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
[edit]Yoneda embedding
[edit]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[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
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Isbell_duality.svg/220px-Isbell_duality.svg.png)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Nerve_and_realization_%28ver._left_kan_extension%29.svg/220px-Nerve_and_realization_%28ver._left_kan_extension%29.svg.png)
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][22][23][10][24]
The functors of Isbell duality are such that and .[22][25][note 2]
See also
[edit]References
[edit]- ^ a b c d e f (Baez 2022)
- ^ (Di Liberti 2020, 2. Isbell duality)
- ^ a b (Lawvere 1986, p. 169)
- ^ (Rutten 1998)
- ^ (Melliès & Zeilberger 2018)
- ^ (Willerton 2013)
- ^ (Space and quantity in nlab)
- ^ a b (Yoneda embedding in nlab)
- ^ (Valence 2017, Corollaire 2)
- ^ a b (Isbell duality in nlab)
- ^ (Valence 2017, Définition 67)
- ^ (Di Liberti & Loregian 2019, Definition 5.12)
- ^ (Riehl 2016, Theorem 3.4.6.)
- ^ (Starr 2020, Example 4.7.)
- ^ (Opposite functors in nlab)
- ^ (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
[edit]- 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
- Gutierres, Gonçalo; Hofmann, Dirk (2013), "Approaching Metric Domains", Applied Categorical Structures, 21 (6): 617–650, arXiv:1103.4744, doi:10.1007/s10485-011-9274-z, S2CID 254225188
- 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
- Imamura, Yuki (2022), "Grothendieck Enriched Categories", Applied Categorical Structures, 30 (5): 1017–1041, arXiv:2105.05108, doi:10.1007/s10485-022-09681-1
Footnote
[edit]- ^ 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]
- ^ For the symbol Lan, see left Kan extension.
External links
[edit]- Loregian, Fosco (2018), "Kan extensions" (PDF), tetrapharmakon.github.io
- Di Liberti, Ivan; Loregian, Fosco (2019). "On the unicity of formal category theories". arXiv:1901.01594 [math.CT].
- Sorokin, Alex (2022), Derived profunctors, Boston, Massachusetts : Northeastern University, doi:10.17760/D20467288, hdl:2047/D20467288
- 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
- "Opposite functors", ncatlab.org