A Categorical Realization of the (2-)Category of Monoids via Sch{ü}tzenberger Categories and Strict Factorization Systems

Abstract

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

arXiv comment15 pages