Isbell duality

In mathematics, 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]}

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

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

and the co-Yoneda embedding ^{[1] [11]} (a.k.a. dual Yoneda embedding ^[12] ) 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 ${\displaystyle {\mathcal {O}}}$ (“ring of functions”) and ${\displaystyle \mathrm {Spec} }$ (“spectrum”): 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] [11] [17] [18]}

Applying the nerve construction, the functors ${\displaystyle {\mathcal {O}}\dashv \mathrm {Spec} }$ of Isbell duality are such that ${\displaystyle {\mathcal {O}}\cong \mathrm {Lan_{Y}Z} }$ and ${\displaystyle \mathrm {Spec} \cong \mathrm {Lan_{Z}Y} }$. ^{[17] [19] [note 1]}

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. ( Awodey 2006 , Definition 8.1.)
11. ( Isbell duality in nlab )
12. ( Day & Lack 2007 , §9. Isbell conjugacy)
13. ( Di Liberti 2020 , Remark 2.3 (The (co)nerve construction).)
14. ( Kelly 1982 , Proposition 4.33)
15. ( Riehl 2016 , Remark 6.5.9.) harv error: no target: CITEREFRiehl2016 (help)
16. ( Imamura 2022 , Theorem 2.4)
17. ( Di Liberti 2020 , Remark 2.4)
18. ( Fosco 2021 )
19. ( 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 , doi:10.70930/tac/r1jknjot
• 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

Barr, Michael; Kennison, John F.; Raphael, R. (2009), "Isbell duality for modules", Theory and Applications of Categories, 22: 401–419, doi:10.70930/tac/1zcfxg2x

• 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) 106379, 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 , doi:10.70930/tac/6l0334s5
• 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
• 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 , doi:10.70930/tac/rkp3zgxc

Footnote

1. 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]
• Loregian, Fosco (2018), "Kan extensions" (PDF), tetrapharmakon.github.io, archived from the original (PDF) on 9 January 2024
• 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