On distributivity in higher algebra I: The universal property of bispans

TL;DR

This paper develops a universal property for bispans in higher categories, enabling a unified framework to describe complex algebraic structures like rings, spectra, and K-theory with both additive and multiplicative transfers.

Contribution

It introduces an $(4,4)$-category of bispans characterized by a universal property, generalizing span-based functoriality to include distributivity and dual pushforwards.

Findings

Universal property for bispans in higher categories.

Application to symmetric monoidal $4$-categories and semiring structures.

Encoding of additive and multiplicative transfers in equivariant and motivic spectra.

Abstract

Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\infty$-)categories of spans (or correspondences). In this paper we study the more complicated setup where we have two pushforwards (an "additive" and a "multiplicative" one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\infty,2)$-categories of bispans, characterized by a universal property: they corepresent functors out of $\infty$-categories of spans where the pullbacks have left adjoints and certain canonical 2-morphisms (encoding base change and distributivity) are invertible. This gives a universal way to obtain functors from bispans, which amounts to upgrading "monoid-like" structures to "ring-like" ones. For example, symmetric monoidal $\infty$-categories can be described as product-preserving functors from spans of finite sets, and if the tensor product is compatible with finite coproducts our universal property gives the canonical semiring structure using the coproduct and tensor product. More interestingly, we encode the additive and multiplicative transfers on equivariant spectra as a functor from bispans in finite $G$-sets, extend the norms for finite \'etale maps in motivic spectra to a functor from certain bispans in schemes, and make $\mathrm{Perf}(X)$ for $X$ a spectral Deligne--Mumford stack a functor of bispans using a multiplicative pushforward for finite \'etale maps in addition to the usual pullback and pushforward maps. Combining this with the polynomial functoriality of $K$-theory constructed by Barwick, Glasman, Mathew, and Nikolaus, we obtain norms on algebraic $K$-theory spectra.