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 is states as follows; "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 $𝒜$ into the category of presheaves $[{𝒜}^{op},𝒱]$ on $𝒜$, taking $X\in 𝒜$ to the contravariant representable functor:

$Y(h^{\bullet }):𝒜\to [{𝒜}^{op},𝒱]$

$X\mapsto \mathrm{h}\mathrm{o}\mathrm{m}(-,X).$

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

$Z({h_{\bullet }}^{op}):𝒜\to {[𝒜,𝒱]}^{op}$

$X\mapsto \mathrm{h}\mathrm{o}\mathrm{m}(X,-).$

Isbell duality

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Origin of symbols $𝒪$ (“ring of functions”) and $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$ (“spectrum”): Lawvere (1986, p. 169) says that; "$𝒪$" assigns to each general space the algebra of functions on it, whereas "$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$" assigns to each algebra its “spectrum” which is a general space.

File:Nerve and realization (ver. left kan extension).svg

note:In order for this commutative diagram to hold, it is required that $𝒜$ is small and E is co-complete.

Every functor $F:{𝒜}^{\mathrm{o}\mathrm{p}}\to 𝒱$ has an Isbell conjugate of a functor $F^{\ast }:𝒜\to 𝒱$, given by

$F^{\ast }(X)=\mathrm{h}\mathrm{o}\mathrm{m}(F,y(X)).$

In contrast, every functor $G:𝒜\to 𝒱$ has an Isbell conjugate of a functor $G^{\ast }:{𝒜}^{\mathrm{o}\mathrm{p}}\to 𝒱$ given by

$G^{\ast }(X)=\mathrm{h}\mathrm{o}\mathrm{m}(z(X),G).$

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 embedding；

Let $𝒱$ be a symmetric monoidal closed category, and let $𝒜$ be a small category enriched in $𝒱$.

The Isbell duality is an adjunction between the functor categories; $(𝒪⊣\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}):[{𝒜}^{op},𝒱]\underset{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}{\stackrel{𝒪}{\rightleftarrows }}{[𝒜,𝒱]}^{op}$.

Applying the nerve construction, the functors $𝒪⊣\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$ of Isbell duality are such that $𝒪\cong \mathrm{L}\mathrm{a}{\mathrm{n}}_{\mathrm{Y}}\mathrm{Z}$ and $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\cong \mathrm{L}\mathrm{a}{\mathrm{n}}_{\mathrm{Z}}\mathrm{Y}$.

See also

• Kan extension
• Limit (category theory)
• Isbell completion
• Profunctor

Bibliography

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