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]} In addition, Lawvere ^[7] is states as follows; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics". ^[8]

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] [9] [10] [11]}

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

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

and the co-Yoneda embedding ^{[1] [12] [9] [13]} (a.k.a. contravariant Yoneda embedding ^{[14] [note 1]} or the dual Yoneda embedding ^[21] ) is a contravariant functor from a small category ${\displaystyle {\mathcal {A}}}$ into the opposite of 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:

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

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

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.

note:In order for this commutative diagram to hold, it is required that ${\displaystyle {\mathcal {A}}}$ is small and E is co-complete.

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

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

These two functors are not typically inverses, or even natural isomorphisms. Isbell duality asserts that the relationship between these two functors is an adjunction. ^[1]

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 functor 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] [3] [12] [26] [27] [28]}

The functors ${\displaystyle {\mathcal {O}}\dashv \mathrm {Spec} }$ of Isbell duality are such that ${\displaystyle {\mathcal {O}}\cong \mathrm {Lan_{Y}Z} }$ and ${\displaystyle \mathrm {Spec} \cong \mathrm {Lan_{Z}Y} }$. ^{[26] [29] [note 2]}

See also

• Kan extension
• Limit (category theory)
• Isbell completion
• Profunctor

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. ( Lawvere 1986 , p. 169)
8. ( Space and quantity in nlab )
9. ( Yoneda embedding in nlab )
10. ( Valence 2017 , Corollaire 2)
11. ( Awodey 2006 , Definition 8.1.)
12. ( Isbell duality in nlab )
13. ( Valence 2017 , Définition 67)
14. ( Di Liberti & Loregian 2019 , Definition 5.12)
15. (Riehl 2016, Theorem 3.4.11.)
16. (Leinster 2004, (c) and (c').)
17. (Riehl 2016, Definition 1.3.11.)
18. (Starr 2020, Example 4.7.)
19. (Opposite functors in nlab)
20. (Pratt 1996, §.4 Symmetrizing the Yoneda embedding)
21. ( Day & Lack 2007 , §9. Isbell conjugacy)
22. ( Di Liberti 2020 , Remark 2.3 (The (co)nerve construction).)
23. ( Kelly 1982 , Proposition 4.33)
24. ( Riehl 2016 , Remark 6.5.9.)
25. ( Imamura 2022 , Theorem 2.4)
26. ( Di Liberti 2020 , Remark 2.4)
27. ( Fosco 2021 )
28. ( Valence 2017 , Définition 68)
29. ( 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
• Awodey, Steve (2006), Category Theory, doi:10.1093/acprof:oso/9780198568612.001.0001, ISBN   978-0-19-856861-2
• Baez, John C. (2022), "Isbell Duality" (PDF), Notices Amer. Math. Soc., 70: 140–141, arXiv: 2212.11079 , doi:10.1090/noti2602
• 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
• Imamura, Yuki (2022), "Grothendieck Enriched Categories", Applied Categorical Structures, 30 (5): 1017–1041, arXiv: 2105.05108 , doi:10.1007/s10485-022-09681-1
• 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
• 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
• 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]
• 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. ^{[15] [16]} See variance of functor, pre/post-composition, ^[17] and opposite functor. ^{[18] [19]} It follows from Yoneda lemma that ${\displaystyle \left[{\mathcal {A}},{\mathcal {V}}\right]^{op}\left(Y,{\mathcal {A}}(X,-)\right)\simeq Y(X)}$. In addition, this pair of Yoneda embeddings is collectively called the two Yoneda embeddings. ^[20]
2. For the symbol Lan, see left Kan extension.

External links

• Di Liberti, Ivan; Loregian, Fosco (2019), "On the unicity of formal category theories", arXiv: 1901.01594 [math.CT]
• Leinster, Tom (2004), "An ABC of Category Theory ch.4 Representability" (PDF), maths.ed.ac.uk
• Loregian, Fosco (2018), "Kan extensions" (PDF), tetrapharmakon.github.io
• Starr, Jason (2020), "Some Notes on Category Theory in MAT 589 Algebraic Geometry" (PDF), math.stonybrook.edu
• 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)
• "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