Localization Theory in an Infinity Topos
Abstract
We develop the theory of reflective subfibrations on an ∞-topos E. A reflective subfibration L on E is a pullback-compatible assignment of a reflective subcategory D_X ⊆ E/X with associated localization functor L_X, for every X in E. Reflective subfibrations abound in homotopy theory, albeit often disguised, e.g., as stable factorization systems. The added properties of a reflective subfibration L on E compared to a mere reflective subcategory of E are crucial for most of our results. For example, we can prove that L-local maps (i.e., those maps p in D_X for some X in E) admit a classifying map. The existence of such a classifying map is a powerful tool that we exploit to show that there is a reflective subfibration L' whose local maps are exactly the L-separated maps, that is, those maps with L-local diagonal. We investigate some interactions between L and L' and explain when the two reflective subfibrations coincide. Finally, we show the existence of reflective subfibrations associated to sets of maps in E and describe some of their properties.