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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