The universal property of bispans
Rune Haugseng (Norwegian University of Science and Technology (NTNU))
Commutative semirings can be described in terms of bispans of finite sets, meaning spans with an extra forward leg; if we instead take bispans in finite G-sets we get Tambara functors, which are the structure on $\pi_0$ of $G$-equivariant commutative ring spectra. Motivated by applications of the $\infty$-categorical upgrade of such descriptions to motivic and equivariant ring spectra, I will discuss the universal property of $(\infty,2)$-categories of bispans. 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. This is joint work with Elden Elmanto.