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 $\infty$-categorical upgrade of such descriptions to motivic and equivariant ring spectra, I will discuss the universal property of $(\infty, 2)$-categories of bispans. 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. This is joint work with Elden Elmanto.

In a forthcoming work, Jordan Ellenberg, Matthew Satriano, and David Zureick-Brown introduce a new theory of heights on algebraic stacks. This theory extends the classical theory of heights on algebraic varieties. Moreover, Ellenberg, Satriano, and Zureick-Brown have formulated a stacky version of the Manin conjecture which predicts the distribution of rational points on a suitable algebraic stack with respect to a suitable stacky height. This conjecture when applied to the classifying stack of a finite group G recovers a version of Malle’s conjecture for the group G; Malle’s conjecture predicts the asymptotic distribution of number fields of bounded discriminant with Galois group G.

I will give an introduction to this circle of ideas in the case of stacky curves. In this setting the theory is simpler and can often be explicitly described. In particular, one can explicitly describe the stacky height function associated to the anti-canonical bundle of a stacky projective line with chosen number of half points. With this explicit description in hand I will discuss some recent work with Stanely Xiao which verifies a new instance of the Ellenberg, Satriano, and Zureick-Brown conjecture and discuss some open problems and ongoing investigations in this area.