  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. For many years, a similar categorical framework
has been lacking for non-abelian (co)homology, the subject of which includes
the categories of non-abelian groups, of rings, Lie algebras, etc. The notion
of semi-abelian category due to Janelidze, Marki and Tholen allows a unified
study of many important homological properties of such categories. For
instance, in any semi-abelian category, the classical diagram lemmas (the Short
Five Lemma, the 3x3 Lemma, the Snake Lemma, Noether's Isomorphism Theorems)
hold. The point of this thesis is, that the semi-abelian framework is indeed
perfectly suited for the study of non-abelian (co)homology and the
corresponding homotopy theory. It describes the foundations for a categorical
homology theory that unifies many basic aspects of the classical (co)homology
theories of e.g., groups, Lie algebras and crossed modules. Starting from
homology of (proper) chain complexes and of simplicial objects, a semi-abelian
version of Barr-Beck cotriple homology is introduced. A systematic study of the
theory of Baer invariants in this context yields categorical versions of Hopf's
formula (describing the second homology object in terms of commutators) and the
Stallings-Stammbach sequence. We obtain a Universal Coefficient Theorem, a
version of the Hochschild-Serre sequence, and an interpretation of the second
cohomology group in terms of isomorphism classes of central extensions. Also,
Quillen model category structures for homotopy of simplicial objects and
internal categories are studied, and are shown to be compatible with the
respective semi-abelian notions of homology.
