Monoid extensions and the Grothendieck construction

TL;DR

This paper explores how the Grothendieck construction, a categorical tool, can be applied to classify various types of monoid extensions, extending known group extension theories to monoids.

Contribution

It introduces a generalized Grothendieck construction and demonstrates its application to classify monoid extensions, including Schreier and weakly Schreier types.

Findings

Classification of certain monoid extensions using the Grothendieck construction

Connection between group extension theory and monoid extensions

Framework for understanding monoid extensions categorically

Abstract

In category theory circles it is well-known that the Schreier theory of group extensions can be understood in terms of the Grothendieck construction on indexed categories. However, it is seldom discussed how this relates to extensions of monoids. We provide an introduction to the generalised Grothendieck construction and apply it to recover classifications of certain classes of monoid extensions (including Schreier and weakly Schreier extensions in particular).