Tensor–hom adjunction

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

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

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.

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

${\displaystyle {\mathcal {C}}=\mathrm {Mod} _{S}\quad {\text{and}}\quad {\mathcal {D}}=\mathrm {Mod} _{R}.}$

Fix an ${\displaystyle (R,S)}$-bimodule ${\displaystyle X}$ and define functors ${\displaystyle F\colon {\mathcal {D}}\rightarrow {\mathcal {C}}}$ and ${\displaystyle G\colon {\mathcal {C}}\rightarrow {\mathcal {D}}}$ as follows:

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

${\displaystyle G(Z)=\operatorname {Hom} _{S}(X,Z)\quad {\text{for }}Z\in {\mathcal {C}}}$

Then ${\displaystyle F}$ is left adjoint to ${\displaystyle G}$. This means there is a natural isomorphism

${\displaystyle \operatorname {Hom} _{S}(Y\otimes _{R}X,Z)\cong \operatorname {Hom} _{R}(Y,\operatorname {Hom} _{S}(X,Z)).}$

This is actually an isomorphism of abelian groups. More precisely, if ${\displaystyle Y}$ is an ${\displaystyle (A,R)}$-bimodule and ${\displaystyle Z}$ is a ${\displaystyle (B,S)}$-bimodule, then this is an isomorphism of ${\displaystyle (B,A)}$-bimodules. This is one of the motivating examples of the structure in a closed bicategory.¹

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

${\displaystyle \varepsilon :FG\to 1_{\mathcal {C}}}$

has components

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

given by evaluation: For

${\displaystyle \phi \in \operatorname {Hom} _{S}(X,Z)\quad {\text{and}}\quad x\in X,}$

${\displaystyle \varepsilon (\phi \otimes x)=\phi (x).}$

The components of the unit

${\displaystyle \eta :1_{\mathcal {D}}\to GF}$

${\displaystyle \eta _{Y}:Y\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)}$

are defined as follows: For ${\displaystyle y}$ in ${\displaystyle Y}$,

${\displaystyle \eta _{Y}(y)\in \operatorname {Hom} _{S}(X,Y\otimes _{R}X)}$

is a right ${\displaystyle S}$-module homomorphism given by

${\displaystyle \eta _{Y}(y)(t)=y\otimes t\quad {\text{for }}t\in X.}$

The counit and unit equations can now be explicitly verified. For ${\displaystyle Y}$ in ${\displaystyle {\mathcal {D}}}$,

${\displaystyle \varepsilon _{FY}\circ F(\eta _{Y}):Y\otimes _{R}X\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)\otimes _{R}X\to Y\otimes _{R}X}$

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

${\displaystyle \varepsilon _{FY}\circ F(\eta _{Y})(y\otimes x)=\eta _{Y}(y)(x)=y\otimes x.}$

Likewise,

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}:\operatorname {Hom} _{S}(X,Z)\to \operatorname {Hom} _{S}(X,\operatorname {Hom} _{S}(X,Z)\otimes _{R}X)\to \operatorname {Hom} _{S}(X,Z).}$

For ${\displaystyle \phi }$ in ${\displaystyle \operatorname {Hom} _{S}(X,Z)}$,

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )}$

is a right ${\displaystyle S}$-module homomorphism defined by

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )(x)=\varepsilon _{Z}(\phi \otimes x)=\phi (x)}$

and therefore

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )=\phi .}$

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

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

This allows us to interpret the isomorphism of hom-sets

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

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

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