Crossed modules of monoids I. Relative categories

TL;DR

This paper develops the theory of relative pullbacks in monoid categories, establishing the foundation for studying internal categories, crossed modules, and simplicial monoids relative to spans, with applications to small categories and bimonoids.

Contribution

It introduces the concept of relative categories in monoids via pullbacks relative to spans, linking internal categories, crossed modules, and simplicial monoids in a unified framework.

Findings

Defined relative pullbacks in monoid categories.

Established equivalences among internal categories, crossed modules, and simplicial monoids.

Applied theory to small categories and bimonoids in symmetric monoidal categories.

Abstract

This is the first 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 first part the theory of relative pullbacks is worked out leading to the definition of a relative category.