|Location:||MSRI: Simons Auditorium|
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 cartesian fibrations of (oo,2)-categories.
In this talk, we 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, we will prove some statements made by Gaitsgory and Rozemblyum concerning locally cartesian fibrations and (oo,2)-categories fibred over (oo,1)-categories, with the intent of substantiating the validity of our definition.