nLab
coherence theorem for symmetric monoidal bicategories

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The coherence theorem for symmetric monoidal bicategories, like many coherence theorems, has several forms (or, alternatively, refers to several different theorems):

  1. Every diagram of constraint 2-cells in a free symmetric monoidal bicategory commutes; in other words, any two parallel composites of constraint 2-cells are equal. Moreover, two parallel composites of constraint 1-cells are isomorphic if and only if they have the same underlying permutation, in which case they are uniquely isomorphic.

  2. Every symmetric monoidal bicategory is equivalent to a strict symmetric monoidal bicategory.

References

  • Nick Gurski and Angelica Osorno?, “Infinite loop spaces, and

    coherence for symmetric monoidal bicategories“, arXiv

  • Bruce Bartlett, “Quasistrict symmetric monoidal 2-categories via wire diagrams”, arXiv

  • Bruce Bartlett, “Quasistrict symmetric monoidal 2-categories via wire diagrams”, n-Cafe

Despite the terminology, the result by Chris Schommer-Pries explained by Bruce Bartlett goes further than the earlier result of Gurski and Osorno. The result in the n-Cafe post seems to go even further.

Last revised on February 21, 2015 at 23:49:19. See the history of this page for a list of all contributions to it.