Homotopy-coherent distributivity and the universal property of bispans

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 some cases 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). For example, commutative semirings can be described in terms of bispans of finite sets, while bispans in finite $G$-sets can be used to encode Tambara functors, which are the structure on $\pi_0$ of $G$-equivariant commutative ring spectra.

Motivated by applications of the ∞-categorical upgrade of such descriptions to motivic and equivariant ring spectra, I will discuss the universal property of $(\infty,2)$-categories of bispans [1]. This gives a universal way to obtain functors from bispans, which amounts to upgrading 'monoid-like' structures to 'ring-like' ones. In the talk I will focus on the simplest case of bispans in finite sets, where this gives a new construction of the semiring structure on a symmetric monoidal $\infty$-category whose tensor product commutes with coproducts.

References:

[1] Elden Elmanto and Rune Haugseng, On distributivity in higher algebra I: The universal property of bispans, arXiv:2010.15722.

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)