Cohomology theory in 2-categories

TL;DR

This paper develops a cohomology theory for a 2-categorical analogue of abelian categories, including symmetric categorical groups, enabling the construction of long cohomology sequences in this higher categorical context.

Contribution

It introduces a new axiomatic, self-dual framework for cohomology in 2-categories, generalizing classical abelian category results to higher categories.

Findings

Constructed a long cohomology 2-exact sequence from extensions of complexes.

Provided a simplified, axiomatic approach analogous to abelian categories.

Included symmetric categorical groups as key examples.

Abstract

Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups $\mathrm{SCG}$ are being investigated. In this paper, we consider a 2-categorical analogue of an abelian category, in such a way that it contains $\mathrm{SCG}$ as an example. As a main theorem, we construct a long cohomology 2-exact sequence from any extension of complexes in such a 2-category. Our axiomatic and self-dual definition will enable us to simplify the proofs, by analogy with abelian categories.