nLab
coherence theorem for symmetric monoidal bicategories

Context

Monoidal categories

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.

Revised on February 21, 2015 23:49:19 by John Baez (99.11.156.244)