A Generalization of the Ehresmann-Schein-Nambooripad Theorem to Two-Sided Ehresmann Semigroupoids
Rafael Haag, Thaísa Tamusiunas
TL;DR
The work develops a unified ESN-type framework that embeds both inverse semigroups and Ehresmann semigroups within two-sided Ehresmann semigroupoids and their associated local biordered Ehresmann categories. It constructs mutual translations $S o C(S)$ and $C o S( rak C)$, proving mutual inverses and thus object-level isomorphisms that generalize the classical ESN correspondence. By analyzing morphisms, the authors establish three parallel isomorphisms: (i) $(2,1,1)$-morphisms correspond to inductive functors between local biordered categories; (ii) $igvee$-premorphisms correspond to ordered functors for restriction semigroupoids; and (iii) $igwedge$-premorphisms correspond to inductive prefunctors. The results recover known ESN-type correspondences for restriction and inverse cases as well as inverse semigroupoids, while providing new complete and local formulations via complete local biordered Ehresmann categories and locally inductive structures, with clear implications for the study of partial symmetries and their categorical encodings.
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.