Isbell duality

In mathematics, Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (named after John R. Isbell) 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 {\displaystyle {\mathcal {A}}} into the category of presheaves [ A o p , V ] {\displaystyle \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]} on A {\displaystyle {\mathcal {A}}} , taking X ∈ A {\displaystyle X\in {\mathcal {A}}} to the contravariant representable functor:

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

X ↦ h o m ( − , X ) . {\displaystyle X\mapsto \mathrm {hom} (-,X).}

and the co-Yoneda embedding (a.k.a. dual Yoneda embedding) is a contravariant functor from a small category A {\displaystyle {\mathcal {A}}} into the opposite of the category of co-presheaves [ A , V ] o p {\displaystyle \left[{\mathcal {A}},{\mathcal {V}}\right]^{op}} on A {\displaystyle {\mathcal {A}}} , taking X ∈ A {\displaystyle X\in {\mathcal {A}}} to the covariant representable functor:

z ( h ∙ o p ) : A → [ A , V ] o p {\displaystyle z\;({h_{\bullet }}^{op}):{\mathcal {A}}\rightarrow \left[{\mathcal {A}},{\mathcal {V}}\right]^{op}}

X ↦ h o m ( X , − ) . {\displaystyle X\mapsto \mathrm {hom} (X,-).}

Isbell duality

Every functor F : A o p → V {\displaystyle F\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}} has an Isbell conjugate of a functor F ∗ : A → V {\displaystyle F^{\ast }\colon {\mathcal {A}}\to {\mathcal {V}}} , given by

F ∗ ( X ) = h o m ( F , y ( X ) ) . {\displaystyle F^{\ast }(X)=\mathrm {hom} (F,y(X)).} In contrast, every functor G : A → V {\displaystyle G\colon {\mathcal {A}}\to {\mathcal {V}}} has an Isbell conjugate of a functor G ∗ : A o p → V {\displaystyle G^{\ast }\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}} given by

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