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(∞,2)-Topoi and descent. (j. w. Louis Martini)

The goal of this talk is to introduce the notion of a Grothendieck (∞,2)-topos as a presentable (∞,2)-category satisfying a categorified version of the descent axiom for (∞,1)-topoi of Rezk-Lurie, which we call fibrational descent. As the name indicates, fibrational descent axiomatizes the structure of internal fibrations in an (∞,2)-category and it is closely related to the straightening-unstraightening equivalence of Grothendieck-Lurie. After presenting the main definition, I will give an overview of several different ways of characterising 2-topoi, which includes a 2-dimensional version of Giraud's theorem and categorified Lawvere-Tierney axioms. Moreover, I will show how the theory of internal categories in an (∞,1)-topos (as develop by Martini and Wolf) can be embedded into our formalism as (∞,1)-localic 2-topoi. If time permits, I will explain how to construct a version of the Yoneda embedding in an (∞,2)-topos and a theory of partially lax Kan extensions.