Inspired by [CORS18], we develop the theory of reflective subfibrations on an ∞-topos E. A reflective subfibration L• on E is a pullbackcompatible assignment of a reflective subcategory D X ⊆ 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 D X ) admit a classifying map, and we introduce Lseparated maps, that is, those maps with L-local diagonal. L-separated maps are the local class of maps for a reflective subfibration L ′• on E. We prove this fact in the companion paper [Ver]. In this paper, we investigate some interactions between L• and L ′• and explain when the two reflective subfibrations coincide.• 7.1. Further interactions between L • and L ′ • 7.2. Self-separated reflective subfibrations References
We prove that, given any reflective subfibration L• on an ∞-topos E, there exists a reflective subfibration L ′• on E whose local maps are the Lseparated maps, that is, the maps whose diagonals are L-local. This is the companion paper to [Ver].
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