Isbell duality

Short description: Construction of enriched category theory

Isbell conjugacy or Isbell duality (named after John R. Isbell) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986. That is a duality between covariant and contravariant representable presheaves associated with an objects of categories under the Yoneda embedding.

Definition

The Yoneda embedding

[math]\displaystyle{ Y:\mathcal{A} \rightarrow \left[\mathcal{A}^{op}, \mathcal{V} \right] }[/math]

[math]\displaystyle{ X \mapsto \mathrm{hom} (-,X). }[/math]

and the co-Yoneda embedding or the dual Yoneda embedding

[math]\displaystyle{ Z:\mathcal{A} \rightarrow \left[\mathcal{A}, \mathcal{V} \right]^{op} }[/math]

[math]\displaystyle{ X \mapsto \mathrm{hom} (X,-). }[/math]

Let [math]\displaystyle{ \mathcal{V} }[/math] be a symmetric monoidal closed category, and let [math]\displaystyle{ \mathcal{A} }[/math] be a small category enriched in [math]\displaystyle{ \mathcal{V} }[/math].

The Isbell conjugacy is an adjunction between the categories; [math]\displaystyle{ \left(\mathcal{O} \dashv \mathrm{Spec} \right) \colon \left[\mathcal{A}^{op}, \mathcal{V} \right] {\underset{\mathrm{Spec}}{\overset{\mathcal{O}}{\rightleftarrows}}} \left[\mathcal{A}, \mathcal{V} \right]^{op} }[/math].

The functors [math]\displaystyle{ \mathcal{O} \dashv \mathrm{Spec} }[/math] of Isbell duality are such that [math]\displaystyle{ \mathcal{O} \cong \mathrm{Lan_{Y}Z} }[/math] and [math]\displaystyle{ \mathrm{Spec} \cong \mathrm{Lan_{Z}Y} }[/math].

See also

• Kan extension

Bibliography

• Avery, Tom; Leinster, Tom (2021), "Isbell conjugacy and the reflexive completion", Theory and Applications of Categories 36: 306–347 
• Baez, John C. (2022), "Isbell Duality", Notices Amer. Math. Soc. 70: 140–141, https://www.ams.org/journals/notices/202301/noti2602/noti2602.html?adat=January%202023&trk=2602&galt=none&cat=column&pdfissue=202301&pdffile=rnoti-p140.pdf 
• Day, Brian J.; Lack, Stephen (2007), "Limits of small functors", Journal of Pure and Applied Algebra 210 (3): 651–663, doi:10.1016/j.jpaa.2006.10.019 .
• Di Liberti, Ivan (2020), "Codensity: Isbell duality, pro-objects, compactness and accessibility", Journal of Pure and Applied Algebra 224 (10), doi:10.1016/j.jpaa.2020.106379 
• Fosco, Loregian (22 July 2021), (Co)end Calculus, Cambridge University Press, doi:10.1017/9781108778657, ISBN 9781108746120, https://books.google.com/books?id=cfIuEAAAQBAJ&dq&pg=PA90 
• Gutierres, Gonçalo; Hofmann, Dirk (2013), "Approaching Metric Domains", Applied Categorical Structures 21 (6): 617–650, doi:10.1007/s10485-011-9274-z 
• Shen, Lili; Zhang, Dexue (2013), "Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions", Theory and Applications of Categories 28 (20): 577-615, http://www.tac.mta.ca/tac/volumes/28/20/28-20.pdf 
• 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 
• {{citation

| last = Kelly | first = Gregory Maxwell
| isbn = 0-521-28702-2
| mr = 651714
| publisher = Cambridge University Press, Cambridge-New York
| series = London Mathematical Society Lecture Note Series
| title = Basic concepts of enriched category theory
| volume = 64

• "Taking categories seriously (p. 169)", Revista Colombiana de Matemáticas 20 (3–4): 147–178, 1986, http://eudml.org/doc/181771 
• Melliès, Paul-André; Zeilberger, Noam (2018), "An Isbell duality theorem for type refinement systems", Mathematical Structures in Computer Science 28 (6): 736–774, doi:10.1017/S0960129517000068 
• 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, doi:10.1016/j.aim.2018.10.007 
• Wood, R.J (1982), "Some remarks on total categories", Journal of Algebra 75 (2): 538–545, doi:10.1016/0021-8693(82)90055-2 

External links

• Loregian, Fosco (2018), Kan extensions, https://tetrapharmakon.github.io/stuff/REFCARDS_kan.pdf 
• Di Liberti, Ivan; Loregian, Fosco (2019). "On the unicity of formal category theories". arXiv:1901.01594 [math.CT].
• Isbell duality, https://ncatlab.org/nlab/show/Isbell+duality 
• Yoneda embedding, https://ncatlab.org/nlab/show/Yoneda+embedding 
• co-Yoneda lemma, https://ncatlab.org/nlab/show/co-Yoneda+lemma