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 ${𝒜}$ into the category of presheaves $[{{𝒜}}^{op},{𝒱}]$ on ${𝒜}$, taking $X\in {𝒜}$ to the contravariant representable functor: $Y(h^{\bullet }):{𝒜}\to [{{𝒜}}^{op},{𝒱}]$ $X\mapsto \mathrm{h}\mathrm{o}\mathrm{m}(-,X).$ 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 ${𝒜}$ into the category of co-presheaves ${[{𝒜},{𝒱}]}^{op}$ on ${𝒜}$, taking $X\in {𝒜}$ to the covariant representable functor: $Z({h_{\bullet }}^{op}):{𝒜}\to {[{𝒜},{𝒱}]}^{op}$ $X\mapsto \mathrm{h}\mathrm{o}\mathrm{m}(X,-).$ Every functor $F:{{𝒜}}^{\mathrm{o}\mathrm{p}}\to {𝒱}$ has an Isbell conjugate $F^{\ast }:{𝒜}\to {𝒱}$, given by $F^{\ast }(X)=\mathrm{h}\mathrm{o}\mathrm{m}(F,y(X)).$ In contrast, every functor $G:{𝒜}\to {𝒱}$ has an Isbell conjugate $G^{\ast }:{{𝒜}}^{\mathrm{o}\mathrm{p}}\to {𝒱}$ given by $G^{\ast }(X)=\mathrm{h}\mathrm{o}\mathrm{m}(z(X),G).$

Isbell duality

File:Isbell duality.svg

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

File:Nerve and realization (ver. left kan extension).svg

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 ${𝒱}$ be a symmetric monoidal closed category, and let ${𝒜}$ be a small category enriched in ${𝒱}$. The Isbell duality is an adjunction between the categories; $({𝒪}⊣\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}):[{{𝒜}}^{op},{𝒱}]\underset{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}{\stackrel{{𝒪}}{\rightleftarrows }}{[{𝒜},{𝒱}]}^{op}$. The functors ${𝒪}⊣\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$ of Isbell duality are such that ${𝒪}\cong \mathrm{L}\mathrm{a}{\mathrm{n}}_{\mathrm{Y}}\mathrm{Z}$ and $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\cong \mathrm{L}\mathrm{a}{\mathrm{n}}_{\mathrm{Z}}\mathrm{Y}$.

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