Theory and Applications of Categories, Vol. 27, No. 9, 2012, pp. 189–209.

HIGHER CENTRAL EXTENSIONS VIA COMMUTATORS

DIANA RODELO AND TIM VAN DER LINDEN

Abstract. We prove that all semi-abelian categories with the the Smith is Huq property satisfy the Commutator Condition (CC): higher central extensions may be characterised in terms of binary (Huq or Smith) commutators. In fact, even Higgins commutators suffice. As a consequence, in the presence of enough projectives we obtain explicit Hopf formulae for homology with coefficients in the abelianisation functor, and an interpretation of cohomology with coefficients in an abelian object in terms of equivalence classes of higher central extensions. We also give a counterexample against (CC) in the semi-abelian category of (commutative) loops.

Introduction

The concept of higher centrality is a cornerstone in the recent approach to homology and cohomology of non-abelian algebraic structures based on categorical Galois theory [5, 35]. Through higher central extensions, the Brown–Ellis–Hopf formulae [11, 14] which express homology objects as a quotient of commutators have been made categorical [17, 19, 20], which greatly extends their scope while simplifying the study of concrete cases (see, for instance, [13]). Higher central extensions are also essential in the study of relative commutators [22, 23] and are classified by cohomology groups [49].

To take full advantage of these results, sufficiently explicit characterisations of higher centrality are essential. On the one hand, the higher Hopf formulae are valid in any semiabelian category [39] with enough projectives, but these formulae only become concrete once the relevant concept of higher centrality is appropriately characterised, ideally in terms of classical binary commutators. Indeed, the main result of [20] says that in a semi-abelian monadic category A, for any n-presentation F of Z,

H_{n 1} Z, ab \frac{F_{n}, F_{n}  _{i n} Ker f_{i}}{L_{n} F}.  (A)

Coefficients are chosen in the abelianisation functor ab: A Ab A . Here F_{n} is the initial object of F and the f_{i} are the initial arrows. The object F_{n}, F_{n} is the Huq commutator

The first author was supported by CMUC/FCT (Portugal) and the FCT Grant PTDC/MAT/120222/2010 through the European program COMPETE/FEDER. The second author works as chargé de recherches for Fonds de la Recherche Scientifique–FNRS and would like to thank CMUC for its kind hospitality during his stays in Coimbra.

Received by the editors 2012-03-26 and, in revised form, 2012-10-04. Published on 2012-10-08 in the volume of articles from CT2011. 2010 Mathematics Subject Classification: 18G50, 18G60, 18G15, 20J, 55N. Key words and phrases: Higgins, Huq, Smith commutator; higher central extension; semi-abelian,

exact Mal’tsev category; Hopf formula; (co)homology.

\begin{matrix}c \\ ◯ \end{matrix} Diana Rodelo and Tim Van der Linden, 2012. Permission to copy for private use granted.

of F_{n} with itself, which makes the numerator entirely explicit. But the denominator is not; rather, L_{n} F is the smallest normal subobject of F_{n} which, when divided out, makes F central with respect to Ab A in the sense of categorical Galois theory. Nevertheless, in all known examples also this object may be expressed in terms of commutators.

On the other hand, given an object Z in a semi-abelian category, its cohomology with coefficients in an abelian object A classifies the higher central extensions of Z by A, provided those higher central extensions admit a characterisation in terms of Huq commutators. Thus far, such precise characterisations of higher central extensions were only available in concrete cases.

A semi-abelian category A satisfies the Smith is Huq Condition (SH) when two equivalence relations (Smith) commute if and only if their normalisations (Huq) commute. Under (SH) we may remedy the lack of characterisation mentioned above. We prove that an n-fold extension in a semi-abelian category A with (SH) is central with respect to the abelian objects in A if and only if a certain join of binary Huq commutators vanishes. This gives us the following refined version of the main theorem of [49].

Theorem. Let Z be an object and A an abelian object in a semi-abelian category with (SH). Then for every n 1 we have an isomorphism H^{n 1} Z, A Centr^{n} Z, A .

Examples of semi-abelian categories with (SH) are all action representative semiabelian categories [6, 4] and all action accessible ones [10], all strongly semi-abelian categories [7], all Moore categories [25, 47], all categories of interest [45, 44], but not all varieties of Ω-groups: the category of digroups is a counterexample [3, 7]. Hence our results are valid, e.g., in the categories of groups, Lie and Leibniz algebras, (pre)crossed modules and associative algebras.

The above can be made slightly more precise as follows. We shall say that an n-fold extension F in a semi-abelian category A is H-central when

_{i I} Ker f_{i} , _{i n I} Ker f_{i} 0

for all I n. Here the f_{i} are the initial arrows of the n-fold extension F, the commutators are either Huq or Higgins commutators, and we write 0 , n 0, …, n 1 . The category A satisfies the Commutator Condition (CC) when H-centrality is equivalent to centrality with respect to Ab A in the Galois-theory sense. This means that the denominator L_{n} F of (A) may be expressed as the join

_{I n} _{i I} Ker f_{i} , _{i n I} Ker f_{i} .

