nLab
tensor power

Contents

Context

Monoidal categories

monoidal categories

With symmetry

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,)(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 nnth 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.

Last revised on April 8, 2019 at 11:39:27. See the history of this page for a list of all contributions to it.