Isbell conjugacy (named after John R. Isbell) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986.^[1]

Definition

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

The Isbell conjugacy is an adjunction between the categories ${\displaystyle {\mathcal {V}}^{{\mathcal {A}}^{op}}}$ and ${\displaystyle ({\mathcal {V}}^{\mathcal {A}})^{op}}$ arising from the Yoneda embedding ${\displaystyle Y:{\mathcal {A}}\rightarrow {\mathcal {V}}^{{\mathcal {A}}^{op}}}$ and the dual Yoneda embedding ${\displaystyle Z:{\mathcal {A}}\rightarrow ({\mathcal {V}}^{\mathcal {A}})^{op}}$.

References

Bibliography

• Kelly, Gregory Maxwell (1982), Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, ISBN 0-521-28702-2, MR 0651714.
• 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.
• Avery, Tom; Leinster, Tom (2021), "Isbell conjugacy and the reflexive completion", Theory and Applications of Categories, 36: 306–347, arXiv:2102.08290
• Baez, John C. (2022), "Isbell Duality", Notices Amer. Math. Soc., 70: 140–141, arXiv:2212.11079
• 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
• 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
• 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
• 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, S2CID 15424936
• "Isbell duality". ncatlab.org.