  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 this paper we
study the more complicated setup where 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). We
show that there exist $(\infty,2)$-categories of bispans, characterized by a
universal property: they corepresent functors out of $\infty$-categories of
spans where the pullbacks have left adjoints and certain canonical 2-morphisms
(encoding base change and distributivity) are invertible. This gives a
universal way to obtain functors from bispans, which amounts to upgrading
"monoid-like" structures to "ring-like" ones. For example, symmetric monoidal
$\infty$-categories can be described as product-preserving functors from spans
of finite sets, and if the tensor product is compatible with finite coproducts
our universal property gives the canonical semiring structure using the
coproduct and tensor product. More interestingly, we encode the additive and
multiplicative transfers on equivariant spectra as a functor from bispans in
finite $G$-sets, extend the norms for finite \'etale maps in motivic spectra to
a functor from certain bispans in schemes, and make $\mathrm{Perf}(X)$ for $X$
a spectral Deligne--Mumford stack a functor of bispans using a multiplicative
pushforward for finite \'etale maps in addition to the usual pullback and
pushforward maps. Combining this with the polynomial functoriality of
$K$-theory constructed by Barwick, Glasman, Mathew, and Nikolaus, we obtain
norms on algebraic $K$-theory spectra.
