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Duality is an adjunction between a category of co/presheaf under the co/Yoneda embedding. From Wikipedia, the free encyclopedia
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] Also, Lawvere (1986, p. 169) says that; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics".[7]
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][8][9]
and the co-Yoneda embedding[1][10][8][11] (a.k.a. contravariant Yoneda embedding[12][note 1] or the dual Yoneda embedding[18]) is a contravariant functor (a covariant functor from the opposite category) from a small category into the category of co-presheaves on , taking to the covariant representable functor:
Every functor has an Isbell conjugate[1] , given by
In contrast, every functor has an Isbell conjugate[1] given by
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 categories; .[3][1][23][24][10][25]
The functors of Isbell duality are such that and .[23][26][note 2]
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