Normed equivariant ring spectra and higher Tambara functors
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arXiv
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| Format: | Preprint |
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2024
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| author | Cnossen, Bastiaan Haugseng, Rune Lenz, Tobias Linskens, Sil |
| author_facet | Cnossen, Bastiaan Haugseng, Rune Lenz, Tobias Linskens, Sil |
| contents | In this paper we extend equivariant infinite loop space theory to take into account multiplicative norms: For every finite group $G$, we construct a multiplicative refinement of the comparison between the $\infty$-categories of connective genuine $G$-spectra and space-valued Mackey functors, first proven by Guillou-May, and use this to give a description of connective normed equivariant ring spectra as space-valued Tambara functors.
In more detail, we first introduce and study a general notion of homotopy-coherent normed (semi)rings, and identify these with product-preserving functors out of a corresponding $\infty$-category of bispans. In the equivariant setting, this identifies space-valued Tambara functors with normed algebras with respect to a certain normed monoidal structure on grouplike $G$-commutative monoids in spaces. We then show that the latter is canonically equivalent to the normed monoidal structure on connective $G$-spectra given by the Hill-Hopkins-Ravenel norms. Combining our comparison with results of Elmanto-Haugseng and Barwick-Glasman-Mathew-Nikolaus, we produce normed ring structures on equivariant algebraic K-theory spectra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_08399 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normed equivariant ring spectra and higher Tambara functors Cnossen, Bastiaan Haugseng, Rune Lenz, Tobias Linskens, Sil Algebraic Topology Category Theory K-Theory and Homology 18N70, 55P91 (Primary), 18F25, 18N60, 55P47 (Secondary) In this paper we extend equivariant infinite loop space theory to take into account multiplicative norms: For every finite group $G$, we construct a multiplicative refinement of the comparison between the $\infty$-categories of connective genuine $G$-spectra and space-valued Mackey functors, first proven by Guillou-May, and use this to give a description of connective normed equivariant ring spectra as space-valued Tambara functors. In more detail, we first introduce and study a general notion of homotopy-coherent normed (semi)rings, and identify these with product-preserving functors out of a corresponding $\infty$-category of bispans. In the equivariant setting, this identifies space-valued Tambara functors with normed algebras with respect to a certain normed monoidal structure on grouplike $G$-commutative monoids in spaces. We then show that the latter is canonically equivalent to the normed monoidal structure on connective $G$-spectra given by the Hill-Hopkins-Ravenel norms. Combining our comparison with results of Elmanto-Haugseng and Barwick-Glasman-Mathew-Nikolaus, we produce normed ring structures on equivariant algebraic K-theory spectra. |
| title | Normed equivariant ring spectra and higher Tambara functors |
| topic | Algebraic Topology Category Theory K-Theory and Homology 18N70, 55P91 (Primary), 18F25, 18N60, 55P47 (Secondary) |
| url | https://arxiv.org/abs/2407.08399 |