Every motive is the motive of a stable $\infty$-category
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915198223253504 |
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| author | Ramzi, Maxime Sosnilo, Vladimir Winges, Christoph |
| author_facet | Ramzi, Maxime Sosnilo, Vladimir Winges, Christoph |
| contents | We define a class of motivic equivalences of small stable $\infty$-categories $W_{\mathrm{mot}}$ and show that the Dwyer--Kan localization functor $\mathrm{Cat}^{\mathrm{perf}}_\infty \to \mathrm{Cat}^{\mathrm{perf}}_\infty[W_{\mathrm{mot}}^{-1}]$ is the universal localizing invariant in the sense of Blumberg--Gepner--Tabuada. In particular, we show that every object in its target $\mathcal{M}_{\mathrm{loc}}$ can be represented as $\mathcal{U}_{\mathrm{loc}}(\mathcal{C})$ for some small stable $\infty$-category $\mathcal{C}$. As another consequence, and using work of Efimov, we improve the universal property of $\mathcal{M}_{\mathrm{loc}}$ and show that any $\aleph_1$-finitary localizing invariant factors uniquely through it. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_11338 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Every motive is the motive of a stable $\infty$-category Ramzi, Maxime Sosnilo, Vladimir Winges, Christoph K-Theory and Homology Algebraic Topology Category Theory We define a class of motivic equivalences of small stable $\infty$-categories $W_{\mathrm{mot}}$ and show that the Dwyer--Kan localization functor $\mathrm{Cat}^{\mathrm{perf}}_\infty \to \mathrm{Cat}^{\mathrm{perf}}_\infty[W_{\mathrm{mot}}^{-1}]$ is the universal localizing invariant in the sense of Blumberg--Gepner--Tabuada. In particular, we show that every object in its target $\mathcal{M}_{\mathrm{loc}}$ can be represented as $\mathcal{U}_{\mathrm{loc}}(\mathcal{C})$ for some small stable $\infty$-category $\mathcal{C}$. As another consequence, and using work of Efimov, we improve the universal property of $\mathcal{M}_{\mathrm{loc}}$ and show that any $\aleph_1$-finitary localizing invariant factors uniquely through it. |
| title | Every motive is the motive of a stable $\infty$-category |
| topic | K-Theory and Homology Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2503.11338 |