Misplaced Pages

Isbell duality

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Duality is an adjunction between a category of co/presheaf under the co/Yoneda embedding.

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 A {\displaystyle {\mathcal {A}}} into the category of presheaves [ A o p , V ] {\displaystyle \left} on A {\displaystyle {\mathcal {A}}} , taking X ∈ A {\displaystyle X\in {\mathcal {A}}} to the contravariant representable functor:

Y ( h ∙ ) : A → [ A o p , V ] {\displaystyle Y\;(h^{\bullet }):{\mathcal {A}}\rightarrow \left}

X ↦ h o m ( − , X ) . {\displaystyle 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 from a small category A {\displaystyle {\mathcal {A}}} into the opposite of the category of co-presheaves [ A , V ] o p {\displaystyle \left^{op}} on A {\displaystyle {\mathcal {A}}} , taking X ∈ A {\displaystyle X\in {\mathcal {A}}} to the covariant representable functor:

Z ( h ∙ o p ) : A → [ A , V ] o p {\displaystyle Z\;({h_{\bullet }}^{op}):{\mathcal {A}}\rightarrow \left^{op}}

X ↦ h o m ( X , − ) . {\displaystyle X\mapsto \mathrm {hom} (X,-).}

Every functor F : A o p → V {\displaystyle F\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}} has an Isbell conjugate F ∗ : A → V {\displaystyle F^{\ast }\colon {\mathcal {A}}\to {\mathcal {V}}} , given by

F ∗ ( X ) = h o m ( F , y ( X ) ) . {\displaystyle F^{\ast }(X)=\mathrm {hom} (F,y(X)).}

In contrast, every functor G : A → V {\displaystyle G\colon {\mathcal {A}}\to {\mathcal {V}}} has an Isbell conjugate G ∗ : A o p → V {\displaystyle G^{\ast }\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}} given by

G ∗ ( X ) = h o m ( z ( X ) , G ) . {\displaystyle G^{\ast }(X)=\mathrm {hom} (z(X),G).}

Isbell duality

Origin of symbols O {\displaystyle {\mathcal {O}}} and S p e c {\displaystyle \mathrm {Spec} } : Lawvere (1986, p. 169) says that; " O {\displaystyle {\mathcal {O}}} " assigns to each general space the algebra of functions on it, whereas " S p e c {\displaystyle \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 V {\displaystyle {\mathcal {V}}} be a symmetric monoidal closed category, and let A {\displaystyle {\mathcal {A}}} be a small category enriched in V {\displaystyle {\mathcal {V}}} .

The Isbell duality is an adjunction between the functor categories; ( O ⊣ S p e c ) : [ A o p , V ] ⇄ O S p e c [ A , V ] o p {\displaystyle \left({\mathcal {O}}\dashv \mathrm {Spec} \right)\colon \left{\underset {\mathrm {Spec} }{\overset {\mathcal {O}}{\rightleftarrows }}}\left^{op}} .

The functors O ⊣ S p e c {\displaystyle {\mathcal {O}}\dashv \mathrm {Spec} } of Isbell duality are such that O ≅ L a n Y Z {\displaystyle {\mathcal {O}}\cong \mathrm {Lan_{Y}Z} } and S p e c ≅ L a n Z Y {\displaystyle \mathrm {Spec} \cong \mathrm {Lan_{Z}Y} } .

See also

References

  1. ^ (Baez 2022)
  2. (Di Liberti 2020, 2. Isbell duality)
  3. ^ (Lawvere 1986, p. 169)
  4. (Rutten 1998)
  5. (Melliès & Zeilberger 2018)
  6. (Willerton 2013)
  7. (Lawvere 1986, p. 169)
  8. (Space and quantity in nlab)
  9. ^ (Yoneda embedding in nlab)
  10. (Valence 2017, Corollaire 2)
  11. (Awodey 2006, Definition 8.1.)
  12. ^ (Isbell duality in nlab)
  13. (Valence 2017, Définition 67)
  14. (Di Liberti & Loregian 2019, Definition 5.12)
  15. (Riehl 2016, Theorem 3.4.11.)
  16. (Leinster 2004, (c) and (c').)
  17. (Riehl 2016, Definition 1.3.11.)
  18. (Starr 2020, Example 4.7.)
  19. (Opposite functors in nlab)
  20. (Pratt 1996, §.4 Symmetrizing the Yoneda embedding)
  21. (Day & Lack 2007, §9. Isbell conjugacy)
  22. (Di Liberti 2020, Remark 2.3 (The (co)nerve construction).)
  23. (Kelly 1982, Proposition 4.33)
  24. (Riehl 2016, Remark 6.5.9.)
  25. (Imamura 2022, Theorem 2.4)
  26. ^ (Di Liberti 2020, Remark 2.4)
  27. (Fosco 2021)
  28. (Valence 2017, Définition 68)
  29. (Di Liberti & Loregian 2019, Lemma 5.13.)

Bibliography

Footnote

  1. Note that: the contravariant Yoneda embedding written in the article is replaced with the opposite category for both domain and codomain from that written in the textbook. See variance of functor, pre/post-composition, and opposite functor. In addition, this pair of Yoneda embeddings is collectively called the two Yoneda embeddings.
  2. For the symbol Lan, see left Kan extension.

External links

Stub icon

This category theory-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: