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Symmetry shifting for monoidal bicategories

Raffael Stenzel

Abstract

We show that every braiding on a monoidal bicategory induces a monoidal structure on its bicategory of monoids, such that if the former is sylleptic or symmetric then the latter is braided or symmetric, respectively. This extends a classic theorem of Joyal and Street for monoidal categories. The proof presented in this paper is an application of the $\infty$-operadic Additivity Theorem and thereby averts any considerable calculations.

Symmetry shifting for monoidal bicategories

Abstract

We show that every braiding on a monoidal bicategory induces a monoidal structure on its bicategory of monoids, such that if the former is sylleptic or symmetric then the latter is braided or symmetric, respectively. This extends a classic theorem of Joyal and Street for monoidal categories. The proof presented in this paper is an application of the -operadic Additivity Theorem and thereby averts any considerable calculations.
Paper Structure (10 sections, 24 theorems, 52 equations)

This paper contains 10 sections, 24 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Outline of the technical aspects
  4. Organisation of the paper
  5. Monoidal $\infty$-categories and monoids therein
  6. $\infty$-Operads
  7. Algebras for $\infty$-operads
  8. Monoidal categories and monoids therein
  9. Monoidal bicategories and monoids therein
  10. Future work

Key Result

Proposition 2.1

Let $\mathcal{C}^{\otimes}$ be a symmetric monoidal $\infty$-category and $\mathcal{V}$ be an $\infty$-operad. Then the $\infty$-category $\mathrm{Alg}_{\mathcal{V}}(\mathcal{C}^{\otimes})$ inherits a symmetric monoidal structure from that of $\mathcal{C}^{\otimes}$.

Theorems & Definitions (63)

  • Proposition 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Theorem 2.7: lurieha
  • Lemma 2.8
  • proof
  • ...and 53 more