Isbell duality

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Short description: Construction of enriched category theory

Isbell conjugacy or Isbell duality (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]

Definition

Isbell duality.svg

The Yoneda embedding

[math]\displaystyle{ Y:\mathcal{A} \rightarrow \left[\mathcal{A}^{op}, \mathcal{V} \right] }[/math]

[math]\displaystyle{ X \mapsto \mathrm{hom} (-,X). }[/math]

and the co-Yoneda embedding[1][6] or the dual Yoneda embedding[7]

[math]\displaystyle{ Z:\mathcal{A} \rightarrow \left[\mathcal{A}, \mathcal{V} \right]^{op} }[/math]

[math]\displaystyle{ X \mapsto \mathrm{hom} (X,-). }[/math]

Let [math]\displaystyle{ \mathcal{V} }[/math] be a symmetric monoidal closed category, and let [math]\displaystyle{ \mathcal{A} }[/math] be a small category enriched in [math]\displaystyle{ \mathcal{V} }[/math].

The Isbell conjugacy is an adjunction between the categories; [math]\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} }[/math].[3][1][8][9][6]

The functors [math]\displaystyle{ \mathcal{O} \dashv \mathrm{Spec} }[/math] of Isbell duality are such that [math]\displaystyle{ \mathcal{O} \cong \mathrm{Lan_{Y}Z} }[/math] and [math]\displaystyle{ \mathrm{Spec} \cong \mathrm{Lan_{Z}Y} }[/math].[8][10][note 1]

See also

References

  1. Jump up to: 1.0 1.1 1.2 (Baez 2022)
  2. (Di Liberti 2020)
  3. Jump up to: 3.0 3.1 (Lawvere 1986)
  4. (Rutten 1998)
  5. (Melliès Zeilberger)
  6. Jump up to: 6.0 6.1 (Isbell duality in nlab {{{2}}})
  7. (Day Lack)
  8. Jump up to: 8.0 8.1 (Di Liberti 2020)
  9. (Fosco 2021)
  10. (Di Liberti Loregian)

Bibliography

| last = Kelly | first = Gregory Maxwell
| isbn = 0-521-28702-2
| mr = 651714
| publisher = Cambridge University Press, Cambridge-New York
| series = London Mathematical Society Lecture Note Series
| title = Basic concepts of enriched category theory
| volume = 64

Footnote

  1. For the symbol Lan, see left Kan extension.

External links

  • Loregian, Fosco (2018), Kan extensions, https://tetrapharmakon.github.io/stuff/REFCARDS_kan.pdf 
  • Di Liberti, Ivan; Loregian, Fosco (2019). "On the unicity of formal category theories". arXiv:1901.01594 [math.CT].
  • Isbell duality, https://ncatlab.org/nlab/show/Isbell+duality 
  • Yoneda embedding, https://ncatlab.org/nlab/show/Yoneda+embedding 
  • co-Yoneda lemma, https://ncatlab.org/nlab/show/co-Yoneda+lemma