We will continue on Thursday, October 30, in boardroom (2nd floor) at 1pm by the talk

E. Lanari

Cartesian fibrations of (oo,2)-categories

Abstract:
The problem of dealing with infinitely many coherence constraints in oo-category theory when trying to define oo-functors has lead to a fibrational approach, in which one represents diagrams of the form B-->Cat_oo as a suitable kinds of fibrations over B. While this is a theorem, due to Lurie, in the case of oo-categories (i.e. (oo,1)-categories), so far there has been no combinatorial definition of a cartesian fibration of (oo,2)-categories.

In this talk, I will define cartesian fibrations in this context, prove some of their basic properties and show they are equivalent (under a suitable equivalence of (oo,2)-categories) to the counterpart in the context of categories enriched over marked simplicial sets (where the definition is given, mutatis mutandis, based on what happens with 2-categories). Furthermore, I will prove some statements made by Gaitsgory and Rozemblyum concerning locally cartesian fibrations and (oo,2)-categories fibred over (oo,1)-categories, thus substantiating the validity of our definition.