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Non-representable six-functor formalisms
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abstract
In this article, we study the properties of motivic homotopy category $\mathcal{SH}_{\operatorname{ext}}(\mathcal{X})$ developed by Chowdhury and Khan-Ravi for $\mathcal{X}$ a Nis-loc Stack. In particular, we compare the above construction with Voevodsky's original construction using NisLoc topology. Using the techniques developed by Liu-Zheng and Mann's notion of $\infty$-category of correspondences and abstract six-functor formalisms, we also extend the exceptional functors and extend properties like projection formula, base change and purity to the non-representable situation.
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Abstract Six-Functor Formalisms: Extension to Ind- and Pro- Categories and Functorial Cohomological Purity
A presentable six-functor formalism satisfying cohomological purity extends to Ind- and Pro-categories, defining motivic stable homotopy theory for ind-pro algebraic stacks such as the Hecke stack.
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