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] says; "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.)
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

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• Awodey, Steve (2006), Category Theory, doi:10.1093/acprof:oso/9780198568612.001.0001, ISBN   978-0-19-856861-2
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• 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
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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