It follows from results in [8] and [29] that the Commutator Condition holds for (one-fold) extensions (Subsection 1.6). For double extensions, the Commutator Condition holds as soon as the Smith is Huq Condition does (see Subsection 1.7). Our main concern now becomes to find conditions which imply (CC) in all degrees.

In Section 1 we give a more detailed outline of the mathematical context we shall be working in. Section 2 contains the main result of the paper: Theorem 2.8, which says that the Commutator Condition for double extensions implies the Commutator Condition for all higher degrees. Hence the Commutator Condition is weaker than the Smith is Huq Condition.

Even though (SH) is known to be independent of semi-abelianness, thus far we did not have any examples to show that also (CC) is independent. The known counterexamples (in digroups [3, 7] or in loops [24, 32]) give an action of an object on an abelian object which is not a module. However, when an action is considered as double extension, it cannot be H-central without being central—see Subsection 4.1—which forces us to find a new counterexample. This is done in Section 3 where we show that the category of loops Loop does not satisfy (CC). In fact this counterexample also works in the category of commutative loops CLoop; it gives a new example of a semi-abelian category in which (SH) does not hold.

There are certain further questions which remain unanswered as yet; we give a short overview in Section 4.

1. Preliminaries

In this paper A will always denote a semi-abelian category [39].

1.1. The Huq commutator and the Smith commutator A coterminal pair

[figure: K; k; X; L; l]

of normal monomorphisms (i.e., kernels) in A is said to (Huq-)commute [9, 34] when there is a (necessarily unique) morphism φ such that the diagram

[figure: K; 1_{K}, 0; k; K L; φ; X; L; 0, 1_{L}; l]

is commutative. The Huq commutator k, l ^{Huq} : K, L ^{Huq} X of k and l [8, 3] is the smallest normal subobject of X which should be divided out to make k and l commute, so that k and l commute if and only if K, L ^{Huq} 0. We can define K, L ^{Huq} as the kernel of the (normal epi)morphism X Q , where Q is the colimit of the outer square above.

Given a pair of equivalence relations R, S on a common object X

\xymatrix{R \ar@{=>}[r]^{r_{1}} \ar@{=>}[r]_{r_{2}} & X \ar@{=>}[l]|{Δ_{R}} \ar@{=>}[r]|{Δ_{S}} & S, \ar@{=>}[l]^{s_{1}} \ar@{=>}[l]_{s_{2}}}

consider the induced pullback of r_{2} and s_{1}:

[figure: R _{X} S; π_{S}; π_{R}; S; s_{1}; R; r_{2}; X.]

The equivalence relations R and S centralise each other or (Smith-)commute [50, 46, 9] when there is a (necessarily unique) morphism θ such that the diagram

[figure: R; 1_{R}, Δ_{S}r_{2}; r_{1}; R _{X} S; θ; X; S; Δ_{R}s_{1}, 1_{S}; s_{2}]

is commutative. Like for the Huq commutator, the Smith commutator is the smallest equivalence relation R, S ^{S} on X which, divided out of X, makes R and S commute. It can be obtained through a colimit, similarly to the situation above; see Section 3 for a concrete example. Thus R and S commute if and only if R, S ^{S} Δ_{X}, where Δ_{X} denotes the smallest equivalence relation on X. We say that R is a central equivalence relation when it commutes with ∇_{X}, the largest equivalence relation on X, so that R, ∇_{X} ^{S} Δ_{X}.

1.2. The Smith is Huq Condition It is well known, and easily verified, that if the Smith commutator of two equivalence relations is trivial, then the Huq commutator of their normalisations is also trivial [9]. But, in general, the converse is false; in [3, 7] a counterexample is given in the category of digroups, which is a semi-abelian variety, even a variety of Ω-groups [33]. The requirement that the two commutators vanish together is known as the Smith is Huq Condition (SH) and it is shown in [43] that, for a semi-abelian category, this condition holds if and only if every star-multiplicative graph is an internal groupoid, which is important in the study of internal crossed modules [37]. Moreover, the Smith is Huq Condition is also known to hold for pointed strongly protomodular categories [9] (in particular, for any Moore category [25, 47]) and in action accessible categories [10] (in particular, for any category of interest [44, 45]).

1.3. Extensions We write Arr^{n} A for the category of n-fold arrows in A.

A zero-fold extension in A is an object of A and a (one-fold) extension is a regular epimorphism in A. For n 2, an n-fold extension is an object c, f of Arr^{n} A (a morphism of Arr^{n 1} A ) as in

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

such that the morphisms c, d, f, g and the universally induced comparison morphism d, c : X D _{Z} C to the pullback of f with g are n 1 -fold extensions. A two-fold extension is also called a double extension. The n-fold extensions determine a full subcategory Ext^{n} A of Arr^{n} A ; we write Ext A Ext^{1} A .

