Isbell duality
In mathematics, Isbell conjugacy 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 Yoneda embedding is a covariant functor from a small category into the category of presheaves on, taking to the contravariant representable functor:and the co-Yoneda embedding is a contravariant functor from a small category into the opposite of the category of co-presheaves on, taking to the covariant representable functor:
Isbell duality
Every functor has an Isbell conjugate of a functor, given byIn contrast, every functor has an Isbell conjugate of a functor given by
These two functors are not typically inverses, or even natural isomorphisms. Isbell duality asserts that the relationship between these two functors is an adjunction.
Isbell duality is the relationship between Yoneda embedding and co-Yoneda embedding；
Let be a symmetric monoidal closed category, and let be a small category enriched in.
The Isbell duality is an adjunction between the functor categories;.
Applying the nerve construction, the functors of Isbell duality are such that and.