Abstract Six-Functor Formalisms: Extension to Ind- and Pro- Categories and Functorial Cohomological Purity

Abstract

In this article, we study two consequences of abstract six-functor formalisms. Firstly, we show that an abstract six-functor formalism can be extended to specific Ind- and Pro- categories of geometric setups. As an application, we can define the motivic stable homotopy theory for ind-pro-algebraic stacks such as the Hecke stack. Secondly, we also show that Cohomological Purity is functorial using the multisimplicial language developed by Liu-Zheng.