Let $\mathcal{A}$ and $\mathcal{B}$ be Waldhausen or exact categories, so that we can take the $K$-theory spectrum of $\mathcal{A}$ and $\mathcal{B}$. An exact functor $F: \mathcal{A} \to \mathcal{B}$ induces a morphism of $K$-theory spectra $K(\mathcal{A}) \to K(\mathcal{B})$. 


Under what conditions does a non-exact functor $\mathcal{A} \to \mathcal{B}$ induce a morphism of $K$-theory *spaces* $\Omega^\infty K(\mathcal{A}) \to \Omega^\infty K(\mathcal{B})$? 


To give an example, suppose $\mathcal{A}$ is the category of finite sets and $\mathcal{B}$ is the category of finite $C\_p$-sets. Then there is a functor $\mathcal{A} \to \mathcal{B}$ given by multiplicative induction (it sends a set $S$ to $\mathrm{Map}(C\_p, S)$) and that should induce a morphism on $K$-theory spaces. This gives a map $\Omega^\infty S^0 \to \Omega^\infty S^0\_{C\_p}$, which should arise from the Hill-Hopkins-Ravenel "multiplicative norm" construction. (This sort of construction is used in Segal's paper on the Kahn-Priddy theorem in the 1974 "New developments in topology.")


Has this been written down somewhere? 



When do non-exact functors induce morphisms on $K$-theory?
As suggested by Dustin Clausen in his answer, polynomial functors induce maps on $K$-theory. In the setting of stable $\infty$-categories, you proved this in your [joint work](https://arxiv.org/abs/2102.00936) with Barwick, Glasman, and Nikolaus.


