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| Format: | Preprint |
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2017
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| Online Access: | https://arxiv.org/abs/1703.02849 |
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| _version_ | 1866913177081479168 |
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| author | Kato, Yuki |
| author_facet | Kato, Yuki |
| contents | In this paper, we develop motivic derived algebraic geometry, an enhancement of derived algebraic geometry adapted to the $\mathbb{A}^1$-homotopy theory of Morel and Voevodsky. We construct motivic model categories by imposing descent for a Grothendieck topology and invariance with respect to an interval object, and use them to formulate motivic versions of $\infty$-categories, $\infty$-topoi, and classifying $\infty$-topoi. We then define motivic spectral schemes and motivic spectral Deligne--Mumford stacks in terms of structured motivic $\infty$-topoi. The main result establishes the existence of a motivic stackification functor: a geometric morphism between compatible motivic classifying \(\infty\)-topoi induces a pullback functor on structured motivic topoi, and this functor admits a left adjoint relative to the underlying motivic $\infty$-topos. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1703_02849 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Motivic classifying $\infty$-topoi and spectral stacks Kato, Yuki Category Theory 18G55 (primary), 14F42, 14A30, 18N60 (secondary) In this paper, we develop motivic derived algebraic geometry, an enhancement of derived algebraic geometry adapted to the $\mathbb{A}^1$-homotopy theory of Morel and Voevodsky. We construct motivic model categories by imposing descent for a Grothendieck topology and invariance with respect to an interval object, and use them to formulate motivic versions of $\infty$-categories, $\infty$-topoi, and classifying $\infty$-topoi. We then define motivic spectral schemes and motivic spectral Deligne--Mumford stacks in terms of structured motivic $\infty$-topoi. The main result establishes the existence of a motivic stackification functor: a geometric morphism between compatible motivic classifying \(\infty\)-topoi induces a pullback functor on structured motivic topoi, and this functor admits a left adjoint relative to the underlying motivic $\infty$-topos. |
| title | Motivic classifying $\infty$-topoi and spectral stacks |
| topic | Category Theory 18G55 (primary), 14F42, 14A30, 18N60 (secondary) |
| url | https://arxiv.org/abs/1703.02849 |