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

2023/11/10 by Emmanuel Jerez, Jerez, Emmanuel · 1 citation
Mathematics · Medicine · #18A32 #18B40 #20C99 Secondary 18A30 #43A65 #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications #Primary 16W22

paper · pdf · doi:10.48550/arxiv.2311.06223

openalex publication_date 2023/11/10 · openalex created_date 2023/11/14 · openalex updated_date 2026/07/28

Abstract

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: Grpd → PA and Ψ: PA → Grpd, with the property that ΨΦ≅ 1_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.

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