Isbell Conjugacy

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. Also, says that; "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 $\mathcal$ into the category of presheaves \left mathcal^, \mathcal \right/math> on $\mathcal$, taking $X \in \mathcal$ to the contravariant representable functor:

Y \; (h^) :\mathcal \rightarrow \left mathcal^, \mathcal \right/math>

$X \mapsto \mathrm (-,X).$

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 $\mathcal$ into the category of co-presheaves \left mathcal, \mathcal \right on $\mathcal$, taking $X \in \mathcal$ to the covariant representable functor:

Z \; (^): \mathcal \rightarrow \left mathcal, \mathcal \right

$X \mapsto \mathrm (X,-).$

Every functor $F \colon \mathcal^\mathrm\to \mathcal$ has an Isbell conjugate $F^ \colon \mathcal \to \mathcal$, given by

$F^ (X) = \mathrm (F , y(X)).$

In contrast, every functor $G \colon \mathcal \to \mathcal$ has an Isbell conjugate $G^ \colon \mathcal^\mathrm \to \mathcal$ given by

$G^ (X) = \mathrm (z(X) , G).$

Isbell duality

Isbell duality is the relationship between Yoneda embedding and co-Yoneda embedding；

Let $\mathcal$ be a symmetric monoidal closed category, and let $\mathcal$ be a small category enriched in $\mathcal$.

The Isbell duality is an adjunction between the categories; \left(\mathcal \dashv \mathrm \right) \colon \left mathcal^, \mathcal \right
\left mathcal, \mathcal \right.

The functors $\mathcal \dashv \mathrm$ of Isbell duality are such that $\mathcal \cong \mathrm$ and $\mathrm \cong \mathrm$.For the symbol Lan, see left Kan extension.

See also

*Kan extension
* Limit (category theory)

References

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Footnote

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Category theory

{{categorytheory-stub
Adjoint functors