Continuous six-functor formalism on locally compact Hausdorff spaces

Qingchong Zhu

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arXiv · 2507.13537

Continuous six-functor formalism on locally compact Hausdorff spaces

Qingchong Zhu

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Abstract

We show that the functor sending a locally compact Hausdorff space $X$ to the $\infty$-category of spectral sheaves $\mathrm{Shv}(X; \mathrm{Sp})$ is initial among all continuous six-functor formalisms on the category of locally compact Hausdorff spaces. Here, continuous six-functor formalisms are those valued in dualizable presentable stable $\infty$-categories and satisfying canonical descent, profinite descent, and hyperdescent. As an application, we generalize Efimov's computation of the algebraic $K$-theory of sheaves to all localizing invariants on continuous six-functor formalisms. Our results show that localizing invariants behave analogously to compactly supported sheaf cohomology theories when evaluated on continuous six-functor formalisms on locally compact Hausdorff spaces.

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Qingchong Zhu. 2025-08-12. Continuous six-functor formalism on locally compact Hausdorff spaces. https://arxiv.org/abs/2507.13537

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