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Isbell duality

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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]

Definition

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Yoneda embedding

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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[19]) 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

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Origin of symbols and : Lawvere (1986, p. 169) 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.
note:In order for this commutative diagram to hold, it is required that E is co-complete.[20][21][22][23]

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][24][25][10][26]

The functors of Isbell duality are such that and .[24][27][note 2]

See also

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References

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  1. ^ a b c d e f (Baez 2022)
  2. ^ (Di Liberti 2020, 2. Isbell duality)
  3. ^ a b (Lawvere 1986, p. 169)
  4. ^ (Rutten 1998)
  5. ^ (Melliès & Zeilberger 2018)
  6. ^ (Willerton 2013)
  7. ^ (Space and quantity in nlab)
  8. ^ a b (Yoneda embedding in nlab)
  9. ^ (Valence 2017, Corollaire 2)
  10. ^ a b (Isbell duality in nlab)
  11. ^ (Valence 2017, Définition 67)
  12. ^ (Di Liberti & Loregian 2019, Definition 5.12)
  13. ^ (Riehl 2016, Theorem 3.4.11.)
  14. ^ (Leinster 2004, (c) and (c').)
  15. ^ (Riehl 2016, Definition 1.3.11.)
  16. ^ (Starr 2020, Example 4.7.)
  17. ^ (Opposite functors in nlab)
  18. ^ (Pratt 1996, §.4 Symmetrizing the Yoneda embedding)
  19. ^ (Day & Lack 2007, §9. Isbell conjugacy)
  20. ^ (Di Liberti 2020, Remark 2.3 (The (co)nerve construction).)
  21. ^ (Kelly 1982, Proposition 4.33)
  22. ^ (Riehl 2016, Remark 6.5.9.)
  23. ^ (Imamura 2022, Theorem 2.4)
  24. ^ a b (Di Liberti 2020, Remark 2.4)
  25. ^ (Fosco 2021)
  26. ^ (Valence 2017, Définition 68)
  27. ^ (Di Liberti & Loregian 2019, Lemma 5.13.)

Bibliography

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Footnote

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  1. ^ 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][14] See variance of functor, pre/post-composition,[15] and opposite functor.[16][17] In addition, this pair of Yoneda embeddings is collectively called the two Yoneda embeddings.[18]
  2. ^ For the symbol Lan, see left Kan extension.
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