Isbell duality

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 ${\mathcal {A}}$ into the category of presheaves $\left[{\mathcal {A}}^{op},{\mathcal {V}}\right]$ on ${\mathcal {A}}$, taking $X\in {\mathcal {A}}$ to the contravariant representable functor:

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

$X\mapsto \mathrm {hom} (-,X).$

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 ${\mathcal {A}}$ into the category of co-presheaves $\left[{\mathcal {A}},{\mathcal {V}}\right]^{op}$ on ${\mathcal {A}}$, taking $X\in {\mathcal {A}}$ to the covariant representable functor:

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

$X\mapsto \mathrm {hom} (X,-).$

Every functor $F\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}$ has an Isbell conjugate $F^{\ast }\colon {\mathcal {A}}\to {\mathcal {V}}$, given by

$F^{\ast }(X)=\mathrm {hom} (F,y(X)).$

In contrast, every functor $G\colon {\mathcal {A}}\to {\mathcal {V}}$ has an Isbell conjugate $G^{\ast }\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}$ given by

$G^{\ast }(X)=\mathrm {hom} (z(X),G).$

Isbell duality

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

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

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

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

The Isbell duality is an adjunction between the categories; $\left({\mathcal {O}}\dashv \mathrm {Spec} \right)\colon \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]{\underset {\mathrm {Spec} }{\overset {\mathcal {O}}{\rightleftarrows }}}\left[{\mathcal {A}},{\mathcal {V}}\right]^{op}$.

The functors ${\mathcal {O}}\dashv \mathrm {Spec}$ of Isbell duality are such that ${\mathcal {O}}\cong \mathrm {Lan_{Y}Z}$ and $\mathrm {Spec} \cong \mathrm {Lan_{Z}Y}$.

See also

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

Bibliography

• Avery, Tom; Leinster, Tom (2021), "Isbell conjugacy and the reflexive completion" (PDF), Theory and Applications of Categories, 36: 306–347, arXiv:2102.08290
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External links

• Di Liberti, Ivan; Loregian, Fosco (2019). "On the unicity of formal category theories". arXiv:1901.01594 [math.CT].
• Leinster, Tom (2004), "An ABC of Category Theory ch.4 Representability" (PDF), maths.ed.ac.uk
• Loregian, Fosco (2018), "Kan extensions" (PDF), tetrapharmakon.github.io
• Sorokin, Alex (2022), Derived profunctors, Boston, Massachusetts : Northeastern University, doi:10.17760/D20467288, hdl:2047/D20467288
• Starr, Jason (2020), "Some Notes on Category Theory in MAT 589 Algebraic Geometry" (PDF), math.stonybrook.edu
• Valence, Arnaud (2017), Esquisse d'une dualité géométrico-algébrique multidisciplinaire : la dualité d'Isbell, Thèse en cotutelle en Philosophie – Étude des Systèmes, soutenue le 30 mai 2017. (PDF)
• "Isbell duality", ncatlab.org
• "space and quantity", ncatlab.org
• "Yoneda embedding", ncatlab.org
• "co-Yoneda lemma", ncatlab.org
• "copresheaf", ncatlab.org
• "Natural transformations and presheaves: Remark 1.28. (presheaves as generalized spaces)", ncatlab.org
• "Opposite functors", ncatlab.org