2025/09/17 by Davide Ferri, Ferri, Davide
Mathematics · #18M05 #20L05 #Advanced Operator Algebra Research #Category Theory (math.CT) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2509.13973
openalex publication_date 2025/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Groupoids are the oidification of groups, and they are largely used in topology and representation theory. We consider here the category Gpd of all groupoids with all morphisms, and the category GpdΛ of groupoids over a fixed set of vertices Λ, with morphisms fixing Λ. In GpdΛ, a First Isomorphism Theorem is already well known; see Ávila, Marín, and Pinedo (2020). Famously, the First Isomorphism Theorem fails to hold in Gpd. However, we retrieve here a universally lifted version of the First Isomorphism Theorem in Gpd, through the definition of virtual kernels. Semidirect products of a group by a groupoid are well known. We define crossed products in Gpd, and prove that they are equivalent to split epimorphisms, i.e. that they are the `categorial' notion of semidirect product in Gpd in the sense of Bourn and Janelidze (1998). We observe that in GpdΛ crossed products and semidirect products are essentially equivalent, under mild assumptions, and our Split Lemma in Gpd collapses to a much simpler Split Lemma in GpdΛ that appears in Metere and Montoli (2010) and Ibort and Marmo (2023).