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.^[1][2] That is a duality between covariant and contravariant representable presheaves associated with an objects of categories under the Yoneda embedding.^[3]

Definition

The Yoneda embedding

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

${\displaystyle X\mapsto \mathrm {hom} (-,X).}$

and the co-Yoneda embedding^[4][5] or the dual Yoneda embedding^[6]

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

${\displaystyle X\mapsto \mathrm {hom} (X,-).}$

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; ${\displaystyle \left({\mathcal {O}}\dashv \mathrm {Spec} \right)\colon \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]{\underset {\mathrm {Spec} }{\overset {\mathcal {O}}{\rightleftarrows }}}\left[{\mathcal {A}},{\mathcal {V}}\right]^{op}}$^{[1][4][7][8][5]}.

See also

• Kan extension

References

Bibliography

• 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
• 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
• 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, London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, ISBN 0-521-28702-2, MR 0651714.
• Lawvere, F. W. (1986), "Taking categories seriously (p. 169)", Revista Colombiana de Matemáticas, 20 (3–4): 147–178, MR 0948965
• 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
• 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
• 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" (PDF), tetrapharmakon.github.io
• Ivan Di Liberti; Loregian, Fosco (2019). "On the unicity of formal category theories". arXiv:1901.01594 [math.CT].
• "Isbell duality", ncatlab.org
• "Yoneda embedding", ncatlab.org
• "co-Yoneda lemma", ncatlab.org