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 π’œ into the category of presheaves [π’œop,𝒱] on π’œ, taking Xβˆˆπ’œ to the contravariant representable functor:

y(hβˆ™):π’œβ†’[π’œop,𝒱]

X↦hom(βˆ’,X).

and the co-Yoneda embedding (a.k.a. dual Yoneda embedding) is a contravariant functor from a small category π’œ into the opposite of the category of co-presheaves [π’œ,𝒱]op on π’œ, taking Xβˆˆπ’œ to the covariant representable functor:

z(hβˆ™op):π’œβ†’[π’œ,𝒱]op

X↦hom(X,βˆ’).

Isbell duality

Origin of symbols π’ͺ (β€œring of functions”) and Spec (β€œspectrum”): Lawvere (1986, p.Β 169)[failed verification] says that; "π’ͺ" assigns to each general space the algebra of functions on it, whereas "Spec" assigns to each algebra its β€œspectrum” which is a general space.
note:In order for this commutative diagram to hold, it is required that π’œ is small and E is co-complete.

Every functor F:π’œop→𝒱 has an Isbell conjugate of a functor Fβˆ—:π’œβ†’π’±, given by

Fβˆ—(X)=hom(F,y(X)).

In contrast, every functor G:π’œβ†’π’± has an Isbell conjugate of a functor Gβˆ—:π’œop→𝒱 given by

Gβˆ—(X)=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.

Isbell duality is the relationship between Yoneda embedding and co-Yoneda embeddingοΌ›

Let 𝒱 be a symmetric monoidal closed category, and let π’œ be a small category enriched in 𝒱.

The Isbell duality is an adjunction between the functor categories; (π’ͺ⊣Spec):[π’œop,𝒱]⇄π’ͺSpec[π’œ,𝒱]op.

Applying the nerve construction, the functors π’ͺ⊣Spec of Isbell duality are such that π’ͺβ‰…Lanyz and Specβ‰…Lanzy.

See also

Footnote

  1. ↑ For the symbol Lan, see left Kan extension.