Global 2-rings and genuine refinements
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914515499614208 |
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| author | Gepner, David Linskens, Sil Pol, Luca |
| author_facet | Gepner, David Linskens, Sil Pol, Luca |
| contents | We introduce the notion of a naive global 2-ring: a functor from the opposite of the $\infty$-category of global spaces to presentably symmetric monoidal stable $\infty$-categories. By passing to global sections, every naive global 2-ring decategorifies to a multiplicative cohomology theory on global spaces, i.e. a naive global ring. We suggest when a naive global 2-ring deserves to be called \emph{genuine}. As evidence, we associate to such a global 2-ring a family of equivariant cohomology theories which satisfy a version of the change of group axioms introduced by Ginzburg, Kapranov and Vasserot. We further show that the decategorified multiplicative global cohomology theory associated to a genuine global $2$-ring canonically refines to an $\mathbb{E}_\infty$-ring object in global spectra. As we show, two interesting examples of genuine global 2-rings are given by quasi-coherent sheaves on the torsion points of an oriented spectral elliptic curve and Lurie's theory of tempered local systems. In particular, we obtain global spectra representing equivariant elliptic cohomology and tempered cohomology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05124 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global 2-rings and genuine refinements Gepner, David Linskens, Sil Pol, Luca Algebraic Topology 55N91 (Primary), 55N34 (Secondary) We introduce the notion of a naive global 2-ring: a functor from the opposite of the $\infty$-category of global spaces to presentably symmetric monoidal stable $\infty$-categories. By passing to global sections, every naive global 2-ring decategorifies to a multiplicative cohomology theory on global spaces, i.e. a naive global ring. We suggest when a naive global 2-ring deserves to be called \emph{genuine}. As evidence, we associate to such a global 2-ring a family of equivariant cohomology theories which satisfy a version of the change of group axioms introduced by Ginzburg, Kapranov and Vasserot. We further show that the decategorified multiplicative global cohomology theory associated to a genuine global $2$-ring canonically refines to an $\mathbb{E}_\infty$-ring object in global spectra. As we show, two interesting examples of genuine global 2-rings are given by quasi-coherent sheaves on the torsion points of an oriented spectral elliptic curve and Lurie's theory of tempered local systems. In particular, we obtain global spectra representing equivariant elliptic cohomology and tempered cohomology. |
| title | Global 2-rings and genuine refinements |
| topic | Algebraic Topology 55N91 (Primary), 55N34 (Secondary) |
| url | https://arxiv.org/abs/2407.05124 |