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 Hom functor and the tensor product functor respectively commute with arbitrary limits and colimits that exist in their domain category, but in general fail to commute with (even finite) colimits and limits: their failure to preserve short exact sequences 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.