The Szendrei Expansion of Restriction Semigroupoids
Rafael Haag, Wesley G. Lautenschlaeger, Thaísa Tamusiunas
TL;DR
The paper addresses the unification of restriction semigroupoid structures by introducing left restriction semigroupoids and proving a Wagner-Preston–type representation into the category of partial maps between fibers, $PT^{op}(\r pi)$, for some surjection $\pi: X \to Y$. It then constructs the Szendrei expansion $Sz(S)$ for a left restriction semigroupoid $S$ and proves a universal factorization property: any premorphism $\varphi:S\to T$ factors uniquely as $\varphi=\overline{\varphi}\iota$ with a restriction morphism $\overline{\varphi}:Sz(S)\to T$. This extends known correspondences for groups and inverse semigroups to left restriction semigroupoids and provides a global-to-partial translate for partial actions. Additionally, the expansion is shown to be an expansion in the sense of Bierger 1984, with specialization to restriction categories yielding analogous universal properties.
Abstract
We introduce the concept of a restriction semigroupoid S, which unifies the notion of restriction semigroups and restriction categories within a single structure. We prove a representation theorem, showing that every restriction semigroupoid can be embedded into a determined category of partial maps. Furthermore, we construct the Szendrei expansion Sz(S) of S and establish that each premorphism between two restriction semigroupoids S and T is uniquely factorized by a morphism between the Szendrei expansion Sz(S) and T.