Abelian categories in dimension 2

Details Abelian categories in dimension 2 Abelian categories in dimension 2

2008-09-10

arxiv.org/abs/0809.1760v1

Mathematics

Category Theory

x + 268 pages. This is the English version of my PhD thesis; the original French version is available at http://edoc.bib.ucl

Scientific paper

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.

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