Equivariant Functors and Sheaves

Abstract

In this thesis we study two main topics which culminate in a proof that four distinct definitions of the equivariant derived category of a smooth algebraic group G acting on a variety X are in fact equivalent. In the first part of this thesis we introduce and study equivariant categories on a quasi-projective variety X. These are a generalization of the equivariant derived category of Lusztig and are indexed by certain pseudofunctors that take values in the 2-category of categories. This 2-categorical generalization allow us to prove rigorously and carefully when such categories are additive, monoidal, triangulated, and admit t-structures, among other properties. We also define equivariant functors and natural transformations before using these to prove how to lift adjoints to the equivariant setting. We also give a careful foundation of how to manipulate t-structures on these equivariant categories for future use and with an eye towards future applications.In the final part of this thesis we prove a four-way equivalence between the different formulations of the equivariant derived category of ℓ-adic sheaves on a quasi-projective variety X. We show that the equivariant derived category of Lusztig is equivalent to the equivariant derived category of Bernstein-Lunts and the simplicial equivariant derived category. As a final step we show that these equivariant derived categories are equivalent to the derived ℓ-adic category on the algebraic stack [X/G] of Behrend. We also provide an isomorphism of the simplicial equivariant derived category on the variety X with the simplicial equivariant derived category on the simplicial presentation of [X/G], as well as prove explicit equivalences between the categories of equivariant ℓ-adic sheaves, local systems, and perverse sheaves with the classical incarnations of