Duality is an adjunction between a category of co/presheaf under the co/Yoneda embedding.
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]
and the co-Yoneda embedding[1][10][8][11] (a.k.a. contravariant Yoneda embedding[12][note 1] or the dual Yoneda embedding[17]) 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
Isbell duality is the relationship between Yoneda embedding and co-Yoneda embedding;
Rutten, J.J.M.M. (1998), "Weighted colimits and formal balls in generalized metric spaces", Topology and Its Applications, 89 (1–2): 179–202, doi:10.1016/S0166-8641(97)00224-1
Sturtz, K. (2019). "Erratum and Addendum: The factorization of the Giry monad". arXiv:1907.00372 [math.CT].
Pratt, Vaughan (1996), "Broadening the denotational semantics of linear logic", Electronic Notes in Theoretical Computer Science, 3: 155–166, doi:10.1016/S1571-0661(05)80415-3
^Note that: the contravariant Yoneda embedding written in the article is replaced with the opposite category for both domain and codomain from that written in the textbook.[13] See opposite functor.[14][15] In addition, this pair of Yoneda embeddings is collectively called the two Yoneda embeddings.[16]