Structure and classification of monoidal groupoids

TL;DR

This paper develops a 3-dimensional cohomological framework to classify monoidal groupoids with abelian isotropy groups, extending Schreier-Grothendieck theory to a non-abelian setting.

Contribution

It introduces a novel 3D cohomology approach for classifying monoidal groupoids, especially those with abelian isotropy groups, using Leech's cohomology.

Findings

Classification theorems for monoidal groupoids with abelian isotropy groups

Development of non-abelian factor set theory in a 3D context

Application of Leech's cohomology to classify monoidal functors

Abstract

The structure of monoidal categories in which every arrow is invertible is analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck theory of non-abelian factor sets for their classification. In particular, we state and prove precise classification theorems for those monoidal groupoids whose isotropy groups are all abelian, as well as for their homomorphisms, by means of Leech's cohomology groups of monoids.