Tensor-hom adjunction


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.

General statementEdit

Say R and S are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules):


Fix an (R,S)-bimodule X and define functors F: DC and G: CD as follows:


Then F is left adjoint to G. This means there is a natural isomorphism


This is actually an isomorphism of abelian groups. More precisely, if Y is an (A, R) bimodule and Z is a (B, S) bimodule, then this is an isomorphism of (B, A) bimodules. This is one of the motivating examples of the structure in a closed bicategory.[1]

Counit and unitEdit

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




For   in  ,


is a right  -module homomorphism defined by


and therefore


The Ext and Tor functorsEdit

The Hom functor   commutes with arbitrary limits, while the tensor product   functor commutes with arbitrary colimits that exist in their domain category. However, in general,   fails to commute with colimits, and   fails to commute with limits; this failure occurs even among finite limits or colimits. This failure to preserve short exact sequences motivates the definition of the Ext functor and the Tor functor.

See alsoEdit


  1. ^ May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory. A.M.S. p. 253. ISBN 0-8218-3922-5.