Homology and homotopy in semi-abelian categories

The theory of abelian categories proved very useful, providing an ax-iomatic framework for homology and cohomology of modules over a ring (in particular, abelian groups) [5]. A similar framework has been lacking for non-abelian (co)homology, the subject of which includes the categories of groups and Lie algebras etc. The point of my thesis is that semi-abelian categories (in the sense of Janelidze, Márki and Tholen) provide a suitable context for non-abelian (co)homology and the corresponding homotopy theory. A semi-abelian category is pointed, Barr exact and Bourn protomodular with binary coproducts [6], [1]. Examples include all abelian categories; all varieties of Ω-groups (i.e. varieties of universal algebras with an underlying group structure), in particular the categories of groups, non-unital rings, Lie algebras, crossed modules; internal versions of these varieties; Heyting algebras ; the dual of the category of pointed objects in any topos, in particular the dual of the category of pointed sets. The main difference between an abelian category and a semi-abelian one is that in a latter category, not every monomophism is a kernel: e.g. in the case of groups, not every subgroup is a normal subgroup. Keeping this difference into account, it is possible to consider homology of chain complexes and of simplicial objects [3], [4]. Using techniques from commutator theory and generalizing Barr and Beck's cotriple homology, a general version of Hopf's formula and the Stallings-Stammbach sequence in homology and their cohomological counterparts are obtained. Results, well-known in the case of groups and Lie algebras, are recovered in this general context: e.g. the second cohomology group classifies central extensions. Although homotopy theory of chain complexes seems to be problematic, Quillen model category structures for simplicial objects and internal categories exist that are compatible with this notion of homology [2].