# nLab tensor power

### Context

#### Monoidal categories

monoidal categories

# Contents

## 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

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

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

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

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 February 4, 2014 00:41:27 by Urs Schreiber (89.204.137.228)