2019/04/25 by Sebastien Vasey, Vasey, Sebastien
Mathematics · #03C45 #03C48 #03C52 #03C55 #03C75 #03E05 #03E55 (Secondary) #18C35 (Primary) #Advanced Topology and Set Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.1904.11307
openalex publication_date 2019/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality localized to a given category, as well as stable independence, a generalization of pushouts and model-theoretic forking that may interest mathematicians at large. We give many examples, including recently discovered connections with homotopy theory and homological algebra. We also discuss concrete versions of accessible categories (such as abstract elementary classes), and how they allow nontrivial `element by element' constructions. We conclude with a new proof of the equivalence between saturated and homogeneous which does not use the coherence axiom of abstract elementary classes.