An n-fold arrow may be considered as a diagram 2^{n} A in A, a cube of dimension n; in particular, n-fold extensions are pictured as n-cubes. Given such an n-fold extension F, we shall write F_{n} for its initial object and f_{i}: F_{n} F_{n i}, i n, for the initial arrows. The extension property of F implies that for any choice of i n, the induced square in A

[figure: F_{n}; f_{i}; F_{n i}; lim_{i J n} F_{J}; lim_{J n i} F_{J}]

is still a double extension [49].

1.4. Central extensions We write Ab A for the full subcategory of A determined by the abelian objects, that is, those objects which admit an internal abelian group structure. Let ab: A Ab A denote the abelianisation functor, left adjoint to the inclusion of Ab A in A. It sends an object X of A to the abelian object ab X X X, X ^{Huq}. We define centrality of (higher) extensions with respect to the Birkhoff subcategory Ab A of A [38, 8].

An extension f: X Z is called trivial when the induced naturality square

\xymatrix{X \ar@{|->}[r]^{f} \ar@{|->}[d]_{η_{X}} & Z \ar@{|->}[d]^{η_{Z}} \\ ab X \ar@{|->}[r]_{ab f} & ab Z}

is a pullback, and f is central when there exists an extension g: Y Z such that the pullback of f along g is trivial. In our context we can take g f so that f is central if and only if either projection of its kernel pair is trivial (central extensions coincide with normal extensions).

The full subcategory CExt_{Ab A}^{1} A of Ext^{1} A determined by those extensions which are central is again reflective. Inductively, we get a reflective subcategory CExt_{Ab A}^{n} A of Ext^{n} A containing the n-fold central extensions (relative to Ab A ) of A, n 1. Each level gives rise to a notion of central extension which determines the next level—see [20, Theorem 4.6] and [17] where this is worked out in detail. In particular, for every n 1 we have a reflector, the centralisation functor

centr_{n}: Ext^{n} A CExt_{Ab A}^{n} A ,

left adjoint to the inclusion of CExt_{Ab A}^{n} A in Ext^{n} A .

1.5. The Commutator Condition (CC) Given an n-fold extension F with initial object F_{n} and initial arrows f_{i}: F_{n} F_{n i}, we write k_{i}: K_{i} Ker f_{i} F_{n} for all i n. We say that F is H-central when

_{i I} K_{i}, _{i n I} K_{i} ^{Huq} 0  (B)

for all I n. Here the commutators are either Huq or Higgins commutators (Subsection 1.8); this explains the “H” (see Lemma 2.5). The category A satisfies the Commutator Condition (CC) when H-centrality is equivalent to centrality with respect to Ab A in the Galois-theory sense (Subsection 1.4).

The condition (CC) falls apart in one version for each degree of extension n: the category A satisfies (CCn) when an n-fold extension in it is H-central if and only if it is central. The principal result in this work is to show that (CC2) implies (CC) (Theorem 2.8).

1.6. One-fold extensions and (CC1) Recall [35] that an extension f: X Z in the category of groups is central (with respect to Ab) if and only if Ker f , X 0. This result was adapted to a semi-abelian context in [26, 8]: the one-fold central extensions (in the sense of Galois theory) may be characterised through the Smith commutator of equivalence relations as those extensions f: X Z such that X _{Z} X, ∇_{X} ^{S} Δ_{X} , where X _{Z} X denotes the kernel pair of f. This means that X _{Z} X is a central equivalence relation on X (Subsection 1.1). A characterisation closer to the group case appears in Proposition 2.2 of [29] where the condition is reformulated in terms of the Huq commutator of normal subobjects so that it becomes Ker f , X ^{Huq} 0. Hence f is central if and only if it is H-central, so that (CC1) is true in any semi-abelian category.

1.7. Double central extensions and (CC2) One level up, the double central extensions of groups vs. abelian groups were first characterised in [36]. A double extension (of Z) is a pushout square of regular epimorphisms

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

Let us write K Ker c , L Ker d for the kernels of c and d and R X _{C} X, S X _{D} X for the respective kernel pairs. Then (C) is central when

K, L 0 K L, X .

General versions of this characterisation were given in [28] for Mal’tsev varieties, then in [48] for semi-abelian categories and finally in [21] for exact Mal’tsev categories: the double extension (C) is central if and only if

R, S ^{S} Δ_{X} R S, ∇_{X} ^{S}.

This means that the span X, d, c is a special kind of pregroupoid in the slice category A Z (see [40] for the definition of a pregroupoid).

The problem we are now confronted with is that the correspondence between the Huq commutator of normal monomorphisms and the Smith commutator of equivalence relations which exists in level one is no longer there when we go up in degree. However, we know that always K L, X ^{Huq} 0 if and only if R S, ∇_{X} ^{S} Δ_{X} by (CC1). Furthermore, R, S ^{S} Δ_{X} implies K, L ^{Huq} 0, so when (C) is central it is also Hcentral. On the other hand, the Smith is Huq Condition says that K, L ^{Huq} 0 implies R, S ^{S} Δ_{X}, so that the two concepts of centrality are equivalent—and hence (CC2) holds under (SH).

