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

TL;DR

The paper develops a 2-categorical framework for monoids by replacing ${\mathbf{Mon}}$ with small categories carrying a unital, thin, complete strict factorization system, via the Schützenberger category ${\mathbb{D}}(M)$ and the reconstruction functor $\Sigma$. It proves a pair of adjoint equivalences ${\mathbb{D}} \dashv {\Sigma}$ that realize ${\mathbf{Mon}} \simeq {\mathbf{uc-CSFS}}$ and ${\mathbf{Mon_s}} \simeq {\mathbf{uc-CSFS_s}}$, enabling a 2-categorical Morita theory. Natural transformations (2-cells) between semi-pointed functors correspond to intertwining elements (conjugations), yielding a robust notion of Morita equivalence in the 2-category ${\mathbf{Mon_s^2}}$. The main contribution is that adjoint equivalences in this 2-category precisely recover Morita equivalence of monoids, with explicit constructions using idempotents $e$ to produce equivalences between $M$ and $eMe$; this provides a direct, elementary 2-categorical account of Morita theory beyond topos-theoretic approaches.

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.

Figures (2)

• Figure 1.1: ${\mathbb{D}}(M)$ and composition
• Figure 1.2: Strict factorization in ${\mathbb{D}}(M)$

Theorems & Definitions (24)

• Example 1.1
• Example 1.2
• Example 1.3
• Example 1.4
• Example 1.5
• Example 1.6
• Example 1.7
• Proposition 1.8
• proof
• Corollary 1.9