Towards Categorical Behaviour of Groups Abstract 1. Introduction

Feature Article

Towards categorical behaviour of groups

Maria Manuel Clementino

CMUC/Departamento de Matem´atica, Universidade de Coimbra 3001-454 Coimbra, PORTUGAL

Abstract

We present a brief introduction to the study of homological categories, which en- compass many algebraically-like categories, namely the category of groups, and also categories of topological algebras.

1. Introduction of the homology and cohomology of a topological space. This, in turn, suggested the problem of describing those categories in which the values of such a homology the- Category Theory started more than 60 years ago, when ory could lie. After discussions with Eilenberg, this was Eilenberg and Mac Lane wrote a paper on natural done by Mac Lane [18, 19]. His notion of an “abelian transformations [11], having as role model a well-known bicategory” was clumsy, and the subject languished until example of natural equivalence: Buchsbaum’s axiomatic study [10] and the discovery by For any vector space V over a field K, consider its dual Grothendieck [15] that categories of sheaves (of abelian V ∗ (i.e. the vector space of linear functionals from V groups) over a topological space were abelian categories to K). If V is finite-dimensional, then V ∗ has the same but not categories of modules, and that homological alge- dimension as V , and so we can conclude that they are bra in these categories was needed for a complete treat- isomorphic, although there is no natural way of defining ment of sheaf cohomology (Godement [14]). With this such isomorphism. This contrasts with the case of V ∗∗, impetus, abelian categories joined the establishment.” ∗ the dual of V . There is a (injective) linear transfor- Moreover, quoting now Borceux [2]: ∗∗ mation φV : V → V , which assigns to each x ∈ V the linear transformationx ˆ : V ∗ → K, f 7→ xˆ(f) := f(x). An elementary introduction to the theory of abelian categories culminates generally with the proof of the basic Moreover, if V is finite-dimensional, φV turns out to be an isomorphism, defining this way a natural equivalence. diagram lemmas of homological algebra: the five lemma, the nine lemma, the snake lemma, and so on. This gives That is, φV is not a mere equivalence between V and ∗∗ evidence of the power of the theory, but leaves the reader V but it is part of a collection φ = (φV )V of equivalences. To make this idea precise, in [11] Eilenberg and with the misleading impression that abelian categories Mac Lane defined categories, functors (between cate- constitute the most natural and general context where gories) and then natural transformations (between func- these results hold. This is indeed misleading, since all tors). those lemmas are valid as well – for example – in the category of all groups, which is highly non-abelian. Shortly after, the use of Category Theory proved to However, in contrast with the smooth genesis of abelian be useful in several areas of Mathematics. The notion categories, besides several attempts to identify relevant of abelian category – encompassing abelian groups and, categorical features of the category of groups (cf. [16] more generally, modules – became prominent. Quoting for an account on the subject), it took a few decades Mac Lane [20, Notes on Abelian Categories, page 209]: until the right ingredient was identified. This was due “Shortly after the discovery of categories, Eilenberg and to Bourn [6], who defined protomodular category and Steenrod [12] showed how the language of categories and showed that a simple categorical condition (see condi- functors could be used to give an axiomatic description tion (2) of Theorem 1) could become a key tool to han-

