Geometry and Topology
Speaker: Ezra Getzler (Northwestern University)
"Complete Segal spaces in the theory of derived stacks"
Time: 15:30
Room: Zoom Meeting ID: 958 6908 4555
Derived stacks play the same role in the theory of moduli that projective resolutions play in the study of modules. In the work of Toen and Vezzosi, Lurie, and Pridham, derived stacks are realized as fibrant objects in a combinatorial model category. We prefer to think of derived stacks as the objects of a category of fibrant objects, the right fibrations of simplicial derived schemes with a fixed base $B_*$.
An important model for $\infty$-categories is Rezk's theory of complete Segal spaces. Their definition actually makes sense for simplicial objects in any category of fibrant objects. In joint work with Kai Behrend, we show that the complete Segal spaces in a category of fibrant objects are the objects of a category of fibrant objects if the category of fibrant objects satisfying an additional axiom:
If $f \colon X \to Y$ is a trivial fibration and $g \colon Y \to Z$ is a morphism such that $gf$ is a fibration, then $g$ is a fibration.
Many (if not all) categories of fibrant objects satisfy this axiom: for example, it holds for Kan complexes.
There is a functor from quasicategories to complete Segal spaces, which shows that these two theories are closely related. Boavida de Brito and Rasekh extend this to a functor from cartesian fibrations to complete Segal spaces in the category of right fibrations. Thus, the above result may be interpreted as a concrete realization of cartesian fibrations in derived geometry.
Mathematics Calendar 
Week of March 21, 2021
