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A higher Mackey functor description of algebras over an $N_{\infty}$-operad
Published 19 Feb 2024 in math.AT | (2402.12447v1)
Abstract: 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.
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