1_{H^{3}(Z,A)}, since for any central internal groupoid G, with inversion map v and direction (A, τ), (γ^{∘}φ){G} is equal to {G} through ((i^{∘}d, v, i^{∘}c), i), the G-component of the unit of the adjunction K

[figure: X; d; c; (i∘d,v,i∘c); Y; f; i; i; Z; R[c] ×_{X} R[d]; dom; cod; X.; f∘d; id]

5.5. Remark. We know that d_{Z}^{−1}(A, τ) is a symmetric monoidal category with property J by Remark 3.1. The arguments in Remark 5.2 show how the addition on H^{3}(Z, A) is transported to an abelian group structure on Centr^{2}(Z, A) as described in Remark 5.2, (4). This makes the connection between the canonical abelian group structure from Proposition 4.4 and Corollary 4.5 and the Baer sum on d_{Z}^{−1}(A, τ) explicit.

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Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal

Centro de Matemática, Universidade de Coimbra, 3001-454 Coimbra, Portugal

Vakgroep Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium

Email: drodelo@ualg.pt
      tvdlinde@vub.ac.be

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