Tensor-hom adjunction explained

- ⊗ X and hom-functor \operatorname{Hom}(X, - ) form an adjoint pair:\operatorname{Hom}(Y ⊗ X, Z) \cong \operatorname{Hom}(Y,\operatorname{Hom}(X,Z)).

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 statement

Say R and S are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules):

l{C} = Mod _{S} and l{D} = Mod _{R} .

Fix an

(R,S) -bimodule X and define functors F \colon l D → l C and G \colon l C → l D as follows:F(Y) = Y ⊗ _{R} X for Y \in l{D} G(Z) = \operatorname{Hom} _{S} (X, Z) for Z \in l{C}

Then

F is left adjoint to G . This means there is a natural isomorphism\operatorname{Hom} _{S} (Y ⊗ _{R} X, Z) \cong \operatorname{Hom} _{R} (Y , \operatorname{Hom} _{S} (X, Z)).

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

\varepsilon : FG \to 1 _{l{C} }

has components

\varepsilon _{Z} : \operatorname{Hom} _{S} (X, Z) ⊗ _{R} X \to Z

given by evaluation: For

\phi \in \operatorname{Hom} _{S} (X, Z) and x \in X, \varepsilon(\phi ⊗ x) = \phi(x).

The components of the unit

η : 1 _{l{D} } \to GFη _{Y} : Y \to \operatorname{Hom} _{S} (X, Y ⊗ _{R} X)

are defined as follows: For

y in Y ,η _{Y(y)} \in \operatorname{Hom} _{S} (X, Y ⊗ _{R} X)

is a right

S -module homomorphism given by η _{Y(y)(t)} = y ⊗ t for t \in X.

The counit and unit equations can now be explicitly verified. For

Y in l{D} , \varepsilon _{FY} \circ F( η _{Y)} : Y ⊗ _{R} X \to \operatorname{Hom} _{S} (X , Y ⊗ _{R} X) ⊗ _{R} X \to Y ⊗ _{R} X

is given on simple tensors of

Y ⊗ X by\varepsilon _{FY} \circ F( η _{Y)(y} ⊗ x) = η _{Y(y)(x)} = y ⊗ x.

Likewise,

G(\varepsilon _{Z)\circη} _{GZ} : \operatorname{Hom} _{S} (X, Z) \to \operatorname{Hom} _{S} (X, \operatorname{Hom} _{S} (X , Z) ⊗ _{R} X) \to \operatorname{Hom} _{S} (X, Z).

For

\phi in \operatorname{Hom} _{S} (X, Z) , G(\varepsilon _{Z)\circη} _{GZ} (\phi)

is a right

S -module homomorphism defined byG(\varepsilon _{Z)\circη} _{GZ} (\phi)(x) = \varepsilon _{Z} (\phi ⊗ x) = \phi(x)

and therefore

G(\varepsilon _{Z)\circη} _{GZ} (\phi) = \phi.

The Ext and Tor functors

\hom(X, - ) commutes with arbitrary limits, while the tensor product - ⊗ X functor commutes with arbitrary colimits that exist in their domain category. However, in general, \hom(X, - ) fails to commute with colimits, and - ⊗ X 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

• Currying
• Eckmann–Hilton_duality
• Ext functor
• Tor functor
• Change of rings

Notes and References

1. Book: Parametrized Homotopy Theory . May . J.P. . Sigurdsson . J. . 2006 . A.M.S. . 0-8218-3922-5 . 253.