Equivariant Functors and Sheaves

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, admit $t$-structures, among and more. 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 $\ell$-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. We then show that these equivariant derived categories are equivalent to the derived $\ell$-adic category of Behrend on the algebraic stack $[G \backslash X]$. 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 $[G \backslash X]$, as well as prove explicit equivalences between the categories of equivariant $\ell$-adic sheaves, local systems, and perverse sheaves with the classical incarnations of such categories of equivariant sheaves.