Tensor-hom adjunction

In mathematics, the tensor-hom adjunction is that the tensor product and Hom functors and form an adjoint pair:

This is made more precise below. The order "tensor-hom adjunction" is because tensor is the left adjoint, while hom is the right adjoint.

General Statement

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 Unit

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 y in Y,

is a right S-module homomorphism given by

The counit and unit equations can now be explicitly verified. For Y in C,

is given on simple tensors of YX by

Likewise,

For φ in HomS(X, Z),

is a right S-module homomorphism defined by

and therefore

Ext and Tor

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 also

References

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