2025/05/11 by Matí as Menni, Menni, Matí as
Computer Science · Mathematics · #03G30 #14F06 #18B25 #18F10 #Category Theory (math.CT) #Digital Image Processing Techniques #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2505.07131
openalex publication_date 2025/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recent progress on the question of the size of the class of connected and hyperconnected geometric morphisms from a given topos has led to the definition of \em local state classifier. We discuss a historical precedent which leads to the notion of \em non-singular map and we show that, for a topos \cal E with a local state classifier, and each object X therein, the domain of the full subcategory of \cal E/X consisting of non-singular maps over X is a topos, and that the inclusion is the inverse image functor of a hyperconnected geometric morphism. The prospective geometric applications direct our attention to local state classifiers in toposes `of spaces'. We show that, at least in the pre-cohesive topos of reflexive graphs, the local state classifier, which is a colimit by definition, may be characterized as a limit; more specifically, as a variant of a subobject classifier.