Fibred categorical theory of obstruction and classification of morphisms

TL;DR

This paper develops a fibred categorical framework for understanding obstructions and classifications of morphisms, unifying various theories including monoidal functors, group extensions, and algebra extensions with non-abelian kernels.

Contribution

It introduces a new fibred categorical approach that generalizes existing theories of morphism classification and obstruction, with applications to algebra and group theory.

Findings

Unified framework for morphism classification and obstructions

Classification of algebra extensions via Hochschild cohomology

Application to non-abelian kernel algebra extensions

Abstract

We set up a fibred categorical theory of obstruction and classification of morphisms that specializes to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane theory of group extensions. Further applications are provided, as for example a classification of unital associative algebra extensions with non-abelian kernel in terms of Hochschild cohomology.