U. Haifa Topology & Geometry seminar: Wednesday, November 19, 2025. Speaker: Fernando Abellan (Max Planck). Title: “(∞, 2)-Topoi and Descent”.
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.
