The category of partial group actions: quotients, (co)limits and groupoids
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913324453593088 |
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| author | Jerez, Emmanuel |
| author_facet | Jerez, Emmanuel |
| 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 $Γ.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_06223 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The category of partial group actions: quotients, (co)limits and groupoids Jerez, Emmanuel Group Theory Primary 16W22, 20C99 Secondary 18A30, 18A32, 43A65, 18B40 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 $Γ.$ |
| title | The category of partial group actions: quotients, (co)limits and groupoids |
| topic | Group Theory Primary 16W22, 20C99 Secondary 18A30, 18A32, 43A65, 18B40 |
| url | https://arxiv.org/abs/2311.06223 |