Continuous six-functor formalism on locally compact Hausdorff spaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915443112935424 |
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| author | Zhu, Qingchong |
| author_facet | Zhu, Qingchong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13537 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuous six-functor formalism on locally compact Hausdorff spaces Zhu, Qingchong K-Theory and Homology Algebraic Topology Category Theory 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. |
| title | Continuous six-functor formalism on locally compact Hausdorff spaces |
| topic | K-Theory and Homology Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2507.13537 |