Crossed modules of monoids III. Simplicial monoids of Moore length 1

TL;DR

This paper establishes an equivalence between relative simplicial monoids of Moore length 1 and relative categories of monoids, extending the understanding of categorical structures within monoids and their simplicial representations.

Contribution

It proves the equivalence between relative simplicial monoids of Moore length 1 and relative categories of monoids, completing a series of related studies.

Findings

Proves the equivalence via truncation and nerve construction.

Introduces Moore length for simplicial monoids.

Applies to categories of monoids and bimonoids in symmetric monoidal categories.

Abstract

This is the last 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 third part relative simplicial monoids are analyzed. Their Moore length is introduced and the equivalence is proven between relative simplicial monoids of Moore length 1, and relative categories of monoids in Part I. This equivalence is obtained in one direction by truncating a simplicial monoid at the first two degrees; and in the other direction by taking the simplicial nerve of a relative category.