nLab
tensor power

Definition

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

$$V^{\otimes n}:=V\otimes V\otimes \cdots \otimes V(n\mathrm{factors})$$V^{\otimes n} := V \otimes V \otimes \cdots \otimes V \;\; (n factors)

is the $n$the 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

$$(-{)}^{\otimes n}:C\to C$$(-)^{\otimes n} : C \to C

which could be called the $n$th tensor power functor.

Properties

Schur functors

If $C$ is a suitable linear category, the $n$th tensor power functor is a simple example of a Schur functor.

Tensor algebra

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.

Examples

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 $V$.

Related concepts

• free monoid

• cofree coalgebra