An axiomatization of six-functor formalisms
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arXiv
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| Format: | Preprint |
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2023
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| author | Kuijper, Josefien |
| author_facet | Kuijper, Josefien |
| contents | In this paper, we consider some variations on Mann's definition $\infty$-categorical definition of abstract six-functor formalisms. We consider Nagata six-functor formalisms, that have the additional requirement of having Grothendieck and Wirthmüller contexts. We also consider local six-functor formalisms, which in addition to this, take values in presentable stable $\infty$-categories, and have recollements. Using Nagata's compactification theorem, we show that Nagata six-functor formalism on varieties can be given by just specifying adjoint triples for open immersions and for proper morphisms, satisfying certain compatibilities. The existence of recollements is (almost) equivalent to a hypersheaf condition for a Grothendieck topology on the category of ``varieties and spans consisting of an open immersion and a proper map''. Using this characterisation, we show that the category of local six-functor formalisms embeds faithfully into the category of lax symmetric monoidal functors from the category of smooth and complete varieties to the category of presentable stable $\infty$-categories and adjoint triples. We characterise which lax symmetric monoidal functors on complete varieties, taking values in the category of presentable stable $\infty$-categories and adjoint triples, extend to local six-functor formalisms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_11449 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An axiomatization of six-functor formalisms Kuijper, Josefien Algebraic Geometry K-Theory and Homology 14F99 In this paper, we consider some variations on Mann's definition $\infty$-categorical definition of abstract six-functor formalisms. We consider Nagata six-functor formalisms, that have the additional requirement of having Grothendieck and Wirthmüller contexts. We also consider local six-functor formalisms, which in addition to this, take values in presentable stable $\infty$-categories, and have recollements. Using Nagata's compactification theorem, we show that Nagata six-functor formalism on varieties can be given by just specifying adjoint triples for open immersions and for proper morphisms, satisfying certain compatibilities. The existence of recollements is (almost) equivalent to a hypersheaf condition for a Grothendieck topology on the category of ``varieties and spans consisting of an open immersion and a proper map''. Using this characterisation, we show that the category of local six-functor formalisms embeds faithfully into the category of lax symmetric monoidal functors from the category of smooth and complete varieties to the category of presentable stable $\infty$-categories and adjoint triples. We characterise which lax symmetric monoidal functors on complete varieties, taking values in the category of presentable stable $\infty$-categories and adjoint triples, extend to local six-functor formalisms. |
| title | An axiomatization of six-functor formalisms |
| topic | Algebraic Geometry K-Theory and Homology 14F99 |
| url | https://arxiv.org/abs/2309.11449 |