1.8. The Higgins commutator Central extensions, relative to Ab A , may also be characterised in terms of the Higgins commutator [31, 41], which in turn may be obtained through a co-smash product [12] or a cross-effect [15, 2, 30] of the identity functor on A.

Given two objects K and L of A, the co-smash product [12] of K and L

K L Ker \begin{matrix}1_{K} & 0 \\ 0 & 1_{L} \end{matrix} : K L K L

behaves as a kind of “formal commutator” of K and L. (See [31] and [41]; this object is also written K L, or K L when it is interpreted as the second cross-effect of the identity functor 1_{A} evaluated in K, L.) If now k: K X and l: L X are subobjects of an object X, their Higgins commutator K, L X is a subobject of X given by the image of the induced composite morphism

[figure: K L; ι_{K,L}; K L; ^{\begin{matrix}k \\ l \end{matrix}}; X.; K, L]

When K and L are normal subobjects of X and K L X, the Higgins commutator K, L is normal in X so that it coincides with the Huq commutator (Subsection 1.1). In particular, we always have K, X ^{Huq} K, X . In general the Huq commutator is the normal closure of the Higgins commutator. So, K, L K, L ^{Huq} and K, L 0 if and only if K, L ^{Huq} 0. The Higgins commutator may also be used to measure normality of subobjects. In fact, a result in [41] states that K X if and only if K, X K, and is further refined in [31] as follows: the normal closure of K in X may be computed as the join K K, X . In any case, an extension in A such as

[figure: 0; K; k; X; f; Z; 0]

is central if and only if K, X ^{Huq} K, X 0.

1.9. The ternary commutator The Higgins commutator generally does not preserve joins, but the defect may be measured precisely—it is a ternary commutator which can be computed by means of a ternary co-smash product or a cross-effect of order three. Let us extend the definition above: given a third subobject m: M X of the object X, the ternary Higgins commutator K, L, M X is the image of the composite

[figure: K L M; ι_{K,L,M}; K L M; ^{\begin{matrix}k \\ l \\ m \end{matrix}}; X,; K, L, M]

where ι_{K,L,M} is the kernel of

\xymatrix{K L M \ar@{=>}[r]^{^{\begin{matrix}i_{K} & i_{K} & 0 \\ i_{L} & 0 & i_{L} \\ 0 & i_{M} & i_{M} \end{matrix}}} & K L K M L M ;}

i_{k}, i_{L} and i_{M} denote the injection morphisms. The object K L M is the third crosseffect of the identity functor 1_{A} or ternary co-smash product evaluated in K, L and M.

1.10. Proposition. If K, L, M X then

K, L M K, L K, M K, L, M .

Proof. Via the result in [31] or [32].

1.11. (SH) and (CC2) via the ternary commutator It is precisely the availability of this join decomposition which makes the Higgins commutator useful in what follows. This, and the fact that (SH) may be expressed in terms of ternary commutators. By the main result in [32], two normal subobjects K, L X have Smith-commuting denormalisations when K, L 0 K, L, X . Hence the Smith is Huq Condition is equivalent to saying that K, L 0 (they Huq- or Higgins-commute) implies K, L, X 0 (what is missing for them to also Smith-commute).

What we shall be studying here (the Commutator Condition, at first for n 2) is slightly weaker, because next to K, L 0 we shall also assume K L, X 0 to obtain the same conclusion K, L, X 0. This will give us “H-centrality” implies “centrality” (Theorem 2.8) as in our paper [49]. Thus, (SH) (CC2) (CC).

Many other things can be said about these ternary commutators; let us just mention that they are generally not decomposable into iterated binary ones, and refer to [32] for further information.

2. Main result

In this section we prove our main result, Theorem 2.8: (CC2), the Commutator Condition in degree n 2, implies (CC) in all degrees. So (CC) does not explode—in the sense that it would give rise to a new mysterious condition in each dimension separately—but instead stays within bounds, as it is implied by the well-studied condition (SH).

2.1. Degree two We use the same notation as in Subsection 1.7 for double extensions in a semi-abelian category.

2.2. Lemma. Let (C) be a double extension in a semi-abelian category. Then

K, L K L, X

is normal in X, and K, L K L, X K, L ^{Huq} K L, X ^{Huq}.

Proof. This follows from the fact that K, L ^{Huq} K L, X is normal in X while K, L ^{Huq} K, L K, L , X and K, L K L. The second statement is now obvious.

When in A (SH) holds, this implies that the normalisation of R, S ^{S} R S, ∇_{X} ^{S} is K, L K L, X . Hence the centralisation of (C) is its quotient

[figure: \frac{X}{K,L K L,X}; C; D; Z.]

Recall that a double presentation of an object Z is a double extension such as (C) in which the objects X, D and C are (regular epi)-projective.

2.3. Theorem. Let A be a semi-abelian category with enough projectives and such that (SH) holds. Let Z be an object in A and (C) a double presentation of Z. Then

H_{3} Z, ab \frac{K L X, X}{K, L K L, X} .

When, moreover, A is monadic over Set, these homology objects are comonadic Barr–Beck homology objects [1] with respect to the canonical comonad on A.

