2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/168820The 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.x + 268 pages. This is the English version of my PhD thesis; the original French version is available at http://edoc.bib.ucl.ac.be:81/ETD-db/collection/available/BelnUcetd-06112008-231800/Category Theory18E10; 18D05; 18A20; 18A22; 18A32; 18B40; 18D10; 18E05; 18G35text