  Suppose $G$ is a finite group. In this paper, we construct an equivalence
between the $\infty$-category of algebras over an $N_{\infty}$-operad
$\mathcal{O}$ associated to a $G$-indexing system $\mathcal{I}$ and the
corresponding $\infty$-category of higher incomplete $\mathcal{I}$-Mackey
functors with value in spaces. We use the universal property of the incomplete
$(2, 1)$-category of spans of finite $G$-sets $\mathscr{A}_{\mathcal{I}}$ to
construct a functor from $\mathscr{A}_{\mathcal{I}}$ to the $2$-category of
$\mathcal{I}$-normed symmetric monoidal categories of Rubin. We then show that
the left Kan extension of the composition of this functor with the core functor
is an equivalence.
