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Homology and homotopy in semi-abelian categories

maandag, 16 januari, 2006 - 16:00
Campus: Brussels Humanities, Sciences & Engineering campus
Faculteit: Science and Bio-engineering Sciences
Tim Van Der Linden

The theory of abelian categories proved very useful, providing an axiomatic framework for
homology and cohomology of modules over a ring and, in particular, abelian groups. For many
years, a similar 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, Marki 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.
Examples include all abelian categories; all varieties of Omega-groups (i.e., varieties of universal
algebras with a unique constant and an underlying group structure), in particular the categories of
groups, non-unital rings, Lie algebras, crossed modules, commutative algebras; internal versions of
these varieties in an exact category; Heyting algebras and semilattices; compact Hausdorff (profinite)
groups (or semi-abelian algebras); non-unital C* algebras; the dual of the category of pointed objects
in any topos, in particular the dual of the category of pointed sets. Important differences between an
abelian category and a semi-abelian one are that in a latter category, not every monomorphism is a
kernel: e.g., in the case of groups, not every subgroup is a normal subgroup; there is no enrichment:
in the case of groups, the pointwise product of two group homomorphisms need not be a
homomorphism, let alone that this operation defines a group structure; and there are no biproducts:
in the case of groups, binary "cartesian" products and coproducts (i.e., "free products") need not

Keeping these differences into account, it is possible to consider homology of (proper) chain
complexes and of simplicial objects. Using techniques from commutator theory and the theory of
Baer invariants, and generalizing Barr and Beck's cotriple homology, a general version of Hopf's
formula and the Stallings-Stammbach sequence in homology are obtained. These results also have a
cohomological counterpart: as in the case of groups or Lie algebras, the second cohomology group
classifies central extensions, and we acquire a general version of the Hochschild-Serre sequence. We
prove a universal coefficient theorem to explain the connection between homology and cohomology.

On the homotopical level, we show that Quillen model category structures for simplicial objects and
for internal categories (or crossed modules) exist, compatible with these notions of homology.