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 is states as follows; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics".
The (covariant) Yoneda embedding is a covariant functor from a small category
l{A}
\left[l{A}op,l{V}\right]
l{A}
X\inl{A}
Y (h\bullet):l{A} → \left[l{A}op,l{V}\right]
X\mapstohom(-,X).
and the co-Yoneda embedding (a.k.a. contravariant Yoneda embedding or the dual Yoneda embedding) is a contravariant functor (a covariant functor from the opposite category) from a small category
l{A}
\left[l{A},l{V}\right]op
l{A}
X\inl{A}
Z ({h\bullet
X\mapstohom(X,-).
Every functor
F\colonl{A}op\tol{V}
F\ast\colonl{A}\tol{V}
F\ast(X)=hom(F,y(X)).
In contrast, every functor
G\colonl{A}\tol{V}
G\ast\colonl{A}op\tol{V}
G\ast(X)=hom(z(X),G).
Isbell duality is the relationship between Yoneda embedding and co-Yoneda embedding；
Let
l{V}
l{A}
l{V}
The Isbell duality is an adjunction between the categories;
\left(l{O}\dashvSpec\right)\colon\left[l{A}op,l{V}\right]{\underset{Spec
The functors
l{O}\dashvSpec
l{O}\cong
| LanYZ |
Spec\cong
| LanZY |