(Equivariant) commutative ring spectra and infty-categories of bispans

By now, we know various equivalent pointset level and higher categorical models of "coherently commutative monoids." One particularly conceptually simple approach is to define them as product preserving functors out of an $\infty$-category Span(Fin) of spans (aka correspondences) of finite sets. In more fancy terms, this identifies Span(Fin) as the Lawvere theory of (higher categorical) commutative monoids.

In this talk I will explain a multiplicative version of this result, identifying the Lawvere theory for (higher categorical) commutative semirings as an $\infty$-category Bispan(Fin) of bispans of finite sets. In particular, this allows to describe connective commutative ring spectra as certain space-valued functors on Bispan(Fin). One of the key ingredients for this comparison is a general criterion for computing localizations of span categories, which is of independent interest.

If time permits, I will also explain how to extend these results to the "genuine" versions of equivariant commutative rings typically considered in equivariant stable homotopy theory.

All of this is joint work with Bastiaan Cnossen, Rune Haugseng, and Sil Linskens.