Proof. This follows from the main result of [17]; see also [20].

2.4. Higher degrees Our purpose is now to prove that the Commutator Condition for double extensions (CC2) implies the Commutator Condition for all n-fold extensions (CC). Consequently, n-fold extensions are central if and only if they are H-central. We shall assume that a Higgins-style characterisation exists for the n 1 -fold central extensions and prove that such a characterisation is also valid for n-fold central extensions. More precisely, we shall prove that under (CC2), the condition (CC n 1 ) implies (CCn).

We begin with a higher-dimensional version of the result above for double extensions which allows us to use either Huq or Higgins commutators in the definition of H-centrality and in (CC). We use the notation from Subsection 1.5.

2.5. Lemma. Let F be an n-fold extension in a semi-abelian category. Then

_{I n} _{i I} K_{i}, _{i n I} K_{i}  _{I n} _{i I} K_{i}, _{i n I} K_{i} ^{Huq}  (D)

(so the join is normal in X).

Proof. The case n 1 is well known; see Subsection 1.8. In general, following the proof of Lemma 2.2, for every I n we have

_{i I} K_{i}, _{i n I} K_{i} ^{Huq}  _{i I} K_{i}, _{i n I} K_{i}  _{i I} K_{i}, _{i n I} K_{i} , F_{n}

while

_{i I} K_{i}, _{i n I} K_{i} , F_{n}  _{i n} K_{i}, F_{n}  _{I n} _{i I} K_{i}, _{i n I} K_{i} ,

which implies the non-trivial inclusion.

Before we prove that (CC2) implies (CCn), so that we can go up in dimension, let us first explain how to go down.

2.6. Proposition. For any n 1, the condition (CC n 1 ) implies (CCn).

Proof. An n-fold arrow F is an n-fold extension if and only if the n 1 -fold arrow F 0 is an n 1 -fold extension. It follows immediately from the definitions that F is H-central precisely when F 0 is H-central, and that F is central if and only if F 0 is a trivial extension. Hence F is central precisely when F 0 is central because every split central extension is trivial in the context considered [20].

2.7. Lemma. If F: X Z is an n-fold H-central extension, then also any of the two projections

π_{1}, π_{2} : X _{Z} X X

in its kernel pair are H-central.

Proof. We prove that G π_{1} is H-central. Consider I n and write kerg_{i}: Ker g_{i} G_{n} for the kernel of g_{i}: G_{n} G_{n i} . Then g_{n} is the “top morphism” of π_{1} : X _{Z} X X ; similarly, write h_{n} for the top morphism of H π_{2}. Now _{i I} kerg_{i} and _{i n I} kerg_{i} commute: to see this, we compose them with the morphisms g_{n} and h_{n}, which form a jointly monic pair. Composing with g_{n} makes one of the intersections—the one containing the kernel of g_{n}—trivial, so already g_{n} _{i I} kerg_{i} and g_{n} _{i n I} kerg_{i} commute. On the other hand, the composites h_{n} _{i I} kerg_{i} and h_{n} _{i n I} kerg_{i} factor through the intersections i I k_{i} and _{i n I} k_{i}, respectively. These two intersections commute because F is Hcentral.

2.8. Theorem. Every semi-abelian category with (CC2) satisfies the Commutator Condition (CC).

Proof. We give a proof by induction on n: we show that under (CC2), for all n 3 the condition (CC n 1 ) implies (CCn).

Let F: X Z be an n-fold H-central extension, i.e., _{i I} K_{i}, _{i n I} K_{i} 0 for all I n. To prove that F is central, we must show that either one of the projections in the kernel pair of F is a trivial n-fold extension. Consider, for i n, the commutative diagram

[figure: K_{n} M_{i}; \overline{k_{i}}; \overline{m_{i}}; M_{i}; π_{1}^{i}; π_{2}^{i}; m_{i}; K_{i}; k_{i}; e_{i}; K_{n}; { \atop k_{n}}; F_{n} _{F_{n 1}} F_{n}; π_{1}; π_{2}; f_{i}^{}; F_{n}; f_{n}; f_{i}; e; F_{n 1}; F_{n i} _{F_{n i,n 1}} F_{n i}; p_{1}; p_{2}; F_{n i}; f_{n 1}; F_{n i,n 1 ,}]  (E)

where { \atop k_{n}} is the kernel of π_{1}, so π_{2}{ \atop k_{n}} k_{n}, while π_{1}^{i}, π_{2}^{i}, e_{i} and f_{i}^{}are the induced morphisms and M_{i} Ker f_{i}^{}.

By the induction hypothesis (CC n 1 ), the first projection π_{1} of the kernel pair of F is a trivial n-fold extension when the naturality square

\xymatrix{F_{n} _{F_{n 1}} F_{n} \ar@{=>}[r]^{π_{1}} \ar@{=>}[d] & F_{n} \ar@{=>}[d] \\ \frac{F_{n} _{F_{n 1}} F_{n}}{_{I n 1}  _{i I} M_{i}, _{i n 1 I} M_{i}} \ar@{=>}[r] & \frac{F_{n}}{_{I n 1}  _{i I} K_{i}, _{i n 1 I} K_{i}}}

is a pullback. Indeed, once (CC n 1 ) is satisfied, the “centralisation” of an n 1 -fold extension is obtained by replacing the “initial” object of the extension with its quotient by the relevant join of commutators. Showing that the above square is a pullback amounts to proving that

_{I n 1} _{i I} M_{i}, _{i n 1 I} M_{i} _{I n 1} _{i I} K_{i}, _{i n 1 I} K_{i} .

