In mathematics, the tensor-hom adjunction is that the tensor product and hom-functor form an adjoint pair:
This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" reflects their relationship: tensor is the left adjoint, while hom is the right adjoint.
Say R and S are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules):
Fix an -bimodule and define functors and as follows:
Then is left adjoint to . This means there is a natural isomorphism
This is actually an isomorphism of abelian groups. More precisely, if is an -bimodule and is a -bimodule, then this is an isomorphism of -bimodules. This is one of the motivating examples of the structure in a closed bicategory.[1]
Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit natural transformations. Using the notation from the previous section, the counit
has components
given by evaluation: For
The components of the unit
are defined as follows: For in ,
is a right -module homomorphism given by
The counit and unit equations can now be explicitly verified. For in ,
is given on simple tensors of by
Likewise,
For in ,
is a right -module homomorphism defined by
and therefore
May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory. A.M.S. p. 253. ISBN 0-8218-3922-5.