1 dle with short exact sequences. In 1990, he presented and u = ker p, v = ker q, if a and c are isomorphisms, b his notion at the International Category Theory Meet- is an isomorphism as well. ing, CT90, but it took almost a decade until the math- ematical community understood the potential of proto- Bourn observed that the Split Short Five Lemma holds modularity. Indeed, the proposal, at CT99 (in Coim- in a pointed category C with pullbacks of split epimor- bra), of studying semi-abelian categories – which are in phisms if and only if the kernel functor particular protomodular categories –, due to Janelidze, K : P tC −→ C /0 × C M´arkiand Tholen [16], was the main step for the recog- f nition of the role of protomodularity in Categorical Al- ( X o / Y ) 7−→ (Ker f → 0,X) gebra. The XXI century began with an explosion of s results on protomodular and semi-abelian categories, is conservative, i.e. reﬂects isomorphisms. Here P tC is being Bourn the main contributor. The monograph by the category of split epimorphisms – or pointed objects Borceux and Bourn [3] contains the main achievements – of C , i.e. pairs (f, s) with f · s = 1; a morphism on the subject. While writing this monograph, the work (f, s) → (f 0, s0) in P tC is a pair of morphisms of C of Borceux (together with the author of this article) on (h, k) making the following diagram topological semi-abelian algebras [4, 5] led him to a new proposal for capturing the essential properties of group- h X / X0 like categories, which became known under the name O O homological category, basically because it turned out to f s f 0 s0 be the right setting to develop Homological Algebra. Y / Y 0 Throughout we will present a brief survey on these con- k tributions. commute (that is k · f = f 0 · h and s0 · k = h · s). To avoid the assumption of C being pointed, one can 2. Protomodularity focus on the second component of this functor, that is, on the functor which assigns to each object (f, s) of P tC the codomain of f (=domain of s): One of the key tools for Homological Algebra is the Short Five Lemma, which holds in abelian categories: p : P tC −→ C Short Five Lemma. Given a commutative diagram (f, s) 7−→ cod f,

u p which is a ﬁbration, the so-called ﬁbration of pointed 0 / K / X / Y / 0 objects of C . a b c If C has split pullbacks, every morphism v : X → Y in v q 0 / K0 / X0 / Y 0 / 0 C induces, via pullback, the change-of-base functor

∗ with exact rows (i.e. p, q are regular epimorphisms and v : P tY C −→ P tX C . u = ker p, v = ker q), if a and c are isomorphisms, b is an isomorphism as well. Proposition 1. [6] Let C be a category with split pullbacks. If C has split pushouts (i.e. admits pushouts This result is still valid in the category rp of groups G of split monomorphisms), then the change-of-base func- and homomorphisms, hence it is not exclusive of abelian tors of the fibration p have left adjoints (i.e. p is also categories. The notion of protomodular category is a cofibration). Conversely, if p is a cofibration and C based on a weaker form of this result, the Split Short admits finite products, then C has split pushouts. Five Lemma, stated below. This statement makes sense only in pointed categories, that is categories with a zero The remarkable novelty of Bourn’s protomodularity is object. the recognition of the role played by these functors: Definition. In a pointed category C , the Split Short Definition. [6] A category with split pullbacks is Five Lemma holds if, for any given commutative dia- C protomodular if the change-of-base functors of the fi- gram bration p : P tC → C are conservative. u s 0 / K / X o Y / 0 p / Protomodularity can be stated alternatively as a very a b c simple condition on pullbacks. v t 0 / K0 / X0 o / Y 0 / 0 Theorem 1. [6] A category C is protomodular if and q only if: in the sense that b · u = v · a, c · p = q · b and b · s = t · c, with p, q split epimorphisms, p · s = 1Y and q · t = 1Y 0 , (1) C has split pullbacks;

2 (2) If in the commutative diagram (2) If C is pointed, then:

/ / (a) f is a monomorphism ⇔ Ker f = 0; 1 p 2 (b) f is a regular epi ⇔ f = coker(ker f). / / the down-arrows are split epimorphisms and 1 and 1 2 are pullbacks, then 2 is also a pullback. There is an interesting approach to normal subobjects in general protomodular categories that we will not de- Moreover, in (2) one can assume only that p is a split scribe here (cf. [3, 7]). epimorphism, provided that C has pullbacks. Theorem 2. Let C be a pointed category with split pullbacks. The following conditions are equivalent: 3. Semi-abelian categories

