2020/06/21 by Jordan Mitchell Barrett, Barrett, Jordan Mitchell, Valentino Vito +1
Computer Science · Mathematics · #03C05 #08A30 #18A32 (Primary) #18C05 (Secondary) #Advanced Algebra and Logic #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings and Algebras (math.RA) #math.CT #math.LO #math.RA #msc:03C05 #msc:08A30 #msc:18A32 #msc:18C05
paper · pdf · doi:10.48550/arxiv.2006.11720
13 pages, no figures; contains expositional improvements from the previous version
openalex publication_date 2020/06/21 · arxiv created 2021/03/26 · arxiv updated 2021/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of F-quotients in a category C, which are relativized to a faithful functor F\colon C → D. The isomorphism theorems of universal algebras generalize to this setting, and we additionally find important links between F-quotients in the concrete category of first-order structures, and quotients defined for model-theoretic equivalence classes. By first working in this categorical setting, some quotient-related results for first-order structures can be naturally obtained. In particular, we are able to prove some isomorphism theorems in the context of model theory directly from their corresponding categorical isomorphism theorems.