On Ehresmann semigroups

TL;DR

This paper introduces a new categorical approach to Ehresmann semigroups using étale actions, characterizes those arising from finite categories, and explores embeddings into inverse semigroups.

Contribution

It presents an alternative categorical framework for Ehresmann semigroups and characterizes their finite-category representations, advancing the understanding of their structure and embeddings.

Findings

Ehresmann semigroups can be described via étale actions of meet semilattices.

Characterization of Ehresmann semigroups from finite categories.

Every restriction semigroup can be embedded into one constructed from a category.

Abstract

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right \'etale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.