Tensor Algebra - Adjunction and Universal Property

Adjunction and Universal Property

The tensor algebra T(V) is also called the free algebra on the vector space V, and is functorial. As with other free constructions, the functor T is left adjoint to some forgetful functor. In this case, it's the functor which sends each K-algebra to its underlying vector space.

Explicitly, the tensor algebra satisfies the following universal property, which formally expresses the statement that it is the most general algebra containing V:

Any linear transformation f : V → A from V to an algebra A over K can be uniquely extended to an algebra homomorphism from T(V) to A as indicated by the following commutative diagram:

Here i is the canonical inclusion of V into T(V) (the unit of the adjunction). One can, in fact, define the tensor algebra T(V) as the unique algebra satisfying this property (specifically, it is unique up to a unique isomorphism), but one must still prove that an object satisfying this property exists.

The above universal property shows that the construction of the tensor algebra is functorial in nature. That is, T is a functor from the K-Vect, category of vector spaces over K, to K-Alg, the category of K-algebras. The functoriality of T means that any linear map from V to W extends uniquely to an algebra homomorphism from T(V) to T(W).