Isbell duality

Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (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. 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".

Definition

Yoneda embedding

The (covariant) Yoneda embedding is a covariant functor from a small category ${\mathcal {A}}$ into the category of presheaves $\left[{\mathcal {A}}^{op},{\mathcal {V}}\right]$ on ${\mathcal {A}}$, taking $X\in {\mathcal {A}}$ to the contravariant representable functor:

$Y\;(h^{\bullet }):{\mathcal {A}}\rightarrow \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]$

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

and the co-Yoneda embedding (a.k.a. contravariant Yoneda embedding or the dual Yoneda embedding) is a contravariant functor (a covariant functor from the opposite category) from a small category ${\mathcal {A}}$ into the category of co-presheaves $\left[{\mathcal {A}},{\mathcal {V}}\right]^{op}$ on ${\mathcal {A}}$, taking $X\in {\mathcal {A}}$ to the covariant representable functor:

$Z\;({h_{\bullet }}^{op}):{\mathcal {A}}\rightarrow \left[{\mathcal {A}},{\mathcal {V}}\right]^{op}$

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

Every functor $F\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}$ has an Isbell conjugate $F^{\ast }\colon {\mathcal {A}}\to {\mathcal {V}}$, given by

$F^{\ast }(X)=\mathrm {hom} (F,y(X)).$

In contrast, every functor $G\colon {\mathcal {A}}\to {\mathcal {V}}$ has an Isbell conjugate $G^{\ast }\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}$ given by

$G^{\ast }(X)=\mathrm {hom} (z(X),G).$

Isbell duality

Origin of symbols ${\mathcal {O}}$ and $\mathrm {Spec}$: Lawvere (1986, p. 169) says that; "${\mathcal {O}}$" assigns to each general space the algebra of functions on it, whereas "$\mathrm {Spec}$" assigns to each algebra its “spectrum” which is a general space.

note:In order for this commutative diagram to hold, it is required that E is co-complete.

Isbell duality is the relationship between Yoneda embedding and co-Yoneda embedding；

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

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

The functors ${\mathcal {O}}\dashv \mathrm {Spec}$ of Isbell duality are such that ${\mathcal {O}}\cong \mathrm {Lan_{Y}Z}$ and $\mathrm {Spec} \cong \mathrm {Lan_{Z}Y}$.

See also

• Kan extension
• Limit (category theory)

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
• 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.
• 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

External links

• 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