AI 中文总结
本文将抽象六函子形式主义扩展至Ind-与Pro-范畴，应用于定义ind-pro-代数栈的动机稳定同伦理论，并借助Liu-Zheng的多单纯语言证明上同调纯性的函子性。
AI 中文摘要
本文研究抽象六函子形式主义的两个推论：其一，证明抽象六函子形式主义可扩展到几何构造的特定归纳（Ind-）范畴与投射（Pro-）范畴，应用上可定义ind-pro-代数栈（如赫克栈）的动机稳定同伦理论；其二，利用Liu-Zheng发展的多单纯语言证明上同调纯性具有函子性。
英文摘要
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.
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