Isbell duality

In mathematics, Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (named after John R. Isbell¹²) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986.³⁴ That is a duality between covariant and contravariant representable presheaves associated with an objects of categories under the Yoneda embedding.⁵⁶ In addition, Lawvere⁷ says; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics".⁸

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: ¹⁹¹⁰

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

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

and the co-Yoneda embedding¹¹¹ (a.k.a. dual Yoneda embedding¹²) 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:

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

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

Every functor ${\displaystyle F\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}}$ has an Isbell conjugate of a functor¹ ${\displaystyle F^{\ast }\colon {\mathcal {A}}\to {\mathcal {V}}}$, 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 functor¹ ${\displaystyle G^{\ast }\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}}$ 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; ${\displaystyle \left({\mathcal {O}}\dashv \mathrm {Spec} \right)\colon \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]{\underset {\mathrm {Spec} }{\overset {\mathcal {O}}{\rightleftarrows }}}\left[{\mathcal {A}},{\mathcal {V}}\right]^{op}}$.^13111718

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 ${\displaystyle \mathrm {Spec} \cong \mathrm {Lan_{z}y} }$.^{1719note 1}