(∞,2)-Topoi and descent

Abstract

Abstract. We set the foundations of a theory of Grothendieck (∞,2)-topoi based on the notion of fibrational descent, which axiomatizes both the existence of a classifying object for fibrations internal to an (∞,2)-category as well as the exponentiability of these fibrations. We give equivalent characterization of (∞,2)-topoi based in categorified Lawvere-Tierney axioms, as well as 2-dimension version of Giraud's theorem. We study internal category-theoretic constructions such as the Yoneda lemma and Kan extensions and produce a categorified localic reflection theorem.