This is made more precise below. The order "tensor-hom adjunction" is because tensor is the left adjoint, while hom is the right adjoint.
Fix an (R,S) bimodule X and define functors F: D → C and G: C → D as follows:
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.
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
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 Y⊗X by
For φ in HomS(X, Z),
is a right S-module homomorphism defined by
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.
- May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory. A.M.S. p. 253. ISBN 0-8218-3922-5.