2023/09/07 by Lisa Orloff Clark, Astrid an Huef, Clark, Lisa Orloff +5
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Discrete mathematics #Double groupoid #Equivalence (formal languages) #FOS: Mathematics #Geometric and Algebraic Topology #Homeomorphism (graph theory) #Homotopy and Cohomology in Algebraic Topology #Mathematics #Operator Algebras (math.OA) #Pure mathematics
paper · pdf · doi:10.48550/arxiv.2309.03413
openalex publication_date 2023/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the equivalence of two definitions of AF groupoid in the literature: one by Renault and the other by Farsi, Kumjian, Pask and Sims. In both definitions, an AF groupoid is an increasing union of more basic groupoids, called elementary groupoids. Surprisingly, the two definitions of elementary groupoid are not equivalent; they coincide if and only if the local homeomorphism that characterises them is a covering map.