Isbell duality

In mathematics, Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (named after John R. IsbellTemplate:R/superscript) 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. In addition, Lawvere says; "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 $𝒜$ into the category of presheaves $[{𝒜}^{op},𝒱]$ on $𝒜$, taking $X\in 𝒜$ to the contravariant representable functor: Template:R/superscript

$y(h^{\bullet }):𝒜\to [{𝒜}^{op},𝒱]$

$X\mapsto \mathrm{h}\mathrm{o}\mathrm{m}(-,X).$

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

$z({h_{\bullet }}^{op}):𝒜\to {[𝒜,𝒱]}^{op}$

$X\mapsto \mathrm{h}\mathrm{o}\mathrm{m}(X,-).$

Isbell duality

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

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

Every functor $F:{𝒜}^{\mathrm{o}\mathrm{p}}\to 𝒱$ has an Isbell conjugate of a functorTemplate:R/superscript $F^{\ast }:𝒜\to 𝒱$, given by

$F^{\ast }(X)=\mathrm{h}\mathrm{o}\mathrm{m}(F,y(X)).$

In contrast, every functor $G:𝒜\to 𝒱$ has an Isbell conjugate of a functorTemplate:R/superscript $G^{\ast }:{𝒜}^{\mathrm{o}\mathrm{p}}\to 𝒱$ given by

$G^{\ast }(X)=\mathrm{h}\mathrm{o}\mathrm{m}(z(X),G).$

These two functors are not typically inverses, or even natural isomorphisms. Isbell duality asserts that the relationship between these two functors is an adjunction.Template:R/superscript

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

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

The Isbell duality is an adjunction between the functor categories; $(𝒪⊣\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}):[{𝒜}^{op},𝒱]\underset{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}{\stackrel{𝒪}{\rightleftarrows }}{[𝒜,𝒱]}^{op}$.Template:R/superscript

Applying the nerve construction, the functors $𝒪⊣\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$ of Isbell duality are such that $𝒪\cong \mathrm{L}\mathrm{a}{\mathrm{n}}_{\mathrm{y}}\mathrm{z}$ and $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\cong \mathrm{L}\mathrm{a}{\mathrm{n}}_{\mathrm{z}}\mathrm{y}$.Template:R/superscript

See also

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

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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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