Tensor-hom adjunction

Summary

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

 

Likewise,

 

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

ReferencesEdit

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