Marsaglia polar method: Difference between revisions

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In [[mathematics]], the '''tensor-hom adjunction''' is that the [[Tensor product of modules|tensor product]] and [[Hom functor|Hom]] [[functor]]s <math>- \otimes X</math> and <math>\operatorname{Hom}(X,-)</math> form an [[adjoint pair]]:
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:<math>\operatorname{Hom}(Y \otimes X, Z) \cong \operatorname{Hom}(Y,\operatorname{Hom}(X,Z)).</math>
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==
Say ''R'' and ''S'' are (possibly noncommutative) [[Ring (mathematics)|rings]], and consider the right [[module (mathematics)|module]] categories (an analogous statement holds for left modules):
 
:<math> \mathcal{C} = Mod_R \quad \text{and} \quad \mathcal{D} = Mod_S.</math>
 
Fix an (''R'',''S'') bimodule ''X'' and define functors ''F'' : ''C'' → ''D'' and ''G'' : ''D'' → ''C'' as follows:
 
:<math>F(Y) = Y \otimes_R X \quad \text{for } Y \in \mathcal{C}</math>
 
:<math>G(Z) = \operatorname{Hom}_S (X, Z) \quad \text{for } Z \in \mathcal{D}</math>
 
Then ''F'' is left [[Adjoint functors|adjoint]] to ''G''. This means there is a [[natural isomorphism]]
 
:<math>\operatorname{Hom}_S (Y \otimes_R X, Z) \cong \operatorname{Hom}_R (Y , \operatorname{Hom}_S (X, Z)).</math>
 
This is actually an isomorphism of [[abelian group]]s.  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]].<ref name="MaySigurdsson">
{{cite book |title=Parametrized Homotopy Theory |last1=May |first1=J.P. |last2=Sigurdsson |first2=J. |year=2006 |publisher=A.M.S. |isbn=0-8218-3922-5 |page=253}}
</ref>
 
==Counit and Unit==
 
Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit [[natural transformation]]s. Using the notation from the previous section, the counit
 
:<math>\varepsilon : FG \to 1_{\mathcal{C}}</math>
 
has [[natural transformation#Definition|component]]s
 
:<math>\varepsilon_Z : \operatorname{Hom}_S (X, Z) \otimes_R X \to Z</math>
 
given by evaluation:  For
 
:<math>\phi \in \operatorname{Hom}_R (X, Z) \quad \text{and} \quad x \in X,</math>
 
:<math>\varepsilon(\phi \otimes x) = \phi(x).</math>
 
The [[natural transformation#Definition|component]]s of the unit
 
:<math>\eta : 1_{\mathcal{D}} \to GF</math>
 
:<math>\eta_Y : Y \to \operatorname{Hom}_S (X, Y \otimes_R X)</math>
 
are defined as follows: For ''y'' in ''Y'',
 
:<math>\eta_Y(y) \in \operatorname{Hom}_S (X, Y \otimes_R X)</math>
 
is a right ''S''-module homomorphism given by
 
:<math>\eta_Y(y)(t) = y \otimes t \quad \text{for } t \in X.</math>
 
The [[Adjoint_pair#Definition_via_counit-unit_adjunction|counit and unit equations]] can now be explicitly verified.  For
''Y'' in ''C'',
 
:<math>
\varepsilon_{FY}\circ F(\eta_Y) :
Y \otimes_R X \to
\operatorname{Hom}_S (X , Y) \otimes_R X \to
Y \otimes_R X
</math>
 
is given on [[Simple_tensor#Tensor_rank|simple tensor]]s of ''Y''⊗''X'' by
 
:<math>\varepsilon_{FY}\circ F(\eta_Y)(y \otimes x) = \eta_Y(y)(x) = y \otimes x.</math>
 
Likewise,
 
:<math>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).
</math>
 
For φ in ''Hom''<sub>S</sub>(''X'', ''Z''),  
 
:<math>G(\varepsilon_Z)\circ\eta_{GZ}(\phi)</math>
 
is a right ''S''-module homomorphism defined by
 
:<math>G(\varepsilon_Z)\circ\eta_{GZ}(\phi)(x) = \varepsilon_{Z}(\phi \otimes x) = \phi(x)</math>
 
and therefore
 
:<math>G(\varepsilon_Z)\circ\eta_{GZ}(\phi) = \phi.</math>
 
==References==
{{reflist}}
* {{Citation|first = Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki | title = Elements of mathematics, Algebra I| publisher = Springer-Verlag | year = 1989|isbn=3-540-64243-9}}
 
[[Category:Category theory|*]]
[[Category:Tensors]]

Latest revision as of 15:27, 17 November 2014

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