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
In a monoidal category with tensor product we say that for a natural number and any object, that
is the th tensor power of .
There is accordingly also the th tensor power of any morphism , being a morphism .
This process defines a functor
which could be called the th tensor power functor.
If is a suitable linear category, the th tensor power functor is a simple example of a Schur functor.
The coproduct of all of the tensor powers of naturally inherits the structure of a monoid in . This is called the tensor algebra of . This is the free monoid object on . For more on this see category of monoids.
Often in the literature this is considered for the case Vect of vector spaces. Given a vector space , the -fold tensor product of this space with itself, , is usually denoted and called the th tensor power of .
Last revised on December 23, 2023 at 19:17:38. See the history of this page for a list of all contributions to it.