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Isbell duality

Duality is an adjunction between a category of co/presheaf under the co/Yoneda embedding. From Wikipedia, the free encyclopedia

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In mathematics, Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (named after John R. Isbell[1][2]) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986.[3][4] That is a duality between covariant and contravariant representable presheaves associated with an objects of categories under the Yoneda embedding.[5][6] In addition, Lawvere[7] is states as follows; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics".[8]

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Definition

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Yoneda embedding

The (covariant) Yoneda embedding is a covariant functor from a small category into the category of presheaves on , taking to the contravariant representable functor: [1][9][10]

and the co-Yoneda embedding[1][11] (a.k.a. dual Yoneda embedding[12]) is a contravariant functor from a small category into the opposite of the category of co-presheaves on , taking to the covariant representable functor:

Isbell duality

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Origin of symbols (“ring of functions”) and (“spectrum”): Lawvere (1986, p. 169)[failed verification] says that; "" assigns to each general space the algebra of functions on it, whereas "" assigns to each algebra its “spectrum” which is a general space.
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note:In order for this commutative diagram to hold, it is required that is small and E is co-complete.[13][14][15][16]

Every functor has an Isbell conjugate of a functor[1] , given by

In contrast, every functor has an Isbell conjugate of a functor[1] given by

These two functors are not typically inverses, or even natural isomorphisms. Isbell duality asserts that the relationship between these two functors is an adjunction.[1]

Isbell duality is the relationship between Yoneda embedding and co-Yoneda embedding;

Let be a symmetric monoidal closed category, and let be a small category enriched in .

The Isbell duality is an adjunction between the functor categories; .[1][3][11][17][18]

Applying the nerve construction, the functors of Isbell duality are such that and .[17][19][note 1]

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