(i) C is protomodular; An abelian category is an additive category, with ker- (ii) The Split Short Five Lemma holds in C . nels and cokernels, and such that every monomorphism is a kernel and every epimorphism is a cokernel. We re- Examples. ([3]) call that an additive category is a pointed category with biproducts (i.e. finite products are biproducts, hence 1. Every additive category with finite limits is pro- also coproducts) and with an additive abelian group tomodular, hence every abelian category is proto- structure in each hom-set so that composition of arrows modular. is bilinear with respect to this addition (cf. [20, 13]). 2. G rp is protomodular. Alternatively, an abelian category can be defined by the 3. The dual category of an elementary topos is pro- following two axioms: tomodular; in particular, S etop is protomodular. 4. If C is protomodular and X ∈ C , then the slice (1) C has finite products, and a zero object, category C /X and the coslice category X\C are protomodular. (2) C has (normal epi, normal mono)-factorizations, i.e. every morphism factors into a cokernel fol- 5. If C is protomodular, all the fibres P tY (C ) = lowed by a kernel. p−1(Y ) of the fibration of points of C are protomodular. Observing that these two conditions imply that is 6. If C is protomodular and finitely complete, then C finitely complete and finitely cocomplete, so that con- its category G rd(C ) of internal groupoids is protomodular as well. dition (1) could be stated self-dually, and that condition (2) is obviously self-dual, it is clear that the notion of In Section 4 we will present a characterization of the abelian category is self-dual, that is: protomodular varieties, i.e. of the varieties of universal op algebras which are protomodular, as categories. C is abelian ⇔ C is abelian. As for G rp, monomorphisms in pointed protomodular categories do not need to be kernels. They behave how- Roughly speaking, to define semi-abelianess Janelidze, ever quite nicely. We select here some of the properties Márkiand Tholen replaced additivity by protomodu- of monomorphisms and regular epimorphisms in proto- larity, in a convenient way. The bridge they used was modular categories. Barr-exactness.

Proposition 2. Let C be a ﬁnitely complete protomod- We recall that a category C is Barr-exact [1] if ular category.

(1) Pulling back reflects monomorphisms, i.e. given (1) C has finite products, a pullback g0 (2) C has pullback-stable (regular epi, mono)- / factorizations, f 0 f g / (3) Every equivalence relation is effective (i.e. the f is a monomorphism provided that f 0 is. kernel pair of some morphism).

3 If C satisﬁes (1) and (2) it is called regular. (3) one (n + 1)-ary term θ satisfying

A category is abelian if, and only if, it is additive θ(α1(x, y), ··· , αn(x, y), y) = x. and Barr-exact. Barr-exactness by itself is not restric- tive enough to capture essential properties of group-like A variety V is semi-abelian if, and only if, it fulfils categories. It includes, for instance, pointed sets and conditions (1)-(3) with e1 = ··· = en = 0. monoids. In case is the variety of groups, in the Theorem we The authors of [16] observed that categories which are V put n = 1, α(x, y) = x − y and θ(x, y) = x + y. It both Barr-exact and protomodular – including of course is clear that any variety which contains a unique con- G rp – are very well-behaved. Namely, for a Barr-exact stant and a group operation is semi-abelian. This is in category with split pushouts, protomodularity is equiv- particular the case of groups, abelian groups, Ω-groups, alent to the existence of semi-direct products (cf. [8]). modules on a ring, rings or algebras without units, Lie They proposed the following: algebras, Jordan algebras. Any semi-abelian variety has Definition. A category C is semi-abelian if it is a Mal’cev operation p, defined as pointed, protomodular and Barr-exact. p(x, y, z) = θ(x, α1(y, z), ··· , αn(y, z)). It is interesting to notice that, although to be semi- (For more examples, see [4].) It is particularly interest- abelian is not self-dual, ing to study the corresponding topological algebras. op C is abelian ⇔ C and C are semi-abelian. Let C be the category of topological algebras for a given semi-abelian variety V . That is, objects of C This result follows essentially from the following are elements of V equipped with a topology making the operations continuous, and morphisms of C are Proposition 3. If C is pointed and both C and C op continuous homomorphisms. Our basic example is of are protomodular, then C has biproducts. course the category T opG rp of topological groups and continuous group homomorphisms. The main ingredi- In semi-abelian categories many algebraic results are ent in the study of classical properties of topological valid, specially results involving the behaviour of exact groups is the existence of the homeomorphisms sequences (see Section 5). We refer to [16] for a very (−)+x interesting incursion into this subject, and to [3, 2] for G / G a thorough study of the properties of semi-abelian cat- (x ∈ G) which, although not living in T opG rp (they egories. are not homomorphisms), show that – topologically – G is homogeneous, i.e. its local properties do not depend on the point x considered. For topological semi-abelian 4. Semi-abelian varieties and algebras one replaces this set of homeomorphisms by a set of sections and retractions, as follows. topological algebras Let A ∈ C . Condition (3) of the Theorem asserts that, for each a ∈ A, the continuous maps The varieties of groups, of loops (or more generally n semi-loops), of cartesian closed (distributive) lattices, ιa : A −→ A of locally boolean distributive lattices, are varieties of x 7−→ (α1(x, a), ··· , αn(x, a)) universal algebras which are semi-abelian categories. In and n fact, for a variety, to be pointed protomodular is equiv- θA : A −→ A alent to be semi-abelian, since it fulfils always the ex- (x1, ··· , xn) 7−→ θ(x1, ··· , xn, a) actness condition. satisfy θa · ιa = 1A, hence present A as a topological re- n In 2003 Bourn and Janelidze [9] characterized semi- tract of A . Condition (2) says that ιa(a) = (0, ··· , 0), abelian (in fact protomodular) varieties as those having which allows the comparison between local properties a finite family of generalized “subtractions” and a gen- at a and at 0. Indeed, from these properties one may eralized “addition”, as stated below. conclude that, for any a ∈ A, each of the sets

