Homology and homotopy in semi-abelian categories Tim Van der Linden Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium E-mail address: [email protected] arXiv:math/0607100v1 [math.CT] 4 Jul 2006 2000 Mathematics Subject Classification. Primary 18G, 55N35, 55P05, 55P10, 18E10; Secondary 20J Introduction The theory of abelian categories proved very useful, providing an axiomatic framework for homology and cohomology of modules over a ring and, in particular, of abelian groups [43, 95, 53, 110]. Out of it (and out of Algebraic Geometry) grew Category Theory. But for many years, a similar categorical framework has been lacking for non-abelian (co)homology, the subject of which includes the categories of groups, rings, Lie algebras etc. The point of this dissertation is that semi- abelian categories (in the sense of Janelidze, M´arki and Tholen [78]) provide a suitable context for the study of non-abelian (co)homology and the corresponding homotopy theory. A semi-abelian category is pointed, Barr exact and Bourn protomodular with binary coproducts [78, 14]. Pointed means that it has a zero object: an initial object that is also terminal. An exact category is regular (finitely complete with pullback-stable regular epimorphisms) and such that any internal equivalence re- lation is a kernel pair [10]. A pointed and regular category is protomodular when the Short Five Lemma holds: For any commutative diagram ker p′ p′ K[p′] 2, 2, E′ 2, 2, B′ u v w K[p] 2, 2, E 2, 2, B ker p p such that p and p′ are regular epimorphisms, u and w being isomorphisms im- plies that v is an isomorphism [18]. Examples include all abelian categories; all varieties of Ω-groups (i.e., varieties of universal algebras with a unique constant and an underlying group structure [68]), in particular the categories of groups, non-unitary rings, Lie algebras, commutative algebras [14], crossed modules and precrossed modules [74]; internal versions of such varieties in an exact category [78]; Heyting semilattices [85]; compact Hausdorff (profinite) groups (or semi-abelian al- gebras) [15, 14], non-unital C∗ algebras [63] and Heyting algebras [19, 78]; the dual of the category of pointed objects in any topos, in particular the dual of the category of pointed sets [78]. In any semi-abelian category, the fundamental diagram lemmas, such as the Short Five Lemma, the 3 × 3 Lemma, the Snake Lemma [21] and Noether’s Iso- morphism Theorems [14] hold. Moreover, there is a satisfactory notion of homology of chain complexes: Any short exact sequence of proper chain complexes induces a long exact homology sequence [51]. It is hardly surprising that such a context would prove suitable for further de- velopment of non-abelian (co)homology. With this dissertation we provide some i ii INTRODUCTION additional evidence. We consider homology of simplicial objects and internal cat- egories, and cotriple (co)homology in the sense of Barr and Beck [9]. Using tech- niques from commutator theory [104, 72, 99, 75, 22, 28, 29] and the theory of Baer invariants [54, 93, 55, 50], we obtain a general version of Hopf’s For- mula [70] and the Stallings-Stammbach Sequence [105, 106] in homology—and their cohomological counterparts, in particular the Hochschild-Serre sequence [65]. We recover results, well-known in the case of groups and Lie algebras: e.g., the fact that the second cohomology group classifies central extensions. And although ho- motopy theory of chain complexes seems to be problematic, Quillen model category structures for simplicial objects and internal categories exist that are compatible with these notions of homology [49, 109]. * * * Chapter 1. Throughout the text, we shall attempt to formulate all results in their “optimal” categorical context. Sometimes this optimal context is the most general one; sometimes it is where most technical problems disappear. (It is, however, always possible to replace the conditions on a category A by “semi-abelian with enough projectives”, unless it is explicitly mentioned that we need “semi-abelian and monadic over Set”, and most of the time “semi-abelian” is sufficient.) In the first chapter we give an overview of the relevant categorical struc- tures: quasi-pointed and pointed, unital and strongly unital, regular and Barr exact, Mal’tsev, Bourn protomodular, sequentiable and homological, and semi-abelian cat- egories. This is what we shall loosely refer to as the “semi-abelian” context. We give examples and counterexamples, fix notations and conventions. We add those (sometimes rather technical) properties that are essential to making the theory work. Chapter 2. In