nLab
coherence theorem for 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 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

Last revised on October 7, 2012 at 20:41:45. See the history of this page for a list of all contributions to it.