The category of partial group actions: quotients, (co)limits and groupoids

We consider the category of partial actions, where the group and the set upon which the group acts can vary. Within this framework, we develop a theory of quotient partial actions and prove that this category is both (co)complete and encompasses the category of groupoids as a full subcategory. In particular, we establish the existence of a pair of adjoint functors, denoted as $\mathrm{\Phi }:\mathbf{\text{Grpd}}\to \mathbf{\text{PA}}$ and $\mathrm{\Psi }:\mathbf{\text{PA}}\to \mathbf{\text{Grpd}}$, with the property that $\mathrm{\Psi }\mathrm{\Phi }\cong 1_{\mathbf{\text{Grpd}}}$. Next, for a given groupoid $\mathrm{\Gamma }$, we provide a characterization of all partial actions that allow the recovery of the groupoid $\mathrm{\Gamma }$ through $\mathrm{\Psi }$. This characterization is expressed in terms of certain normal subgroups of a universal group constructed from $\mathrm{\Gamma }\mathrm{.}$