Homotopical commutative rings and bispans
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910931916685312 |
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| author | Cnossen, Bastiaan Haugseng, Rune Lenz, Tobias Linskens, Sil |
| author_facet | Cnossen, Bastiaan Haugseng, Rune Lenz, Tobias Linskens, Sil |
| contents | We prove that commutative semirings in a cartesian closed presentable $\infty$-category, as defined by Groth, Gepner, and Nikolaus, are equivalent to product-preserving functors from the $(2,1)$-category of bispans of finite sets. In other words, we identify the latter as the Lawvere theory for commutative semirings in the $\infty$-categorical context. This implies that connective commutative ring spectra can be described as grouplike product-preserving functors from bispans of finite sets to spaces. A key part of the proof is a localization result for $\infty$-categories of spans, and more generally for $\infty$-categories with factorization systems, that may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06911 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homotopical commutative rings and bispans Cnossen, Bastiaan Haugseng, Rune Lenz, Tobias Linskens, Sil Category Theory Algebraic Topology 18N70, 18N55 (Primary) We prove that commutative semirings in a cartesian closed presentable $\infty$-category, as defined by Groth, Gepner, and Nikolaus, are equivalent to product-preserving functors from the $(2,1)$-category of bispans of finite sets. In other words, we identify the latter as the Lawvere theory for commutative semirings in the $\infty$-categorical context. This implies that connective commutative ring spectra can be described as grouplike product-preserving functors from bispans of finite sets to spaces. A key part of the proof is a localization result for $\infty$-categories of spans, and more generally for $\infty$-categories with factorization systems, that may be of independent interest. |
| title | Homotopical commutative rings and bispans |
| topic | Category Theory Algebraic Topology 18N70, 18N55 (Primary) |
| url | https://arxiv.org/abs/2403.06911 |