Homotopical commutative rings and bispans Article Swipe

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Bastiaan Cnossen , Rune Haugseng , Tobias Lenz , Sil Linskens ·

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2403.06911

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.

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Commutative ring Commutative property Mathematics Pure mathematics Algebra over a field

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2403.06911
PDF: https://arxiv.org/pdf/2403.06911
OA Status: green
Related Works: 10
OpenAlex ID: https://openalex.org/W4392737491

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DOI: https://doi.org/10.48550/arxiv.2403.06911

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Title: Homotopical commutative rings and bispans

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Type: preprint

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Language: en

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Publication year: 2024

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Publication date: 2024-03-11

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Authors: Bastiaan Cnossen, Rune Haugseng, Tobias Lenz, Sil Linskens

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Landing page: https://arxiv.org/abs/2403.06911

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PDF URL: https://arxiv.org/pdf/2403.06911

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Concepts: Commutative ring, Commutative property, Mathematics, Pure mathematics, Algebra over a field

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