Crossed modules of monoids II. Relative crossed modules

TL;DR

This paper develops the theory of relative crossed modules of monoids, establishing their equivalence with relative internal categories in monoids, and extends the framework to include important examples like small categories and bimonoids.

Contribution

It introduces the concept of relative crossed modules of monoids and proves their equivalence with relative internal categories, generalizing previous structures in monoid theory.

Findings

Defined relative crossed modules of monoids.

Proved their equivalence with relative internal categories.

Included key examples like small categories and bimonoids.

Abstract

This is the second part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans - crossed modules of monoids relative to this class of spans - simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this second part we define relative crossed modules of monoids and prove their equivalence with the relative categories of Part I.