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;
^(Pratt 1996, §.4 Symmetrizing the Yoneda embedding)
^(Day & Lack 2007, §9. Isbell conjugacy)
^(Di Liberti 2020, Remark 2.3 (The (co)nerve construction).)
^(Kelly 1982, Proposition 4.33)
^(Riehl 2016, Remark 6.5.9.)
^(Imamura 2022, Theorem 2.4)
^ a b(Di Liberti 2020, Remark 2.4)
^(Fosco 2021)
^(Valence 2017, Définition 68)
^(Di Liberti & Loregian 2019, Lemma 5.13.)
Bibliography
Avery, Tom; Leinster, Tom (2021), "Isbell conjugacy and the reflexive completion" (PDF), Theory and Applications of Categories, 36: 306–347, arXiv:2102.08290
Baez, John C. (2022), "Isbell Duality" (PDF), Notices Amer. Math. Soc., 70: 140–141, arXiv:2212.11079
Day, Brian J.; Lack, Stephen (2007), "Limits of small functors", Journal of Pure and Applied Algebra, 210 (3): 651–663, arXiv:math/0610439, doi:10.1016/j.jpaa.2006.10.019, MR 2324597, S2CID 15424936.
Di Liberti, Ivan (2020), "Codensity: Isbell duality, pro-objects, compactness and accessibility", Journal of Pure and Applied Algebra, 224 (10), arXiv:1910.01014, doi:10.1016/j.jpaa.2020.106379, S2CID 203626566
Fosco, Loregian (22 July 2021), (Co)end Calculus, Cambridge University Press, arXiv:1501.02503, doi:10.1017/9781108778657, ISBN 9781108746120, S2CID 237839003
Shen, Lili; Zhang, Dexue (2013), "Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions" (PDF), Theory and Applications of Categories, 28 (20): 577–615, arXiv:1307.5625
Isbell, J. R. (1960), "Adequate subcategories", Illinois Journal of Mathematics, 4 (4), doi:10.1215/ijm/1255456274
Isbell, John R. (1966), "Structure of categories", Bulletin of the American Mathematical Society, 72 (4): 619–656, doi:10.1090/S0002-9904-1966-11541-0, S2CID 40822693
Kelly, Gregory Maxwell (1982), Basic concepts of enriched category theory (PDF), London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, ISBN 0-521-28702-2, MR 0651714.[page needed]
Lawvere, F. W. (1986), "Taking categories seriously", Revista Colombiana de Matemáticas, 20 (3–4): 147–178, MR 0948965
Lawvere, F. W. (2005), "Taking categories seriously" (PDF), Reprints in Theory and Applications of Categories (8): 1–24, MR 0948965
Lawvere, F. William (February 2016), "Birkhoff's Theorem from a geometric perspective: A simple example", Categories and General Algebraic Structures with Applications, 4 (1): 1–8
Melliès, Paul-André; Zeilberger, Noam (2018), "An Isbell duality theorem for type refinement systems", Mathematical Structures in Computer Science, 28 (6): 736–774, arXiv:1501.05115, doi:10.1017/S0960129517000068, S2CID 2716529
Riehl, Emily (2016), Category Theory in Context, Dover Publications, Inc Mineola, New York, ISBN 9780486809038
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, Kirk (2018), "The factorization of the Giry monad", Advances in Mathematics, 340: 76–105, arXiv:1707.00488, doi:10.1016/j.aim.2018.10.007
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
Wood, R.J (1982), "Some remarks on total categories", Journal of Algebra, 75 (2): 538–545, doi:10.1016/0021-8693(82)90055-2
Willerton, Simon (2013), "Tight spans, Isbell completions and semi-tropical modules" (PDF), Theory and Applications of Categories, 28 (22): 696–732, arXiv:1302.4370
^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 variance of functor and opposite functor.[14][15] In addition, this pair of Yoneda embeddings is collectively called the two Yoneda embeddings.[16]
Valence, Arnaud (2017), Esquisse d'une dualité géométrico-algébrique multidisciplinaire : la dualité d'Isbell, Thèse en cotutelle en Philosophie – Étude des Systèmes, soutenue le 30 mai 2017. (PDF)
Starr, Jason (2020), "Some Notes on Category Theory in MAT 589 Algebraic Geometry" (PDF), math.stonybrook.edu
"Isbell duality", ncatlab.org
"space and quantity", ncatlab.org
"Yoneda embedding", ncatlab.org
"co-Yoneda lemma", ncatlab.org
"copresheaf", ncatlab.org
"Natural transformations and presheaves: Remark 1.28. (presheaves as generalized spaces)", ncatlab.org