−1 Theorem 3. A variety V of universal algebras is pro- {ιa (U × · · · × U) | U open neighbourhood of 0} tomodular if and only if, for a given n ∈ N, it has: and

(1) n 0-ary terms e1, ··· , en; {θa(U × · · · × U) | U neighbourhood of 0}

(2) n binary terms α1, ··· , αn with αi(x, x) = ei for is a fundamental system of neighbourhoods of a, the for- all i = 1, ··· , n; mer one consisting of open neighbourhoods.

4 A convenient use of these properties guides us straight- As in any pointed category, in a homological category forward to the establishment of most of the classical a sequence of morphisms topological properties known for topological groups. k f (For details see [4, 5].) Here we would like to mention 0 / K / X / Y / 0 one important property, which in fact follows directly from the existence of a Mal’cev operation: is a short exact sequence if k = ker f and f = coker k. Since in a pointed protomodular category every regular In C a morphism is a regular epimorphism if and only epimorphism is the cokernel of its kernel, in a homolog- if it is an open surjection. k f ical category 0 / K / X / Y / 0 is a We remark that a well-known important property of short exact sequence if, and only if, k = ker f and f is topological groups, namely that the profinite topologi- a regular epimorphism. cal groups – i.e. projective limits of finite discrete topological groups – are exactly the compact and totally Furthermore, in a homological category the (regular epi, disconnected groups, in general is not true for semi- mono)-factorization of a morphism is obtained like in abelian topological varieties. For instance, the result abelian categories, i.e. if f = m · e is the (regular epi, fails for topological Ω-groups. It remains an open prob- mono)-factorization of f, then e = coker(ker f). Hence lem to characterize those varieties for which this equal- every kernel has a cokernel and, moreover, every kernel ity holds. (Cf. Johnstone [17, Chapter 6] for more is the kernel of its cokernel. results on the subject.) Using (regular epi, mono)-factorizations, one can define exact sequences as follows. Analyzing now the categorical behaviour of such categories of topological semi-abelian algebras it is easy Definitions. (1) In a homological category a sequence to check that protomodularity is inherited from proto- of morphisms modularity of V , but not exactness, hence they are not f g semi-abelian. Indeed, the kernel pair of a continuous X / Y / Z homomorphism f : G → H between, say, topological is exact if, in the (regular epi, mono)-factorizations of groups, is constructed like in G rp and it inherits the 0 subspace topology of the product topology on G × G. f and g, m = ker e : Hence, any equivalence relation on G provided with a f g X / Y / Z topology which is strictly finer than the subspace topol- B > C = BB || CC {{ ogy of G × G is not a kernel pair, hence equivalence e B! || m e0 C! {{m0 relations are not effective. M M 0 However, regularity is guaranteed, since the (regular (2) A long exact sequence of composable morphisms is epi, mono)-factorization of a morphism f : A → B is exact if each pair of consecutive morphisms forms an obtained via the (regular epi, mono)-factorization in V exact sequence. f A / B B = In a homological category a morphism f : X → Y can BB || f g e B ||m be part of an exact sequence only M X / Y / Z if, in its (regular epi, mono)-factorization f = m · e, equipping M with the quotient topology, which makes e m is a kernel. (Such morphisms are called proper.) necessarily an open map. Since open maps are pullback- Still, in a homological category exact sequences identify stable, the factorization is pullback-stable as claimed. monomorphisms and regular epimorphisms as follows. At this stage one can raise the question: Are pointed Proposition 4. [3] If C is a homological category and regular protomodular categories interesting? The an- f : X → Y is a morphism in C , then: swer is definitely yes. These are the homological categories we will consider in the next section. (1) f is monic if and only if the sequence f 0 / X / Y is exact; 5. Homological categories (2) k = ker f if and only if the sequence k f 0 / K / X / Y is exact; A category is said to be homological if it is pointed, C (3) f is a regular epimorphism if and only if the se- regular and protomodular. f quence X Y 0 is exact; Every semi-abelian category is homological, but there / / are interesting homological categories which are not (4) for a proper morphism f, q = coker f if and only semi-abelian, like T opG rp, and, more generally, any f q category of topological semi-abelian algebras. if X / Y / Q / 0 is exact.