As subobjects of F_{n} _{F_{n 1}} F_{n}, we have

e _{I n 1} _{i I} k_{i} K_{i} , _{i n 1 I} k_{i} K_{i}  _{I n 1} _{i I} ek_{i} K_{i} , _{i n 1 I} ek_{i} K_{i}  (F)
_{I n 1} _{i I} m_{i} M_{i} , _{i n 1 I} m_{i} M_{i} .  (G)

To prove the other inclusion, we shall decompose the subobject (G) as a join using Proposition 1.10.

By protomodularity, the split exact sequence

[figure: 0; K_{n}  _{i I} M_{i}; _{i I} M_{i}; _{i I} K_{i}; 0]

gives us

_{i I} m_{i} M_{i}  _{i I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i}  _{i I} ek_{i} K_{i}

for all I n 1.

For I n 1 we have

_{i I} m_{i} M_{i} , _{i n 1 I} m_{i} M_{i}  _{i I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i}  _{i I} ek_{i} K_{i} , _{i n 1 I} m_{i} M_{i} ,

which decomposes to the join

_{i I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i} , _{i n 1 I} m_{i} M_{i} _{i I} ek_{i} K_{i} , _{i n 1 I} m_{i} M_{i}

_{i I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i} , _{i I} ek_{i} K_{i} , _{i n 1 I} m_{i} M_{i} . (H)

The first term of (H) vanishes by Lemma 2.7 and the assumption that F is H-central. In fact, the intersection _{i I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i} may be written as { \atop k_{n}} K_{n} _{i I} m_{i} M_{i} , i.e., an intersection of kernels of the initial arrows of the first projection of X _{Z} X. Consequently, the commutator

_{i n 1} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i} , F_{n} _{F_{n 1}} F_{n}

vanishes as it is one of the commutators which express the H-centrality of the first projection of the kernel pair X _{Z} X. So by (CC2) also the last term in (H) is trivial, because it is smaller than

_{i I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i} , F_{n} _{F_{n 1}} F_{n}, _{i n 1 I} m_{i} M_{i} 0

as explained in Subsection 1.11.

We now further decompose the second term of (H)

_{i I} ek_{i} K_{i} , _{i n 1 I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i} _{i n 1 I} ek_{i} K_{i}

into the join

_{i I} ek_{i} K_{i} , _{i n 1 I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i} _{i I} ek_{i} K_{i} , _{i n 1 I} ek_{i} K_{i}

_{i I} ek_{i} K_{i} , _{i n 1 I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i} , _{i n 1 I} ek_{i} K_{i} . (I)

The first term of (I) vanishes, as the even larger subobjects

_{i I} m_{i} M_{i}  and  _{i n 1 I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i}

commute, again by Lemma 2.7 and the assumption that F is H-central. By (CC2) also the last term in (I) is trivial, because it is smaller than

_{i I} m_{i} M_{i} , _{i n 1 I} { \atop k_{n}}\overline{m_{i}} K_{n} M_{i} , F_{n} _{F_{n 1}} F_{n} 0.

So all commutators determined by I n 1 in the join (G) are also in the join (F). As I and I n 1 give rise to the same commutator, this finally tells us that the join (G) is smaller than the join (F)—which finishes the proof that when F is H-central, then it is central.

The other implication was almost proved in [49]; the only difference between the result there and the present claim is that there, H-centrality was characterised in terms of Huq commutators, rather than Higgins commutators as in (B). But the two concepts are equivalent by Lemma 2.5.

2.9. Corollary. Every semi-abelian category with (SH) satisfies the Commutator Condition (CC).

2.10. Corollary. Every semi-abelian category with (CCn) for some n 2 satisfies the Commutator Condition (CC).

This immediately gives us explicit versions of Hopf formulae obtained in [17, 20]. Recall that an n-fold extension of an object Z is an n-fold presentation of Z when all its objects, but its terminal object Z, are projective.

2.11. Theorem. Let A be a semi-abelian category with enough projectives such that (SH) holds. Let Z be an object in A and F an n-fold presentation of Z. Then

H_{n 1} Z, ab \frac{F_{n}, F_{n} _{i n} K_{i}}{_{I n} _{i I} K_{i}, _{i n I} K_{i}}.

When, moreover, A is monadic over Set, these homology objects are comonadic Barr– Beck homology objects with respect to the canonical comonad on A.

