In mathematics , the tensor-hom adjunction is 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
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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 .[1]
Counit and unit
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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} η Y : Y → Hom S ( X , Y ⊗ R X ) {\displaystyle \eta _{Y}:Y\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)} are defined as follows: For y {\displaystyle y} in Y {\displaystyle Y} ,
η Y ( y ) ∈ Hom S ( X , Y ⊗ R X ) {\displaystyle \eta _{Y}(y)\in \operatorname {Hom} _{S}(X,Y\otimes _{R}X)} is a right S {\displaystyle S} -module homomorphism given by
η Y ( y ) ( t ) = y ⊗ t for t ∈ X . {\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 Y {\displaystyle Y} in D {\displaystyle {\mathcal {D}}} ,
ε F Y ∘ F ( η Y ) : Y ⊗ R X → Hom S ( X , Y ⊗ R X ) ⊗ R X → Y ⊗ R X {\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 Y ⊗ X {\displaystyle Y\otimes X} by
ε F Y ∘ F ( η Y ) ( y ⊗ x ) = η Y ( y ) ( x ) = y ⊗ x . {\displaystyle \varepsilon _{FY}\circ F(\eta _{Y})(y\otimes x)=\eta _{Y}(y)(x)=y\otimes x.} Likewise,
G ( ε Z ) ∘ η G Z : Hom S ( X , Z ) → Hom S ( X , Hom S ( X , Z ) ⊗ R X ) → Hom S ( X , Z ) . {\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 ϕ {\displaystyle \phi } in Hom S ( X , Z ) {\displaystyle \operatorname {Hom} _{S}(X,Z)} ,
G ( ε Z ) ∘ η G Z ( ϕ ) {\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )} is a right S {\displaystyle S} -module homomorphism defined by
G ( ε Z ) ∘ η G Z ( ϕ ) ( x ) = ε Z ( ϕ ⊗ x ) = ϕ ( x ) {\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )(x)=\varepsilon _{Z}(\phi \otimes x)=\phi (x)} and therefore
G ( ε Z ) ∘ η G Z ( ϕ ) = ϕ . {\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )=\phi .} The Ext and Tor functors
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See also
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References
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May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory . A.M.S. p. 253. ISBN 0-8218-3922-5 .