Chapter 2 we extend the notion of chain complex to the semi-abelian context. As soon as the ambient category is quasi-pointed, exact and protomodular, homology of proper chain complexes—those with boundary operators of which the image is a kernel—is well-behaved: It characterizes exactness. Also, in this case, the two dual definitions of homology (see Definition 2.1.4) coincide. A short exact sequence of proper chain complexes induces a long exact homology sequence. We show that the Moore complex of a simplicial object A in a quasi-pointed exact Mal’tsev category A is always proper; this is the positively graded chain + complex N(A) ∈ PCh A defined by N0A = A0, n−1 NnA = K[∂i : An 2, An−1] i=0 \ ◦ and with boundary operators dn = ∂n i ker ∂i : NnA 2, Nn−1A, for n ≥ 1. When, moreover, A is protomodular, the normalization functor T N : SA 2, PCh+A is exact; hence a short exact sequence of simplicial objects induces a long exact homology sequence. To show this, we use the result that in a regular Mal’tsev category, every simplicial object is Kan [38], and every regular epimorphism of simplicial objects is a Kan fibration. These facts also lead to a proof of Dominique Bourn’s conjecture that for n ≥ 1, the homology objects HnA of a simplicial object A are abelian. INTRODUCTION iii Chapter 3. Extending the work of Fr¨ohlich, Lue and Furtado-Coelho, in Chap- ter 3, we study the theory of Baer invariants in the semi-abelian context. Briefly, this theory concerns expressions in terms of a presentation of an object that are independent of the chosen presentation. For instance, presenting a group G as a quotient F/R of a free group F and a “group of relations” R, [F, F ] R ∩ [F, F ] and [R, F ] [R, F ] are such expressions. Formally, a presentation p: A0 2, 2, A of an object A in a category A is a regular epimorphism. (In a sequentiable category, the kernel ker p: A1 2, 2, A0 of p exists, and then A is equal to the quotient A0/A1.) The category PrA is the full subcategory, determined by the presentations in A, of the category Fun(2, A) of arrows in A. A Baer invariant is a functor B : PrA 2, A that makes homotopic morphisms of presentations equal: Such are (f0,f) and (g0,g): p 2, q f0 A0 2, B0 g0 p q f A 2, B g 2, satisfying f = g. Baer invariants may be constructed from subfunctors of the identity functor of A. For instance, under the right circumstances, a subfunctor V of 1A induces functors PrA 2, A that map a presentation p: A0 2, A to K[p] ∩ V p V p 0 or 0 . V1p K[p] ∩ V0p Choosing, for every object A, a projective presentation p (i.e., with A0 projective), these Baer invariants induce functors A 2, A, which are respectively denoted by ∆V and ∇V . Ideally, A is an exact and sequentiable category and V is a proper subfunctor of 1A that preserves regular epimorphisms: i.e., V is a Birkhoff subfunctor of A. Birkhoff subfunctors correspond bijectively to Birkhoff subcategories in the sense of Janelidze and Kelly [75]: full and reflective, and closed in A under subobjects and regular quotients. Given a Birkhoff subcategory B, the functor V is the kernel of the unit of the adjunction. As a leading example, there is always the subcategory AbA of all abelian objects of A. In this case, one interprets a value V (A) of V as a commutator [A, A]. Baer invariants give rise to exact sequences. For instance, Theorem 3.3.10 states that, given an exact and sequentiable category with enough projectives A, and a Birkhoff subfunctor V of A, any short exact sequence f 0 2, K 2, 2, A 2, B 2, 1 in A induces, in a natural way, an exact sequence ∆V f Uf ∆V (A) ∆V (B) K U(A) U(B) 2, 2, V1f 2, 2, 2, 1. iv INTRODUCTION The symbol V1f that occurs in this sequence can be interpreted as a commutator [K, A] of K with A, relative to V . We relate it to existing notions in the field of categorical commutator theory and the theory of central extensions in the follow- ing manner. In the case of abelianization, the commutator V1 may be described entirely in terms of Smith’s commutator of equivalence relations [104], and the corresponding notion of central extension coincides with the ones considered by Smith [104] and Huq [72]. Relative to an arbitrary Birkhoff subcategory—not just the full subcategory of all abelian objects—the central extensions are the same as those introduced by Janelidze and Kelly [75]. Finally we propose a notion of nilpotency, relative to any Birkhoff subcategory of a semi-abelian category. An object is nilpotent if and only if is lower central series reaches zero (Corollary 3.5.6), and the objects of a given nilpotency class form a Birkhoff subcategory (Proposition 3.5.8).