5 Finally we would like to stress that the key results on [5] F. Borceux and M.M. Clementino, Topologi- short exact sequences are valid in this setting (cf. [3]): cal protomodular algebras, Topology Appl. 153 (2006), 3085–3100. Theorem 4. (Short Five Lemma) For a pointed regular category C , the following conditions are equivalent: [6] D. Bourn, Normalization equivalence, kernel equivalence and aﬃne categories, in: Springer (i) C is homological. Lect. Notes in Math. 1488, pp. 43–62, 1971. (ii) The Short Five Lemma holds, that is, given a commutative diagram [7] D. Bourn, Normal subobjects of topological groups and of topological semi-abelian algebras, u p 0 / K / X / Y / 0 Topology Appl. 153 (2006), 1341–1364.

a b c [8] D. Bourn and G. Janelidze, Protomodularity, q descent and semi-direct products, Theory Appl. 0 v 0 0 0 / K / X / Y / 0 Categ. 4 (1998), 37–46. with exact rows, if a and c are isomorphisms, b is [9] D. Bourn and G. Janelidze, Characterization also an isomorphism. of protomodular varieties of universal algebra, Theorem 5.( 3 × 3 Lemma) Let C be a homological Theory Appl. Categ. 11 (2002), 143–147. category. Consider the commutative diagram [10] David A. Buchsbaum, Exact categories and du- 00 k00 f ality, Trans. Amer. Math. Soc. 80 (1955), 1–34. 0 / K00 / X00 / Y 00 / 0 [11] S. Eilenberg and S. Mac Lane, General theory of u0 v0 w0 0 natural equivalences, Trans. Amer. Math. Soc. k0 f 0 / K0 / X0 / Y 0 / 0 58 (1945), 231–294. u v w [12] S. Eilenberg and N.E. Steenrod, Foundations of k f Algebraic Topology, Princeton Univ. Press 1952. 0 / K / X / Y / 0 where the horizontal lines are short exact sequences and [13] P. Freyd, Abelian Categories: An Introduction v · v0 = 0. Then, if two of the columns are short exact to the Theory of Functors, Harper & Row, New sequences, the third one is also a short exact sequence. York, 1964. Reprints in Theory and Applica- tions of Categories, n◦ 3 (2003), pp. 23-164. The Noether Isomorphisms Theorems are still valid [14] R. Godement, Théorie des Faisceaux, Paris, in homological categories, as well as the Snake Lemma Hermann 1958. (for exact formulations of these results see [3]). Further- more, one can associate to each short exact sequence [15] A. Grothendieck, Sur quelques points d’algèbre of chain complexes the long exact homology sequence, homologique, TôhokuMath. J. 9 (1957), 119– provided that the chain complexes are proper. 221.

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