3. A counterexample

We prove that not every semi-abelian category needs to satisfy the Commutator Condition (CC): for instance, the category of loops and loop homomorphisms Loop doesn’t.

\begin{matrix}1 & 1 & i & i & g & g & h & h & j & j & k & k & l & l & m & m \\ 1 & 1 & i & i & g & g & h & h & j & j & k & k & l & l & m & m \\ i & i & 1 & 1 & h & h & g & g & k & k & j & j & m & m & l & l \\ i & i & 1 & 1 & h & h & g & g & k & k & j & j & m & m & l & l \\ g & g & h & h & 1 & 1 & i & i & l & l & m & m & j & j & k & k \\ g & g & h & h & 1 & 1 & i & i & l & l & m & m & j & j & k & k \\ h & h & g & g & i & i & 1 & 1 & m & m & l & l & k & k & j & j \\ h & h & g & g & i & i & 1 & 1 & m & m & l & l & k & k & j & j \\ j & j & k & k & l & l & m & m & 1 & 1 & i & i & g & g & h & h \\ j & j & k & k & l & l & m & m & 1 & 1 & i & i & g & g & h & h \\ k & k & j & j & m & m & l & l & i & i & 1 & 1 & h & h & g & g \\ k & k & j & j & m & m & l & l & i & i & 1 & 1 & h & h & g & g \\ l & l & m & m & j & j & k & k & g & g & h & h & 1 & 1 & i & i \\ l & l & m & m & j & j & k & k & g & g & h & h & 1 & 1 & i & i \\ m & m & l & l & k & k & j & j & h & h & g & g & i & i & 1 & 1 \\ m & m & l & l & k & k & j & j & h & h & g & g & i & i & 1 & 1 \end{matrix}

Table 1: The loop X with its normal subloops I and Y

This is a refinement of the result from [32] saying that the category Loop does not satisfy (SH). Incidentally, our counterexample also works in the category of commutative loops CLoop, so it is a new example of a semi-abelian category where (SH) is not valid.

Let us recall a few basic notions. A loop is a quasigroup with a neutral element: an algebraic structure X, , , , 1 that satisfies x 1 x 1 x and

y x x y  y x x y
x x y y  x x y y.

We also write xy for the product x y. An associative loop is the same thing as a group. A commutative loop has xy yx for all x, y X—which doesn’t yet imply that X is abelian in Loop: X carries an internal abelian group structure precisely when it is an abelian group, when it is commutative and associative. The defect in being associative is measured by means of the associator elements

x, y, z xy z x yz .

The associator elements are in the ternary commutator X, X, X of X since they are expressions in three variables which vanish as soon as one of the variables is equal to 1.

We take X to be the non-associative commutative loop of which the multiplication table is Table 1. (Any Latin square determines a quasigroup, and a loop is a quasigroup with unit. It is commutative as the multiplication table is symmetric.) We take I to be its normal subloop 1, 1, i, i , as indicated in the multiplication table of X, and H the normal subloop 1, 1, h, h of X. The normal subloop

Y 1, 1, i, i, g, g, h, h H I

of X is actually an abelian group (isomorphic to the cube C_{2}^{3} of the cyclic group of order two C_{2}), so that I and H commute in Y, hence in X. Furthermore, A H I 1, 1 is central in X, as the multiplication on X restricts to a loop homomorphism : A X X . Hence we have the the equalities

H, I 0 H I, X .

On the other hand, the commutator H, I, X is non-trivial, as hi l gl j while h il h m j, so that

1 hi l h il h, i, l H, I, X .

This violates the Commutator Condition for n 2 (Subsection 1.11), since H and I X determine a double extension (of C_{2} X Y) which is H-central but not central.

A direct proof without ternary commutators goes as follows. Let R and S be the respective denormalisations of H and I. Then x, y R (resp. S) when xH yH (resp. xI yI). The Smith commutator R, S ^{S} is the kernel pair of t in the colimit diagram

[figure: R; 1_{R}, Δ_{S}r_{2}; r_{1}; R _{X} S; T; X; t; S; Δ_{R}s_{1}, 1_{S}; s_{2}]

(Subsection 1.1). We claim that t maps h, i, l X to 1, so that the couple h, i, l , 1 is a non-trivial element of R, S ^{S}. This violates the characterisation of double central extensions recalled in Subsection 1.7.

The above colimit may be computed as the pushout

[figure: R S; ^{\begin{matrix}r_{1} \\ s_{2} \end{matrix}}; _{\begin{matrix}1_{R} & Δ_{S}r_{2} \\ Δ_{R}s_{1} & 1_{S} \end{matrix}}; X; t; R _{X} S; T.]

Certainly the formal associator

h, 1 , 1, i , l, l

in R S, where l, l is considered as belonging to R, is mapped to h, i, l in X. On the other hand, the arrow \begin{matrix}1_{R} & Δ_{S}r_{2} \\ Δ_{R}s_{1} & 1_{S} \end{matrix} sends this associator to the element

h, 1, 1 , 1, 1, i , l, l, l

of the pullback R _{X} S. This element is equal to

h, 1, l , 1, 1, l , 1, i, l 1, 1, 1 ,

because any associator containing 1 vanishes, so that indeed 1, 1 h, i, l , 1 R, S ^{S}.

