Tensor-hom adjunction

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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[edit]

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[edit]

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


For φ in HomS(X, Z),

is a right S-module homomorphism defined by

and therefore

Ext and Tor[edit]

The idea that the Hom functor and the tensor product functor don't lift to an exact sequence motivates the definition of the Ext functor and the Tor functor.

See also[edit]


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