Abelian categories in dimension 2
Mathieu Dupont

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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra
