Isbell duality

The (covariant) Yoneda embedding is a covariant functor from a small category $\backslash mathcal\{A\}$ into the category of presheaves $\backslash left[\backslash mathcal\{A\}^\{op\},\; \backslash mathcal\{V\}\; \backslash right]$ on $\backslash mathcal\{A\}$, taking $X\; \backslash in\; \backslash mathcal\{A\}$ to the contravariant representable functor: {{R|Baez2022}}{{harv|Yoneda embedding in nlab}}

{{harv|Valence|2017|loc=Corollaire 2}}{{harv|Awodey|2006|loc= Definition 8.1.}}

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

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

and the co-Yoneda embedding{{R|Baez2022}}{{R|nlab1}}{{R|nlab2}}{{harv|Valence|2017|loc=Définition 67}} (a.k.a. contravariant Yoneda embedding{{harv|Di Liberti |Loregian|2019|loc=Definition 5.12}}{{refn|group=note|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.{{harv|Riehl|2016|loc=Theorem 3.4.11.}}{{harv|Leinster|2004|loc=(c) and (c').}} See variance of functor, pre/post-composition,{{harv|Riehl|2016|loc=Definition 1.3.11.}} and opposite functor.{{harv|Starr|2020|loc=Example 4.7.}}{{harv|Opposite functors in nlab}} It follows from Yoneda lemma that $\backslash left[\backslash mathcal\{A\},\; \backslash mathcal\{V\}\; \backslash right]^\{op\}\; \backslash left(Y,\; \backslash mathcal\{A\}(X,-)\; \backslash right)\; \backslash simeq\; Y(X)$. In addition, this pair of Yoneda embeddings is collectively called the two Yoneda embeddings.{{harv|Pratt|1996|loc=§.4 Symmetrizing the Yoneda embedding}}}} or the dual Yoneda embedding{{harv|Day|Lack|2007|loc=§9. Isbell conjugacy}}) is a contravariant functor from a small category $\backslash mathcal\{A\}$ into the opposite of the category of co-presheaves $\backslash left[\backslash mathcal\{A\},\; \backslash mathcal\{V\}\; \backslash right]^\{op\}$ on $\backslash mathcal\{A\}$, taking $X\; \backslash in\; \backslash mathcal\{A\}$ to the covariant representable functor:

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

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

File:Isbell duality.svg

[[File:Nerve and realization (ver. left kan extension).svg|thumb|note:In order for this commutative diagram to hold, it is required that $\backslash mathcal\{A\}$ is small and E is co-complete.{{harv|Di Liberti

|2020|loc=Remark 2.3 (The (co)nerve construction).}}{{harv|Kelly|1982|loc=Proposition 4.33 }}{{harv|Riehl|2016|loc=Remark 6.5.9.}}{{harv|Imamura|2022|loc=Theorem 2.4 }}]]

Every functor $F\; \backslash colon\; \backslash mathcal\{A\}^\backslash mathrm\{op\}\backslash to\; \backslash mathcal\{V\}$ has an Isbell conjugate of a functor{{R|Baez2022}} $F^\{\backslash ast\}\; \backslash colon\; \backslash mathcal\{A\}\; \backslash to\; \backslash mathcal\{V\}$, given by

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

In contrast, every functor $G\; \backslash colon\; \backslash mathcal\{A\}\; \backslash to\; \backslash mathcal\{V\}$ has an Isbell conjugate of a functor{{R|Baez2022}} $G^\{\backslash ast\}\; \backslash colon\; \backslash mathcal\{A\}^\backslash mathrm\{op\}\; \backslash to\; \backslash mathcal\{V\}$ given by

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

These two functors are not typically inverses, or even natural isomorphisms. Isbell duality asserts that the relationship between these two functors is an adjunction.{{R|Baez2022}}

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

Let $\backslash mathcal\{V\}$ be a symmetric monoidal closed category, and let $\backslash mathcal\{A\}$ be a small category enriched in $\backslash mathcal\{V\}$.

The Isbell duality is an adjunction between the functor categories; $\backslash left(\backslash mathcal\{O\}\; \backslash dashv\; \backslash mathrm\{Spec\}\; \backslash right)\; \backslash colon\; \backslash left[\backslash mathcal\{A\}^\{op\},\; \backslash mathcal\{V\}\; \backslash right]\; \{\backslash underset\{\backslash mathrm\{Spec\}\}\{\backslash overset\{\backslash mathcal\{O\}\}\{\backslash rightleftarrows\}\}\}$

\left[\mathcal{A}, \mathcal{V} \right]^{op}.{{harv|Baez

|2022}}{{R|Lawvere1986}}{{harv|Isbell duality in nlab}}{{harv|Di Liberti

|2020|loc=Remark 2.4}}{{harv|Fosco

|2021}}{{harv|Valence|2017|loc=Définition 68}}

The functors $\backslash mathcal\{O\}\; \backslash dashv\; \backslash mathrm\{Spec\}$ of Isbell duality are such that $\backslash mathcal\{O\}\; \backslash cong\; \backslash mathrm\{Lan\_\{Y\}Z\}$ and $\backslash mathrm\{Spec\}\; \backslash cong\; \backslash mathrm\{Lan\_\{Z\}Y\}$.{{R|DiLiberti2020}}{{harv|Di Liberti |Loregian|2019|loc=Lemma 5.13.}}For the symbol Lan, see left Kan extension.

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

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{{DEFAULTSORT:Isbell duality}}

Category:Category theory

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Category:Adjoint functors