2022/08/29 by George Raptis, Raptis, George, Daniel Schäppi +1
Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2208.13897
openalex publication_date 2022/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a small n-category \mathscrC and an n-topos \mathscrX, we study necessary and sufficient conditions for a functor f \colon \mathscrC → \mathscrX to determine a geometric morphism from \mathscrX to the n-topos P(\mathscrC)n of presheaves on \mathscrC for any n ≥ 1. These results generalize and unify results of Lurie for n=∞ and classical characterizations of flat functors (Diaconescu's theorem) for n=1. Interestingly, for n=∞, our analogue of Diaconescu's theorem requires hypercompleteness. As an application, we show that the ∞-topos associated to an n-site behaves as an n-localic ∞-topos with respect to hypercomplete ∞-topoi.