Localization theory in an $\infty$-topos

Abstract

We develop the theory of reflective subfibrations on an $\infty$-topos $\mathcal{E}$. A reflective subfibration $L_\bullet$ on $\mathcal{E}$ is a pullback-compatible assignment of a reflective subcategory $\mathcal{D}_X\subseteq \mathcal{E}{/X}$, for every $X \in \mathcal{E}$. 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 $\mathcal{D}_X$) 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 $\mathcal{E}$. We prove this fact in the compantion paper "$L'$-localization in an $\infty$-topos". In this paper, we investigate some interactions between $L_\bullet$ and $L'_\bullet$ and explain when the two reflective subfibrations coincide.