nLab
coherence theorem for closed symmetric monoidal categories

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

A kind of coherence theorem for closed symmetric monoidal categories, conjectured by Saunders Mac Lane in the 60’s and proved by Sergei Soloviev in the 90’s, says the following:

Theorem

A diagram in a free closed symmetric monoidal category is commutative if and only if all its instantiations in vector spaces are commutative.

One may use vector spaces over any (fixed) field. In fact, Soloviev’s result is more general that this and provides an axiomatic description of “test categories” for which the theorem holds.

References

  • Sergei Soloviev. “Proof of a Conjecture of S. Mac Lane”. BRICS 1996

Created on May 18, 2016 at 08:10:58. See the history of this page for a list of all contributions to it.