Isbell duality

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Template:Short description Template:CS1 config Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (named after John R. IsbellTemplate:R[1]) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986.[2][3] That is a duality between covariant and contravariant representable presheaves associated with an objects of categories under the Yoneda embedding.[4][5] In addition, Lawvere[6] is states as follows; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics".[7]

Definition

Yoneda embedding

The (covariant) Yoneda embedding is a covariant functor from a small category π’œ into the category of presheaves [π’œop,𝒱] on π’œ, taking X∈π’œ to the contravariant representable functor: Template:R[8][9][10]

Y(h∙):π’œ→[π’œop,𝒱]

X↦hom(−,X).

and the co-Yoneda embeddingTemplate:RTemplate:RTemplate:R[11] (a.k.a. contravariant Yoneda embedding[12]Template:Refn or the dual Yoneda embedding[13]) is a contravariant functor from a small category π’œ into the opposite of the category of co-presheaves [π’œ,𝒱]op on π’œ, taking X∈π’œ to the covariant representable functor:

Z(h∙op):π’œ→[π’œ,𝒱]op

X↦hom(X,−).

Every functor F:π’œop→𝒱 has an Isbell conjugateTemplate:R F∗:π’œ→𝒱, given by

F∗(X)=hom(F,y(X)).

In contrast, every functor G:π’œ→𝒱 has an Isbell conjugateTemplate:R G∗:π’œop→𝒱 given by

G∗(X)=hom(z(X),G).

Isbell duality

Origin of symbols π’ͺ and Spec: Template:Harvtxt says that; "π’ͺ" assigns to each general space the algebra of functions on it, whereas "Spec" assigns to each algebra its β€œspectrum” which is a general space.
note:In order for this commutative diagram to hold, it is required that π’œ is small and E is co-complete.[14][15][16][17]

Isbell duality is the relationship between Yoneda embedding and co-Yoneda embeddingοΌ›

Let 𝒱 be a symmetric monoidal closed category, and let π’œ be a small category enriched in 𝒱.

The Isbell duality is an adjunction between the functor categories; (π’ͺ⊣Spec):[π’œop,𝒱]⇄π’ͺSpec[π’œ,𝒱]op.[18]Template:R[19][20][21][22]

The functors π’ͺ⊣Spec of Isbell duality are such that π’ͺ≅LanYZ and Spec≅LanZY.Template:R[23][note 1]

See also

References

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Bibliography

Footnote

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  1. ↑ Template:Harv
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