A Generalization of the Ehresmann-Schein-Nambooripad Theorem to Two-Sided Ehresmann Semigroupoids

TL;DR

This paper generalizes the Ehresmann-Schein-Nambooripad Theorem to two-sided Ehresmann semigroupoids, establishing a correspondence with local biordered Ehresmann categories, unifying inverse semigroupoids and Ehresmann semigroups.

Contribution

It introduces two-sided Ehresmann semigroupoids and their categorical correspondence, extending previous results to a broader class of algebraic structures.

Findings

Established a correspondence between two-sided Ehresmann semigroupoids and local biordered Ehresmann categories.

Extended the theory to two-sided restriction semigroupoids and characterized their associated categories.

Unified the frameworks of inverse semigroupoids and Ehresmann semigroups under a common categorical perspective.

Abstract

We introduce the notion of two-sided Ehresmann semigroupoids and show that they are in correspondence with a specific class of categories, which we call local biordered Ehresmann categories. This correspondence provides a unified generalization of the Ehresmann-Schein-Nambooripad Theorem for both inverse semigroupoids and Ehresmann semigroups. In particular, two-sided restriction semigroupoids form a distinguished subclass of two-sided Ehresmann semigroupoids, and for this case we describe the associated class of categories, extending earlier results for restriction semigroups.