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.