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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.06223 |
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Table of Contents:
- 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 $Φ: \textbf{Grpd} \to \textbf{PA}$ and $Ψ: \textbf{PA} \to \textbf{Grpd}$, with the property that $ΨΦ\cong 1_{\textbf{Grpd} }$. Next, for a given groupoid $Γ$, we provide a characterization of all partial actions that allow the recovery of the groupoid $Γ$ through $Ψ$. This characterization is expressed in terms of certain normal subgroups of a universal group constructed from $Γ.$