nLab
tensor power

Contents

Definition

In a monoidal category (C,)(C,\otimes) with tensor product \otimes we say that for nn \in \mathbb{N} a natural number and VCV \in C any object, that

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

is the nnthe tensor power of VV.

There is accordingly also the nnth tensor power of any morphism f:VWf : V \to W, being a morphism f n:V nW nf^{\otimes n} : V^{\otimes n} \to W^{\otimes n}.

This process defines a functor

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

which could be called the nnth tensor power functor.

Properties

Schur functors

If CC is a suitable linear category, the nnth tensor power functor is a simple example of a Schur functor.

Tensor algebra

The coproduct of all of the tensor powers of VV naturally inherits the structure of a monoid in CC. This is called the tensor algebra of VV. This is the free monoid object on VV. For more on this see category of monoids.

Examples

Often in the literature this is considered for the case C=C = Vect of vector spaces. Given a vector space VV, the nn-fold tensor product of this space with itself, VVV \otimes \cdots \otimes V, is usually denoted V nV^{\otimes n} and called the nnth tensor power of VV.

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