4. Further remarks

4.1. Modules The Smith is Huq Condition implies that every action of an object on an abelian object A is a module (i.e., an abelian group in the slice category A Z): given any split epimorphism f: X Z , the equality X _{Z} X, X _{Z} X ^{S} Δ_{X} follows from

Ker f , Ker f ^{Huq} A, A ^{Huq} 0.

All known counterexamples against (SH), in digroups [3, 7] or in loops [24, 32], were examples of an action of an abelian object which is not a module, so where the commutator X _{Z} X, X _{Z} X ^{S} is bigger than Δ_{X}. Under (CC) the situation is different: considering f as a double extension

[figure: X; f; f; Z; Z; Z,]

in order to make use of (CC2) we would have to assume the stronger condition

A A, X ^{Huq} A, X ^{Huq} 0;

by (CC1) this already implies the stronger X _{Z} X, ∇_{X} ^{S} Δ_{X}, which defeats the purpose.

4.2. Relative commutators Many of the examples obtained in [20] through explicit calculations now become instances of Theorem 2.11, as do several other examples considered in the literature: groups vs. abelian groups, rings vs. zero rings, and Lie algebras vs. vector spaces, for instance. Nevertheless, there is still a whole class of examples missing, namely all those where the homology is not absolute, i.e., the functor which is being derived is not the abelianisation functor. Higher Hopf formulae exist e.g. for precrossed modules vs. crossed modules, groups vs. groups of a certain nilpotency or solvability class [20], loops vs. groups (in low dimensions) [22], compact groups vs. profinite groups [19] and Leibniz n-algebras vs. Lie n-algebras [13]. We hope to extend the results of the present paper to the relative case so that also these examples may become instances of the general theory. This problem is closely related to the results of [16, 22, 23], as it depends on a suitable notion of relative commutator. In the article [19] a solution is given for reflectors which are protoadditive.

4.3. Equivalence of (CC) and (SH) Another question we did not answer now is whether or not (CC) implies (SH). The problem is that already against (SH) alone the counterexamples are exotic, and now we would have to find a category which does not have (SH) but does satisfy (CC).

4.4. Exact Mal’tsev categories Under (CC), higher central extensions in a semiabelian category may be characterised in terms of binary Huq commutators. So under (SH), this characterisation may be reformulated using binary Smith commutators as follows.

4.5. Corollary. Given an n-fold extension F in a semi-abelian category with (SH), let F_{n} _{F_{n i}} F_{n} denote the kernel pair of f_{i}: F_{n} F_{n i} . Then

_{I n} _{i I} F_{n} _{F_{n i}} F_{n}, _{i n I} F_{n} _{F_{n i}} F_{n} ^{S} Δ_{F_{n}}

if and only if F is central.

We know, however, that when n 2 this characterisation of double central extensions is valid in all exact Mal’tsev categories: in [21], the proof given in the article [48] in a semiabelian context was replaced by a much more efficient one which avoids the use of the Huq commutator and doesn’t need that the category is pointed nor that it is protomodular but works in the exact Mal’tsev context. This naturally leads to the following conjecture:

4.6. Conjecture. The above characterisation of n-fold central extensions is also valid in exact Mal’tsev categories.

That higher central extensions make sense in an exact Mal’tsev context is explained in [18]. The difficulty in 4.6 may be better understood when observing the difference in underlying geometry between the vanishing of the Smith commutators that occur in Corollary 4.5 on the one hand, and the characterisation of higher centrality given in [49]— which is also geometrical in nature, and makes sense in the exact Mal’tsev context—on the other. One could argue that this latter characterisation of higher centrality leads to a “higher-order Smith commutator”. This would be just one n-ary Smith commutator involving higher-order diamonds, instead of a join of several binary Smith commutators, each of which only gives rise to a fragment of the geometry of those higher-order diamonds. The question now essentially becomes whether the characterisation of double central extensions in terms of binary Smith commutators is a coincidence typical for degree two or not.

4.7. Higgins instead of Smith Even when a semi-abelian category does not have the property (CC), the double central extensions in it may still be characterised in terms of Higgins commutators. The only problem is that binary commutators will not suffice, but rather a ternary commutator is needed: the result in [32] says that (C) is central when the join K L, X K, L K, L, X vanishes. An unpublished result by Tomas Everaert, on which the proof of Theorem 2.8 was based, gives the higher-dimensional analogue. It says that an n-fold extension F in a semi-abelian category is central if and only if the join of higher-order Higgins commutators [31]

_{I_{0} I_{k} I n} _{i I_{0}} Ker f_{i} , …, _{i I_{k}} Ker f_{i}

vanishes. The size of the commutators stays bounded, and the join finite, as a commutator in which an entry is repeated is smaller than the commutator with the repetition removed. In fact, all three types of commutators—Higgins, Huq, Smith—may be seen as instances of the weighted commutator introduced by Gran, Janelidze and Ursini ([27], see also [42]).

Acknowledgement

We are grateful to Tomas Everaert and Marino Gran for fruitful discussions on the subject of the paper.

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

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