$L'$-localization in an $\infty$-topos

TL;DR

This paper establishes the existence of a new reflective subfibration in an $ abla$-topos, where local maps are characterized by their diagonals being $L$-local, advancing the theory of localization in higher topos theory.

Contribution

It constructs a reflective subfibration with local maps defined via $L$-separated maps, extending localization concepts in $ abla$-topos theory.

Findings

Existence of a reflective subfibration with $L$-separated local maps.

Characterization of local maps via diagonals being $L$-local.

Provides tools for localization in higher topos contexts.

Abstract

We prove that, given any reflective subfibration $L_\bullet$ on an $\infty$-topos $\mathcal{E}$, there exists a reflective subfibration $L'_\bullet$ on $\mathcal{E}$ whose local maps are the $L$-separated maps, that is, the maps whose diagonals are $L$-local. This is the companion paper to "Localization theory in an $\infty$-topos".