Tensor-hom adjunction

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In mathematics, the tensor-hom adjunction is that the tensor product and Hom functors - \otimes X and \operatorname{Hom}(X,-) form an adjoint pair:

\operatorname{Hom}(Y \otimes X, Z) \cong \operatorname{Hom}(Y,\operatorname{Hom}(X,Z)).

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):

 \mathcal{C} = Mod_R \quad \text{and} \quad \mathcal{D} = Mod_S.

Fix an (R,S) bimodule X and define functors F : CD and G : DC as follows:

F(Y) = Y \otimes_R X \quad \text{for } Y \in \mathcal{C}
G(Z) = \operatorname{Hom}_S (X, Z) \quad \text{for } Z \in \mathcal{D}

Then F is left adjoint to G. This means there is a natural isomorphism

\operatorname{Hom}_S (Y \otimes_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[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

\varepsilon : FG \to 1_{\mathcal{C}}

has components

\varepsilon_Z : \operatorname{Hom}_S (X, Z) \otimes_R X \to Z

given by evaluation: For

\phi \in \operatorname{Hom}_R (X, Z) \quad \text{and} \quad x \in X,
\varepsilon(\phi \otimes x) = \phi(x).

The components of the unit

\eta : 1_{\mathcal{D}} \to GF
\eta_Y : Y \to \operatorname{Hom}_S (X, Y \otimes_R X)

are defined as follows: For y in Y,

\eta_Y(y) \in \operatorname{Hom}_S (X, Y \otimes_R X)

is a right S-module homomorphism given by

\eta_Y(y)(t) = y \otimes t \quad \text{for } t \in X.

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


\varepsilon_{FY}\circ F(\eta_Y) : 
Y \otimes_R X \to 
\operatorname{Hom}_S (X , Y) \otimes_R X \to
Y \otimes_R X

is given on simple tensors of YX by

\varepsilon_{FY}\circ F(\eta_Y)(y \otimes x) = \eta_Y(y)(x) = y \otimes x.

Likewise,

G(\varepsilon_Z)\circ\eta_{GZ} :
\operatorname{Hom}_S (X, Z) \to 
\operatorname{Hom}_S (X, \operatorname{Hom}_S (X , Z) \otimes_R X) \to
\operatorname{Hom}_S (X, Z).

For φ in HomS(X, Z),

G(\varepsilon_Z)\circ\eta_{GZ}(\phi)

is a right S-module homomorphism defined by

G(\varepsilon_Z)\circ\eta_{GZ}(\phi)(x) = \varepsilon_{Z}(\phi \otimes x) = \phi(x)

and therefore

G(\varepsilon_Z)\circ\eta_{GZ}(\phi) = \phi.

References[edit]

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