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The higher algebra and geometry of monoidal bicategories

Raffael Stenzel

TL;DR

The paper provides a higher-categorical generalization of classic 1-dimensional results by identifying braided, sylleptic, and symmetric monoidal bicategories with $\mathsf{E}_k$-monoids in the cartesian $\infty$-category $\mathrm{BiCat}^{\times}$. It builds a bridge between 2-dimensional bicategorical algebra and $\infty$-categorical operad theory via little-cubes geometry, Additivity, and diagonal reflections, enabling controlled extensions of associative structures. The authors develop initial-segment techniques and a robust homotopical framework (via Lack’s model structure and the Gray monoid setting) to prove precise equivalences between bicategorical symmetries and $\mathsf{E}_k$-algebra structures. The results unify and extend Joyal–Street-style descriptions to the bicategorical level, with potential applications to centers, semi-strictification, and higher-categorical coherence phenomena in topology and representation theory.

Abstract

We show that braided, sylleptic and symmetric monoidal bicategories are precisely the $\mathsf{E}_k$-monoids in the cartesian monoidal $(\infty,1)$-category of bicategories for respective integers $k$. To manage the underlying computations, we use the geometry of the little cubes operads to mediate between the 2-dimensional algebra underlying the former and the $\infty$-categorical algebra underlying the latter.

The higher algebra and geometry of monoidal bicategories

TL;DR

The paper provides a higher-categorical generalization of classic 1-dimensional results by identifying braided, sylleptic, and symmetric monoidal bicategories with -monoids in the cartesian -category . It builds a bridge between 2-dimensional bicategorical algebra and -categorical operad theory via little-cubes geometry, Additivity, and diagonal reflections, enabling controlled extensions of associative structures. The authors develop initial-segment techniques and a robust homotopical framework (via Lack’s model structure and the Gray monoid setting) to prove precise equivalences between bicategorical symmetries and -algebra structures. The results unify and extend Joyal–Street-style descriptions to the bicategorical level, with potential applications to centers, semi-strictification, and higher-categorical coherence phenomena in topology and representation theory.

Abstract

We show that braided, sylleptic and symmetric monoidal bicategories are precisely the -monoids in the cartesian monoidal -category of bicategories for respective integers . To manage the underlying computations, we use the geometry of the little cubes operads to mediate between the 2-dimensional algebra underlying the former and the -categorical algebra underlying the latter.
Paper Structure (23 sections, 45 theorems, 167 equations)

This paper contains 23 sections, 45 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Related Work
  4. Future work
  5. Possible alternative approaches
  6. Monoidal $(\infty,1)$-categories and iterated associative structures
  7. Preliminaries
  8. $\infty$-Operads
  9. Algebras for $\infty$-operads
  10. Fragments of commutativity as iterated monoidal structures
  11. Fragments of associativity and unitality
  12. Diagonal reflections of $\mathsf{E}_k$-structures
  13. Iterating fragments of associative structures in $n$-categories
  14. Higher algebras in low dimensional monoidal categories
  15. The cartesian monoidal $\infty$-category of bicategories
  16. ...and 8 more sections

Key Result

Theorem 1.1

There are the following bijections between sets of respective equivalence classes: These bijections are natural with respect to the obvious forgetful maps from bottom to top. The same applies to the corresponding notions of morphisms between them.

Theorems & Definitions (127)

  • Theorem 1.1
  • Example 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Example 2.4
  • Example 2.5
  • Remark 2.7
  • Example 2.8
  • Example 2.9
  • ...and 117 more