Equivalences for the (2-)categories of monoids and unital semigroups

TL;DR

This paper constructs categorical equivalences for monoids and unital semigroups using strict factorization systems, extending to 2-categories to analyze Morita equivalence.

Contribution

It introduces new categorical frameworks and 2-equivalences for monoids and unital semigroups, facilitating Morita theory analysis.

Findings

Established equivalences between categories of monoids and certain structured categories.

Extended equivalences to 2-categories with natural transformations.

Provided a framework for studying Morita equivalence of monoids.

Abstract

We construct a category equivalent to the category $\mathbf{Mon}$ of monoids and monoid homomorphisms, based on categories with strict factorization systems. This equivalence is then extended to the category $\mathbf{Mon_s}$ of unital semigroups and semigroup homomorphisms. By introducing suitable natural transformations, we turn these equivalences into 2-equivalences between 2-categories. The 2-category $\mathbf{Mon_s^2}$ constructed this way proves the good one to study Morita equivalence of monoids.