In mathematics, the tensor-hom adjunction is the statement that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ( X , − ) {\displaystyle \operatorname {Hom} (X,-)} form an adjoint pair:
Hom ( Y ⊗ X , Z ) ≅ Hom ( Y , Hom ( X , Z ) ) . {\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.
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 = M o d S and D = M o d R . {\displaystyle {\mathcal {C}}=\mathrm {Mod} _{S}\quad {\text{and}}\quad {\mathcal {D}}=\mathrm {Mod} _{R}.}
Fix an ( R , S ) {\displaystyle (R,S)} -bimodule X {\displaystyle X} and define functors F : D → C {\displaystyle F\colon {\mathcal {D}}\rightarrow {\mathcal {C}}} and G : C → D {\displaystyle G\colon {\mathcal {C}}\rightarrow {\mathcal {D}}} as follows:
F ( Y ) = Y ⊗ R X for Y ∈ D {\displaystyle F(Y)=Y\otimes _{R}X\quad {\text{for }}Y\in {\mathcal {D}}}
G ( Z ) = Hom S ( X , Z ) for Z ∈ C {\displaystyle G(Z)=\operatorname {Hom} _{S}(X,Z)\quad {\text{for }}Z\in {\mathcal {C}}}
Then F {\displaystyle F} is left adjoint to G {\displaystyle G} . This means there is a natural isomorphism
Hom S ( Y ⊗ R X , Z ) ≅ Hom R ( Y , Hom S ( X , Z ) ) . {\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 Y {\displaystyle Y} is an ( A , R ) {\displaystyle (A,R)} -bimodule and Z {\displaystyle Z} is a ( B , S ) {\displaystyle (B,S)} -bimodule, then this is an isomorphism of ( B , A ) {\displaystyle (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
ε : F G → 1 C {\displaystyle \varepsilon :FG\to 1_{\mathcal {C}}}
has components
ε Z : Hom S ( X , Z ) ⊗ R X → Z {\displaystyle \varepsilon _{Z}:\operatorname {Hom} _{S}(X,Z)\otimes _{R}X\to Z}
given by evaluation: For
ϕ ∈ Hom S ( X , Z ) and x ∈ X , {\displaystyle \phi \in \operatorname {Hom} _{S}(X,Z)\quad {\text{and}}\quad x\in X,}
ε ( ϕ ⊗ x ) = ϕ ( x ) . {\displaystyle \varepsilon (\phi \otimes x)=\phi (x).}
The components of the unit
η : 1 D → G F {\displaystyle \eta :1_{\mathcal {D}}\to GF}
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