Abelian categories in dimension 2

TL;DR

This thesis develops a 2-dimensional analogue of abelian categories using symmetric 2-groups, establishing homology theory and examples like 2-modules and 2-vector spaces, with results dependent on the axiom of choice.

Contribution

It introduces a 2-dimensional framework for abelian categories, extending homology and providing new examples such as 2-modules and 2-vector spaces.

Findings

Homology theory extends to 2-abelian categories.

Existence of long exact sequences in 2-abelian settings.

Internal groupoids form 2-abelian categories iff the axiom of choice holds.

Abstract

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.