# nLab tensor power

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# 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 $n$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

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

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 $V$.

Last revised on July 7, 2019 at 13:10:29. See the history of this page for a list of all contributions to it.