On direct product, semidirect product of groupoids and partial actions

TL;DR

This paper explores constructions of groupoids such as direct and semidirect products, providing conditions for embeddings and equivalences with partial actions, advancing the understanding of groupoid structures and their categorical relationships.

Contribution

It introduces new conditions for embedding groupoids into direct products and characterizes when semidirect products are actually direct, linking partial actions with categorical equivalences.

Findings

Conditions for embedding groupoids into direct products

Criteria for when a semidirect product is direct

Equivalence between partial groupoid actions and star injective functors

Abstract

We present some constructions of groupoids as: direct product, semidirect product, and we give necessary and sufficient conditions for a groupoid to be embedded into a direct product of groupoids. Also, we establish necessary and sufficient conditions to determine when a semidirect product is direct. Moreover, we establish an equivalence between the category of strict partial groupoid actions and the category of star injective functors. Finally, we give a relation of categorical type between the actions groupoids (G, X) and (G, Y), being Y a universal globalization of X.