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] Also, Lawvere (1986, p. 169) says that; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics".^[7]

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: ^[1]^[8]^[9]

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

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

and the co-Yoneda embedding^[1]^[10]^[8]^[11] (a.k.a. contravariant Yoneda embedding^[12]^{[note 1]} or the dual Yoneda embedding^[17]) 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^[1] $F^{\ast }\colon {\mathcal {A}}\to {\mathcal {V}}$ , 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^[1] $G^{\ast }\colon {\mathcal {A}}^{\mathrm {op} }\to {\mathcal {V}}$ 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.^[18]^[19]^[20]^[21]

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; $\left({\mathcal {O}}\dashv \mathrm {Spec} \right)\colon \left[{\mathcal {A}}^{op},{\mathcal {V}}\right]{\underset {\mathrm {Spec} }{\overset {\mathcal {O}}{\rightleftarrows }}}\left[{\mathcal {A}},{\mathcal {V}}\right]^{op}$ .^[3]^[1]^[22]^[23]^[10]^[24]

The functors ${\mathcal {O}}\dashv \mathrm {Spec}$ of Isbell duality are such that ${\mathcal {O}}\cong \mathrm {Lan_{Y}Z}$ and $\mathrm {Spec} \cong \mathrm {Lan_{Z}Y}$ .^[22]^[25]^{[note 2]}

References

^ ^a ^b ^c ^d ^e ^f (Baez 2022)
^ (Di Liberti 2020, 2. Isbell duality)
^ ^a ^b (Lawvere 1986, p. 169)
^ (Rutten 1998)
^ (Melliès & Zeilberger 2018)
^ (Willerton 2013)
^ (Space and quantity in nlab)
^ ^a ^b (Yoneda embedding in nlab)
^ (Valence 2017, Corollaire 2)
^ ^a ^b (Isbell duality in nlab)
^ (Valence 2017, Définition 67)
^ (Di Liberti & Loregian 2019, Definition 5.12)
^ (Riehl 2016, Theorem 3.4.6.)
^ (Starr 2020, Example 4.7.)
^ (Opposite functors in nlab)
^ (Pratt 1996, §.4 Symmetrizing the Yoneda embedding)
^ (Day & Lack 2007, §9. Isbell conjugacy)
^ (Di Liberti 2020, Remark 2.3 (The (co)nerve construction).)
^ (Kelly 1982, Proposition 4.33)
^ (Riehl 2016, Remark 6.5.9.)
^ (Imamura 2022, Theorem 2.4)
^ ^a ^b (Di Liberti 2020, Remark 2.4)
^ (Fosco 2021)
^ (Valence 2017, Définition 68)
^ (Di Liberti & Loregian 2019, Lemma 5.13.)

