With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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:
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.
Created on May 18, 2016 at 12:10:58. See the history of this page for a list of all contributions to it.