  The goal of this thesis is to define a 2-dimensional version of abelian
categories, where symmetric 2-groups play the role that abelian groups played
in 1-dimensional algebra. Abelian and 2-abelian groupoid enriched categories
are defined and it is proved that homology can be developed in them, including
the existence of a long exact sequence of homology corresponding to an
extension of chain complexes. This generalises known results for symmetric
2-groups. The examples include, in addition to symmetric 2-groups, the
2-modules on a 2-ring, which form a 2-abelian groupoid enriched category.
Moreover, internal groupoids, functors and natural transformations in an
abelian category C (in particular, Baez-Crans 2-vector spaces) form a 2-abelian
groupoid enriched category if and only if the axiom of choice holds in C.
