Speaker: Geoff Vooys

Abstract: There are (at least) four known approaches to the equivariant bounded derived category (EDC) known in the literature that people have used in various works. One is due to Bernstein-Lunts in the topological case and was extended rigorously to the scheme case by Pramod Achar.. A second approach to the EDC is the one presented by Lusztig in Cuspidal Local Systems and Graded Hecke Algebras II, which has the benefit of being intimately related to graded Hecke algebra. The third EDC we consider is the equivariant derived category of simplicial sheaves on the simplicial scheme approximating the quotient G \ X (cf., for instance, Deligne's Théorie de Hodge : III). Finally, the last EDC we consider is the one (essentially) defined by Kai Behrend in Derived $\ell$-adic Categories for Algebraic Stacks as a certain 2-colimit of constructible derived categories of $\ell$-adic sheaves on the quotient stack [G/X].

In this talk we will discuss not just what these objects are, but also sketch (emphasis on sketch here; there are a lot of ``no one wants to see this done live'' type details) how they can all be seen to be equivalent. In particular, I will discuss some notions of 2-coskeletal simplicial schemes that may be of interest to anyone interested in higher category theory or other areas of math that use simplicial techniques.

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