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Homotopical commutative rings and bispans

Bastiaan Cnossen, Rune Haugseng, Tobias Lenz, Sil Linskens

TL;DR

This work identifies the Lawvere theory for commutative semirings in cartesian-closed presentable $\infty$-categories as the $(2,1)$-category $\mathrm{Bispan}(\mathbb{F})$ of bispans of finite sets, establishing a natural equivalence $CRig(\mathcal{C}) \simeq Fun^{\times}(\mathrm{Bispan}(\mathbb{F}), \mathcal{C})$ for such $\mathcal{C}$. The approach builds on reinterpreting commutative semirings via Day convolution and localization of the span categories, showing that $\mathrm{Span}(\mathbb{F})^{\otimes}$ localizes to $\mathrm{Bispan}(\mathbb{F})$, with product-preserving functors from $\mathrm{Bispan}(\mathbb{F})$ capturing the semiring structure. A key technical ingredient is a general localization theorem for $\infty$-categories of spans and their factorization systems, which may be of independent interest. The results also yield a concrete description of connective commutative ring spectra as grouplike, product-preserving functors from $\mathrm{Bispan}(\mathbb{F})$ to spaces, and they hint at parametrized generalizations in the $G$-equivariant Tambara setting (to be explored in follow-up work). Practically, this provides a robust, algebraic framework to encode additive and multiplicative structures in higher categories, with clear implications for higher algebra and stable homotopy theory.

Abstract

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.

Homotopical commutative rings and bispans

TL;DR

This work identifies the Lawvere theory for commutative semirings in cartesian-closed presentable -categories as the -category of bispans of finite sets, establishing a natural equivalence for such . The approach builds on reinterpreting commutative semirings via Day convolution and localization of the span categories, showing that localizes to , with product-preserving functors from capturing the semiring structure. A key technical ingredient is a general localization theorem for -categories of spans and their factorization systems, which may be of independent interest. The results also yield a concrete description of connective commutative ring spectra as grouplike, product-preserving functors from to spaces, and they hint at parametrized generalizations in the -equivariant Tambara setting (to be explored in follow-up work). Practically, this provides a robust, algebraic framework to encode additive and multiplicative structures in higher categories, with clear implications for higher algebra and stable homotopy theory.

Abstract

We prove that commutative semirings in a cartesian closed presentable -category, as defined by Groth, Gepner, and Nikolaus, are equivalent to product-preserving functors from the -category of bispans of finite sets. In other words, we identify the latter as the Lawvere theory for commutative semirings in the -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 -categories of spans, and more generally for -categories with factorization systems, that may be of independent interest.
Paper Structure (14 sections, 30 theorems, 62 equations)

This paper contains 14 sections, 30 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Spans
  4. Bispans
  5. Symmetric monoidal $\bm\infty$-categories
  6. Cartesian and cocartesian symmetric monoidal structures
  7. Commutative semirings and spans
  8. (Co)cartesian symmetric monoidal structures via spans
  9. Symmetric monoidal structures on spans
  10. Commutative monoids and Day convolution
  11. Comparison with bispans
  12. Localizations of span $\bm\infty$-categories
  13. Localization of $\bm{\textsf{Span}(\mathbb{F})^{\otimes}}$
  14. Proof of the main theorem

Key Result

Theorem A

Let $\mathcal{C}$ be a cartesian closed presentable $\infty$-category. Then there is a natural equivalence between the $\infty$-category of commutative semirings in $\mathcal{C}$ and the $\infty$-category of product-preserving functors $\textup{\textmd{Bispan}}(\mathbb{F}) \if@display\longrightarrow \rightarrow \mathcal{C}$.

Theorems & Definitions (80)

  • Theorem A
  • Corollary B
  • Theorem C
  • Definition 2.1.1
  • Remark 2.1.2
  • Example 2.1.4
  • Example 2.1.5
  • Theorem 2.1.8: Barwick
  • proof
  • Remark 2.1.9
  • ...and 70 more