Towards $\mathbb{A}^1$-homotopy theory of rigid analytic spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915196838084608 |
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| author | Dahlhausen, Christian Yaylali, Can |
| author_facet | Dahlhausen, Christian Yaylali, Can |
| contents | To any rigid analytic space (in the sense of Fujiwara-Kato) we assign an $\mathbb{A}^1$-invariant rigid analytic homotopy category with coefficients in any presentable category. We show some functorial properties of this assignment as a functor on the category of rigid analytic spaces. Moreover, we show that there exists a full six functor formalism for the precomposition with the analytification functor by evoking Ayoub's thesis. As an application, we identify connective analytic K-theory in the unstable homotopy category with both $\mathbb{Z}\times\mathrm{BGL}$ and the analytification of connective algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_09606 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards $\mathbb{A}^1$-homotopy theory of rigid analytic spaces Dahlhausen, Christian Yaylali, Can Algebraic Topology Algebraic Geometry K-Theory and Homology 14F42, 19E99, 14G22 To any rigid analytic space (in the sense of Fujiwara-Kato) we assign an $\mathbb{A}^1$-invariant rigid analytic homotopy category with coefficients in any presentable category. We show some functorial properties of this assignment as a functor on the category of rigid analytic spaces. Moreover, we show that there exists a full six functor formalism for the precomposition with the analytification functor by evoking Ayoub's thesis. As an application, we identify connective analytic K-theory in the unstable homotopy category with both $\mathbb{Z}\times\mathrm{BGL}$ and the analytification of connective algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra. |
| title | Towards $\mathbb{A}^1$-homotopy theory of rigid analytic spaces |
| topic | Algebraic Topology Algebraic Geometry K-Theory and Homology 14F42, 19E99, 14G22 |
| url | https://arxiv.org/abs/2407.09606 |