Commutative semirings and bispans

Commutative rings are commutative algebra objects in abelian groups, but they can also be viewed as models of a Lawvere theory. If we don't insist on having inverses for addition, this admits a nice description: commutative semirings are product-preserving functors to sets from a category of "bispans" of finite sets. In this talk I will explain an infinity-categorical version of this comparison; in particular, using results of Gepner-Groth-Nikolaus, we can describe connective commutative ring spectra in terms of bispans of finite sets. I will also discuss how this result should eventually extend to the equivariant setting, giving a description of connective genuine commutative G-ring spectra for a finite group G as "homotopical Tambara functors". This is work in progress with Bastiaan Cnossen, Tobias Lenz, and Sil Linskens.