Tensor–hom adjunction

Template:Short description In mathematics, the tensor-hom adjunction is the statement that the tensor product $-\otimes X$ and hom-functor $\mathrm{Hom}(X,-)$ form an adjoint pair:

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

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 for modules

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

$𝒞={\mathrm{M}\mathrm{o}\mathrm{d}}_S\text{and}𝒟={\mathrm{M}\mathrm{o}\mathrm{d}}_R.$

Fix an $(R,S)$-bimodule $X$ and define functors $F:𝒟\to 𝒞$ and $G:𝒞\to 𝒟$ as follows:

$F(Y)=Y{\otimes }_{R}X\text{for }Y\in 𝒟$

$G(Z)={\mathrm{Hom}}_S(X,Z)\text{for }Z\in 𝒞$

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

${\mathrm{Hom}}_S(Y{\otimes }_{R}X,Z)\cong {\mathrm{Hom}}_R(Y,{\mathrm{Hom}}_S(X,Z)).$

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.

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

$\epsilon :FG\to 1_{𝒞}$

has components

${\epsilon }_Z:{\mathrm{Hom}}_S(X,Z){\otimes }_{R}X\to Z$

given by evaluation: For

$\varphi \in {\mathrm{Hom}}_S(X,Z)\text{and}x\in X,$

$\epsilon (\varphi \otimes x)=\varphi (x).$

The components of the unit

$\eta :1_{𝒟}\to GF$

${\eta }_Y:Y\to {\mathrm{Hom}}_S(X,Y{\otimes }_{R}X)$

are defined as follows: For $y$ in $Y$,

${\eta }_Y(y)\in {\mathrm{Hom}}_S(X,Y{\otimes }_{R}X)$

is a right $S$-module homomorphism given by

${\eta }_Y(y)(t)=y\otimes t\text{for }t\in X.$

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

${\epsilon }_{FY}\circ F({\eta }_Y):Y{\otimes }_{R}X\to {\mathrm{Hom}}_S(X,Y{\otimes }_{R}X){\otimes }_{R}X\to Y{\otimes }_{R}X$

is given on simple tensors of $Y\otimes X$ by

${\epsilon }_{FY}\circ F({\eta }_Y)(y\otimes x)={\eta }_Y(y)(x)=y\otimes x.$

Likewise,

$G({\epsilon }_Z)\circ {\eta }_{GZ}:{\mathrm{Hom}}_S(X,Z)\to {\mathrm{Hom}}_S(X,{\mathrm{Hom}}_S(X,Z){\otimes }_{R}X)\to {\mathrm{Hom}}_S(X,Z).$

For $\varphi$ in ${\mathrm{Hom}}_S(X,Z)$,

$G({\epsilon }_Z)\circ {\eta }_{GZ}(\varphi )$

is a right $S$-module homomorphism defined by

$G({\epsilon }_Z)\circ {\eta }_{GZ}(\varphi )(x)={\epsilon }_Z(\varphi \otimes x)=\varphi (x)$

and therefore

$G({\epsilon }_Z)\circ {\eta }_{GZ}(\varphi )=\varphi .$

The Ext and Tor functors

The Hom functor $\mathrm{hom}(X,-)$ commutes with arbitrary limits, while the tensor product $-\otimes X$ functor commutes with arbitrary colimits that exist in their domain category. However, in general, $\mathrm{hom}(X,-)$ fails to commute with colimits, and $-\otimes X$ 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.

In arithmetic

Template:Context Template:See also We can illustrate the tensor-hom adjunction in the category of functions of finite sets. Given a set $N$, its Hom functor takes any set $A$ to the set of functions from $N$ to $A$. The isomorphism class of this set of functions is the natural number $A^N$. Similarly, the tensor product $-\otimes N$ takes a set $A$ to its cartesian product with $N$. Its isomorphism class is thus the natural number $AN$.

This allows us to interpret the isomorphism of hom-sets

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

that universally characterizes the tensor-hom adjunction, as the categorification of the remarkably basic law of exponents

$Z^{YX}=(Z^X{)}^Y.$

See also

• Closed monoidal category
• Eckmann–Hilton duality
• Change of rings

References

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