# nLab tensor power

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Definition

In a monoidal category $\left(C,\otimes \right)$ with tensor product $\otimes$ we say that for $n\in ℕ$ a natural number and $V\in C$ any object, that

${V}^{\otimes n}:=V\otimes V\otimes \cdots \otimes V\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(n\mathrm{factors}\right)$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

$\left(-{\right)}^{\otimes n}:C\to C$(-)^{\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 $V$.

Revised on November 15, 2013 00:35:46 by Urs Schreiber (82.169.114.243)