Mackey functor characterization of n-excisive functors on spectra

Let $\mathcal{M}=\mathcal{F}_s^{\leq n}$ be the category of finite sets and surjections of cardinality at most $n$, as in Example, and let $\mathbf{C}$ be a semiadditive presentable $\infty$-category. Write $\mathbf{Mack}(\mathcal{M},\mathbf{C})$ for the category of $\mathbf{C}$-valued Mackey functors, and let $F:\textbf{Sp}\to\mathbf{C}$ be an $n$-excisive functor. For a set $S$, write $E(S)$ for the value at $S$ of the Mackey functor corresponding to $F$, and let $\mathbb{D}_nF$ denote the $n$th derivative of $F$. Mackey functor characterization conjecture. There is an equivalence

$$\mathbf{Mack}(\mathcal{M},\mathbf{C})\simeq\operatorname{Exc}_n(\textbf{Sp},\mathbf{C}),$$

where the equivalence sends $F$ to a Mackey functor $E$ such that $E(S)$ is equivalent to the cross-effect of $F$ evaluated on the $S$-indexed direct sum of spheres. In particular, when $S$ has $n$ elements, $E(S)$ is equivalent to $\mathbb{D}_nF$ as a spectrum with $\Sigma_n$-action. This conjecture identifies Mackey functors on finite sets and surjections of size at most $n$ with $n$-excisive functors from spectra; a proof was reported as almost complete and was intended for separate work. It is presented as a topological analogue of an algebraic statement and as related to a description of $n$-excisive functors from spaces to spectra, but the paper assumes it without proving it.