Categories of Pseudocones and Equivariant Descent

Abstract

In this monograph we provide an in-depth and systematic study of pseudolimits of pseudofunctors $F:\mathscr{C}^{op} \to \mathfrak{Cat}$ in the $2$-category of categories where $\mathscr{C}$ is a $1$-category and use this to give an explicit and careful study of the category theory used in representation theory, equivariant algebraic geometry, and equivariant algebraic topology and give a unifying language to study equivariant sheaves, equivariant perverse sheaves, and their equivariant derived categories. We show how to use the pseudocone construction $\mathsf{Bicat}(\mathscr{C}^{op},\mathfrak{Cat})(\operatorname{cnst}(1),F)$ in order to derive categorical and homological properties of the pseudolimit of $F$. We explicitly show when the pseudolimit of $F$ is complete, cocomplete, enriched in models of a Lawvere theory, (braided) monoidal, regular, triangulated, admits $t$-structures, and more. We use these various structural results to give a new category-theoretic proof and construction of the equivariant standard and pervese $t$-structures and equivariant six functor formalism for the equivariant derived category $D_G^b(X)$ in both the geometric and topological cases as well as for $D_G^b(X;\overline{\mathbb{Q}}_{\ell})$ in the geometric case. We also show in what sense precise sense we can view the equivariant derived category in terms of localizations. After restricting to the case of group resolution categories, we show the existence of a natural isomorphism $\Theta:\alpha_X^{\ast} \Rightarrow \pi_2^{\ast}$ which satisfies a pseudofunctorial version of the cocycle condition $d_1^{\ast}\Theta = d_2^{\ast}\Theta \circ d_0^{\ast}\Theta$. We also use the pseudocone formalism to give an in-depth analysis of change of groups functors. We use the pseudocone formalism and $\Theta$ to develop a notion of equivariant trace with an eye towards the representation theory of $p$-adic groups.