$(\infty,2)$-Topoi and descent

TL;DR

This paper develops a foundational theory of $( abla,2)$-topoi using fibrational descent, characterizes them via a 2-dimensional Giraud's theorem, and introduces internal Kan extensions and Yoneda embeddings within this framework.

Contribution

It introduces a formal framework for $( abla,2)$-topoi, including a 2-dimensional Giraud's theorem and internal Kan extensions, advancing higher topos theory.

Findings

Proves a 2-dimensional Giraud's theorem for $( abla,2)$-topoi.

Develops the theory of partially lax Kan extensions internal to an $( abla,2)$-topos.

Establishes the existence of an internal Yoneda embedding for $( abla,2)$-topoi.

Abstract

We set the foundations of a theory of Grothendieck $(\infty,2)$-topoi based on the notion of fibrational descent, which axiomatizes both the existence of a classifying object for fibrations internal to an $(\infty,2)$-category as well as the exponentiability of these fibrations. As our main result, we prove a 2-dimensional version of Giraud's theorem which characterizes $(\infty,2)$-topoi as those $(\infty, 2)$-categories that appear as localizations of $\mathfrak{C}\!\operatorname{at}$-valued presheaves in which the localization functor preserves certain partially lax finite limits which we call oriented pullbacks. We develop the basics of a theory of partially lax Kan extensions internal to an $(\infty,2)$-topos, and we show that every $(\infty,2)$-topos admits an internal version of the Yoneda embedding. Our general formalism recovers the theory of categories internal to a $(\infty,1)$-topos (as develop by the second author and Sebastian Wolf) as a full sub-$(\infty,2)$-category of the $(\infty,2)$-category of $(\infty,2)$-topoi. As a technical ingredient, we prove general results on the theory of presentable $(\infty,2)$-categories, including lax cocompletions and 2-dimensional versions of the adjoint functor theorem, which might be of independent interest.