nLab
coherence theorem for 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 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 monoidal bicategory commutes; in other words, any two parallel composites of constraint 2-cells are equal. Moreover, any two parallel composites of constraint 1-cells are uniquely isomorphic.

  2. Every monoidal bicategory is equivalent to a Gray-monoid.

The second version is a direct corollary of the coherence theorem for tricategories. The first can then be deduced from it (not entirely trivially).

References

  • Gordon, Power, Street, Coherence for tricategories, Mem. Amer. Math Soc. 117 (1995) no 558

  • Nick Gurski, Coherence in three-dimensional category theory

Revised on October 7, 2012 20:41:45 by Mike Shulman (192.16.204.218)