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 $(C,\otimes)$ with tensor product $\otimes$ we say that for $n \in \mathbb{N}$ a natural number and $V \in C$ any object, that
is the $n$th tensor power of $V$.
There is accordingly also the $n$th tensor power of any morphism $f : V \to W$, being a morphism $f^{\otimes n} : V^{\otimes n} \to W^{\otimes n}$.
This process defines a functor
which could be called the $n$th tensor power functor.
If $C$ is a suitable linear category, the $n$th tensor power functor is a simple example of a Schur functor.
The coproduct of all of the tensor powers of $V$ naturally inherits the structure of a monoid in $C$. This is called the tensor algebra of $V$. This is the free monoid object on $V$. For more on this see category of monoids.
Often in the literature this is considered for the case $C =$ Vect of vector spaces. Given a vector space $V$, the $n$-fold tensor product of this space with itself, $V \otimes \cdots \otimes V$, is usually denoted $V^{\otimes n}$ and called the $n$th tensor power of $V$.
Last revised on July 7, 2019 at 13:10:29. See the history of this page for a list of all contributions to it.