Title: Equivariant Categories on Varieties
The derived category of sheaves on a variety is a central object of study in arithmetic geometry and the representation theory of algebraic groups. However, while the derived category of a variety is quite important, people who study groups acting on spaces have needed an equivariant version of the derived category so as to give a natural place from which to derive equivariant cohomology. It wasn’t until the nineties that a good definition of the equivariant derived category was discovered in a usable fashion by taking descent data through the derived categories of resolutions of an action on the base space. However, the constructions of equivariant derived categories differ greatly from the equivariant categories of sheaves that precede them and consequently make it difficult to see how to build categories of equivariant objects over a variety in a uniform way.
In this talk I will introduce a formalism that shows how to give a uniform and more general construction of equivariant categories overtop a variety by using certain pseudofunctors to construct equivariant categories. After introducing these categories, I will also show how this construction allows us natural ways to deduce properties of the equivariant category simply by looking at properties of the pseudofunctor. Finally, I’ll show that the functors and natural transformations that arise between equivariant categories that come from equivariant data are induced by pseudonatural transformations and modifications which give us techniques for lifting adjoints in a 2-category of pseudofunctors to what I call equivariant adjoints between equivariant categories. If there is any time left, I’ll also give some comments as to how these techniques allow us to produce triangulations on certain families of equivariant categories that contain the equivariant derived category.