Isbell duality

Isbell conjugacy (a.k.a.

Isbell duality or Isbell adjunction) (named after John R. Isbell[1][2]) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986.

[3][4] That is a duality between covariant and contravariant representable presheaves associated with an objects of categories under the Yoneda embedding.

[5][6] In addition, Lawvere[7] is states as follows; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics".

[8] The (covariant) Yoneda embedding is a covariant functor from a small category

into the category of presheaves

{\displaystyle \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]}

to the contravariant representable functor: [1][9][10][11]

{\displaystyle Y\;(h^{\bullet }):{\mathcal {A}}\rightarrow \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]}

and the co-Yoneda embedding[1][12][9][13] (a.k.a.

contravariant Yoneda embedding[14][note 1] or the dual Yoneda embedding[21]) is a contravariant functor from a small category

into the opposite of the category of co-presheaves

{\displaystyle \left[{\mathcal {A}},{\mathcal {V}}\right]^{op}}

, taking

to the covariant representable functor:

{\displaystyle Z\;({h_{\bullet }}^{op}):{\mathcal {A}}\rightarrow \left[{\mathcal {A}},{\mathcal {V}}\right]^{op}}

{\displaystyle F\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}}

has an Isbell conjugate[1]

In contrast, every functor

has an Isbell conjugate[1]

{\displaystyle G^{\ast }\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}}

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;

{\displaystyle \left({\mathcal {O}}\dashv \mathrm {Spec} \right)\colon \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]{\underset {\mathrm {Spec} }{\overset {\mathcal {O}}{\rightleftarrows }}}\left[{\mathcal {A}},{\mathcal {V}}\right]^{op}}

of Isbell duality are such that

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