Bibliography

Avery, Tom; Leinster, Tom (2021), "Isbell conjugacy and the reflexive completion" (PDF), Theory and Applications of Categories, 36: 306–347, arXiv:2102.08290
Baez, John C. (2022), "Isbell Duality" (PDF), Notices Amer. Math. Soc., 70: 140–141, arXiv:2212.11079
Day, Brian J.; Lack, Stephen (2007), "Limits of small functors", Journal of Pure and Applied Algebra, 210 (3): 651–663, arXiv:math/0610439, doi:10.1016/j.jpaa.2006.10.019, MR 2324597, S2CID 15424936.
Di Liberti, Ivan (2020), "Codensity: Isbell duality, pro-objects, compactness and accessibility", Journal of Pure and Applied Algebra, 224 (10), arXiv:1910.01014, doi:10.1016/j.jpaa.2020.106379, S2CID 203626566
Fosco, Loregian (22 July 2021), (Co)end Calculus, Cambridge University Press, arXiv:1501.02503, doi:10.1017/9781108778657, ISBN 9781108746120, S2CID 237839003
Gutierres, Gonçalo; Hofmann, Dirk (2013), "Approaching Metric Domains", Applied Categorical Structures, 21 (6): 617–650, arXiv:1103.4744, doi:10.1007/s10485-011-9274-z, S2CID 254225188
Shen, Lili; Zhang, Dexue (2013), "Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions" (PDF), Theory and Applications of Categories, 28 (20): 577–615, arXiv:1307.5625
Isbell, J. R. (1960), "Adequate subcategories", Illinois Journal of Mathematics, 4 (4), doi:10.1215/ijm/1255456274
Isbell, John R. (1966), "Structure of categories", Bulletin of the American Mathematical Society, 72 (4): 619–656, doi:10.1090/S0002-9904-1966-11541-0, S2CID 40822693
Kelly, Gregory Maxwell (1982), Basic concepts of enriched category theory (PDF), London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, ISBN 0-521-28702-2, MR 0651714.^{[page needed]}
Lawvere, F. W. (1986), "Taking categories seriously", Revista Colombiana de Matemáticas, 20 (3–4): 147–178, MR 0948965
- Lawvere, F. W. (2005), "Taking categories seriously" (PDF), Reprints in Theory and Applications of Categories (8): 1–24, MR 0948965
Lawvere, F. William (February 2016), "Birkhoff's Theorem from a geometric perspective: A simple example", Categories and General Algebraic Structures with Applications, 4 (1): 1–8
Melliès, Paul-André; Zeilberger, Noam (2018), "An Isbell duality theorem for type refinement systems", Mathematical Structures in Computer Science, 28 (6): 736–774, arXiv:1501.05115, doi:10.1017/S0960129517000068, S2CID 2716529
Riehl, Emily (2016), Category Theory in Context, Dover Publications, Inc Mineola, New York, ISBN 9780486809038
Rutten, J.J.M.M. (1998), "Weighted colimits and formal balls in generalized metric spaces", Topology and Its Applications, 89 (1–2): 179–202, doi:10.1016/S0166-8641(97)00224-1
Sturtz, Kirk (2018), "The factorization of the Giry monad", Advances in Mathematics, 340: 76–105, arXiv:1707.00488, doi:10.1016/j.aim.2018.10.007
- Sturtz, K. (2019). "Erratum and Addendum: The factorization of the Giry monad". arXiv:1907.00372 [math.CT].
Pratt, Vaughan (1996), "Broadening the denotational semantics of linear logic", Electronic Notes in Theoretical Computer Science, 3: 155–166, doi:10.1016/S1571-0661(05)80415-3
Wood, R.J (1982), "Some remarks on total categories", Journal of Algebra, 75 (2): 538–545, doi:10.1016/0021-8693(82)90055-2
Willerton, Simon (2013), "Tight spans, Isbell completions and semi-tropical modules" (PDF), Theory and Applications of Categories, 28 (22): 696–732, arXiv:1302.4370
Imamura, Yuki (2022), "Grothendieck Enriched Categories", Applied Categorical Structures, 30 (5): 1017–1041, arXiv:2105.05108, doi:10.1007/s10485-022-09681-1

Footnote

^ 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.^[13] See opposite functor.^[14]^[15] In addition, this pair of Yoneda embeddings is collectively called the two Yoneda embeddings.^[16]
^ For the symbol Lan, see left Kan extension.

External links

Loregian, Fosco (2018), "Kan extensions" (PDF), tetrapharmakon.github.io
Di Liberti, Ivan; Loregian, Fosco (2019). "On the unicity of formal category theories". arXiv:1901.01594 [math.CT].
Sorokin, Alex (2022), Derived profunctors, Boston, Massachusetts : Northeastern University, doi:10.17760/D20467288, hdl:2047/D20467288
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)
Starr, Jason (2020), "Some Notes on Category Theory in MAT 589 Algebraic Geometry" (PDF), math.stonybrook.edu
"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

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[Baez2022-1] ^ ^a ^b ^c ^d ^e ^f (Baez 2022)

[2] (Di Liberti 2020, 2. Isbell duality)

[Lawvere1986-3] (Lawvere 1986, p. 169)

[4] (Rutten 1998)

[5] (Melliès & Zeilberger 2018)

[6] (Willerton 2013)

[7] (Space and quantity in nlab)

[nlab2-8] (Yoneda embedding in nlab)

[9] (Valence 2017, Corollaire 2)

[nlab1-10] (Isbell duality in nlab)

[11] (Valence 2017, Définition 67)

[12] (Di Liberti & Loregian 2019, Definition 5.12)

[13] (Riehl 2016, Theorem 3.4.6.)

[14] (Starr 2020, Example 4.7.)

[15] (Opposite functors in nlab)

[16] (Pratt 1996, §.4 Symmetrizing the Yoneda embedding)

[18] (Day & Lack 2007, §9. Isbell conjugacy)

[19] (Di Liberti 2020, Remark 2.3 (The (co)nerve construction).)

[20] (Kelly 1982, Proposition 4.33)

[21] (Riehl 2016, Remark 6.5.9.)

[22] (Imamura 2022, Theorem 2.4)

[DiLiberti2020-23] (Di Liberti 2020, Remark 2.4)

[24] (Fosco 2021)

[25] (Valence 2017, Définition 68)

[26] (Di Liberti & Loregian 2019, Lemma 5.13.)

[17] 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.^[13] See opposite functor.^[14]^[15] In addition, this pair of Yoneda embeddings is collectively called the two Yoneda embeddings.^[16]

[27] For the symbol Lan, see left Kan extension.

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[note 2]

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Isbell duality

Contents

Definition

Yoneda embedding

Isbell duality

See also

References

Bibliography

Footnote

External links