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A higher Mackey functor description of algebras over an $N_{\infty}$-operad

Gregoire Marc

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.

A higher Mackey functor description of algebras over an $N_{\infty}$-operad

Abstract

Suppose is a finite group. In this paper, we construct an equivalence between the -category of algebras over an -operad associated to a -indexing system and the corresponding -category of higher incomplete -Mackey functors with value in spaces. We use the universal property of the incomplete -category of spans of finite -sets to construct a functor from to the -category of -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.
Paper Structure (18 sections, 39 theorems, 62 equations)

This paper contains 18 sections, 39 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. General motivations and related work
  3. Structure of the paper
  4. Notation and conventions
  5. Acknowledgements
  6. Preliminaries on $N_{\infty}$-operads
  7. $N_{\infty}$-operads and $G$-indexing systems
  8. The $\infty$-categories of $N_{\infty}$-algebras
  9. Normed symmetric monoidal categories
  10. Preliminaries on $\mathcal{I}$-normed symmetric monoidal categories
  11. Rubin's combinatorial models of $N_{\infty}$-operads
  12. Mackey functors vs $N_{\infty}$-algebras
  13. Orbital subcategories and the $\mathcal{I}$-effective Burnside category
  14. Unfurling for $2$-categories and orbital categories
  15. From ${A}_{\mathcal{I}}$ to $\mathcal{I}\mathbf{SMSt}$
  16. ...and 3 more sections

Key Result

Theorem A

(Theorem THETHH) Suppose $G$ is a finite group. For every $G$-indexing system $\mathcal{I}$ and every $N_{\infty}$-operad $\mathcal{O}$ associated to $\mathcal{I}$, the $\infty$-category obtained from $\operatorname{Alg}_{\mathcal{O}}\left(\operatorname{Top}^G\right)$ by inverting genuine weak equiv

Theorems & Definitions (108)

  • Theorem A
  • Definition 2.1.1
  • Definition 2.1.2: MR4357612
  • Lemma 2.1.3: MR4357612
  • Definition 2.1.4: MR4357612
  • Definition 2.1.5
  • Theorem 2.1.6
  • proof
  • Definition 2.2.1
  • Definition 2.2.2
  • ...and 98 more