2019/07/08 by Marco Vergura, Vergura, Marco
Mathematics · #18E35 #55P60 (Primary) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #math.CT #msc:18E35 #msc:55P60
paper · pdf · doi:10.48550/arxiv.1907.03836
28 pages
arxiv created 2019/07/08 · openalex publication_date 2019/07/08 · arxiv updated 2019/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop the theory of reflective subfibrations on an ∞-topos E. A reflective subfibration L_\bullet on E is a pullback-compatible assignment of a reflective subcategory DX⊆ E/X, for every X ∈ E. Reflective subfibrations abound in homotopy theory, albeit often disguised, e.g., as stable factorization systems. We prove that L-local maps (i.e., those maps that belong to some DX) admit a classifying map, and we introduce the class of L-separated maps, that is, those maps with L-local diagonal. L-separated maps are the local class of maps for a reflective subfibration L'_\bullet on E. We prove this fact in the compantion paper "L'-localization in an ∞-topos". In this paper, we investigate some interactions between L_\bullet and L'_\bullet and explain when the two reflective subfibrations coincide.