Isbell duality

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

Yoneda embedding

The (covariant) Yoneda embedding is a covariant functor from a small category ${\displaystyle {\mathcal {A}}}$ into the category of presheaves ${\displaystyle \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]}$ on ${\displaystyle {\mathcal {A}}}$, taking ${\displaystyle X\in {\mathcal {A}}}$ to the contravariant representable functor: ^{[1] [8] [9]}

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

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

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 ${\displaystyle {\mathcal {A}}}$ into the category of co-presheaves ${\displaystyle \left[{\mathcal {A}},{\mathcal {V}}\right]^{op}}$ on ${\displaystyle {\mathcal {A}}}$, taking ${\displaystyle X\in {\mathcal {A}}}$ to the covariant representable functor:

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

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

Every functor ${\displaystyle F\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}}$ has an Isbell conjugate ^[1] ${\displaystyle F^{\ast }\colon {\mathcal {A}}\to {\mathcal {V}}}$, given by

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

In contrast, every functor ${\displaystyle G\colon {\mathcal {A}}\to {\mathcal {V}}}$ has an Isbell conjugate ^[1] ${\displaystyle G^{\ast }\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}}$ given by

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

Isbell duality

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

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

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 duality is an adjunction between the categories; ${\displaystyle \left({\mathcal {O}}\dashv \mathrm {Spec} \right)\colon \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]{\underset {\mathrm {Spec} }{\overset {\mathcal {O}}{\rightleftarrows }}}\left[{\mathcal {A}},{\mathcal {V}}\right]^{op}}$. ^{[3] [1] [18] [19] [10] [20]}

The functors ${\displaystyle {\mathcal {O}}\dashv \mathrm {Spec} }$ of Isbell duality are such that ${\displaystyle {\mathcal {O}}\cong \mathrm {Lan_{Y}Z} }$ and ${\displaystyle \mathrm {Spec} \cong \mathrm {Lan_{Z}Y} }$. ^{[18] [21] [note 2]}

See also

• Kan extension
• Limit (category theory)

References

1. ( Baez 2022 )
2. ( Di Liberti 2020 , 2. Isbell duality)
3. ( Lawvere 1986 , p. 169)
4. ( Rutten 1998 )
5. ( Melliès & Zeilberger 2018 )
6. ( Willerton 2013 )
7. ( Space and quantity in nlab )
8. ( Yoneda embedding in nlab )
9. ( Valence 2017 , Corollaire 2)
10. ( Isbell duality in nlab )
11. ( Valence 2017 , Définition 67)
12. ( Di Liberti & Loregian 2019 , Definition 5.12)
13. (Riehl 2016, Theorem 3.4.6.)
14. (Starr 2020, Example 4.7.)
15. (Opposite functors in nlab)
16. (Pratt 1996, §.4 Symmetrizing the Yoneda embedding)
17. ( Day & Lack 2007 , §9. Isbell conjugacy)
18. ( Di Liberti 2020 , Remark 2.4)
19. ( Fosco 2021 )
20. ( Valence 2017 , Définition 68)
21. ( Di Liberti & Loregian 2019 , Lemma 5.13.)

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, 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

Footnote

1. 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 opposite functor. ^{[14] [15]} In addition, this pair of Yoneda embeddings is collectively called the two Yoneda embeddings. ^[16]
2. For the symbol Lan, see left Kan extension.

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