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Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$. If I understand correctly, this notation suggests that $\widehat{C}$ consists of "geometric" objects, whereas $\widehat{C^{\mathrm{op}}}$ consists of "algebraic" objects. Then $\mathcal{O}$ associates to some geometric object $X$ the algebraic object of all global functions on $X$, whereas $\mathrm{Spec}$ associates to some algebraic object $A$ the "affine" geometric object associated to $A$. Note both the unit $\eta_X : X \to \mathrm{Spec}(\mathcal{O}(X))$ and the counit $\varepsilon_A : A \to \mathcal{O}(\mathrm{Spec}(A))$ of this adjunction are given by evaluation. As every adjunction we get an equivalence of categories between its fixed points, i.e. those $X$ such that $\eta_X$ is an iso, and those $A$ such that $\varepsilon_A$ is an iso (Isbell-dual objects).

In general this formulation does not make sense, but it is well-known in the following special cases which are also alluded in the nlab article.

1) Algebraic geometry: There is an adjunction $\mathrm{Sch} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}} \mathrm{Ring}^{\mathrm{op}}$. Unit and counit are just evaluation. It restricts to an antiequivalence of categories between affine schemes and rings.

2) Functional analysis: There is an adjunction $\mathrm{Top} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}} {C^*\mathrm{Alg}_1}^{\mathrm{op}}$. Unit and counit are again just evaluation. It restricts to an antiequivalence of categories between compact Hausdorff spaces and commutative unital $C^*$-algebras.

3) Pointless topology: There is an adjunction $\mathrm{Top} {{\mathcal{\Omega} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}} \mathrm{Frm}^{\mathrm{op}} = \mathrm{Loc}$, where $\Omega$ associates to a topological space the frame of its open subsets, and $\mathrm{Spec}$ associates to every locale the space of principal prime ideals. It restricts to an equivalence between sober spaces and spatial locales and is related to Stone duality. This is very, very similar to 2), we just replace $\mathbb{C}$ with the partial order $2$.

Question. Are these adjunctions really special cases of Isbell duality? If not, how are they related? Is there any more general pattern?

I'm also interested in the case of $(2,1)$-categories $C$. Here $\widehat{C}$ should be the category of pseudo-functors $C^{\mathrm{op}} \to \mathrm{Gpd}$. In this setting there is an adjunction similar to 1) between stacks and cocomplete tensor categories, where $\mathcal{O} = \mathrm{Qcoh}$ associates to every stack $X$ the category of quasi-coherent modules, which may be imagined as categorified global functions on $X$. The fixed points are the tensorial stacks which I study currently. Where does this adjunction arise in the literature? It is similar to an adjunction from derived algebraic geometry (Ben-Zvi, Nadler, Prop. 3.1).

There are related conversations between Jim Dolan and Todd Trimble, which already answer my questions partially.

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Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$. If I understand correctly, this notation suggests that $\widehat{C}$ consists of "geometric" objects, whereas $\widehat{C^{\mathrm{op}}}$ consists of "algebraic" objects. Then $\mathcal{O}$ associates to some geometric object $X$ the algebraic object of all global functions on $X$, whereas $\mathrm{Spec}$ associates to some algebraic object $A$ the "affine" geometric object associated to $A$. Note both the unit $\eta_X : X \to \mathrm{Spec}(\mathcal{O}(X))$ and the counit $\varepsilon_A : A \to \mathcal{O}(\mathrm{Spec}(A))$ of this adjunction are given by evaluation. As every adjunction we get an equivalence of categories between its fixed points, i.e. those $X$ such that $\eta_X$ is an iso, and those $A$ such that $\varepsilon_A$ is an iso (Isbell-dual objects).

In general this formulation does not make sense, but it is well-known in the following special cases which are also alluded in the nlab article.

1) Algebraic geometry: There is an adjunction $\mathrm{Sch} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}} \mathrm{Ring}^{\mathrm{op}}$. Unit and counit are just evaluation. It restricts to an antiequivalence of categories between affine schemes and rings.

2) Functional analysis: There is an adjunction $\mathrm{Top} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}} {C^*\mathrm{Alg}_1}^{\mathrm{op}}$. Unit and counit are again just evaluation. It restricts to an antiequivalence of categories between compact Hausdorff spaces and commutative unital $C^*$-algebras.

Question. Are these adjunctions really special cases of Isbell duality? If not, how are they related? Is there any more general pattern?

I'm also interested in the case of $(2,1)$-categories $C$. Here $\widehat{C}$ should be the category of pseudo-functors $C^{\mathrm{op}} \to \mathrm{Gpd}$. In this setting there is an adjunction similar to 1) between stacks and cocomplete tensor categories, where $\mathcal{O} = \mathrm{Qcoh}$ associates to every stack $X$ the category of quasi-coherent modules, which may be imagined as categorified global functions on $X$. The fixed points are the tensorial stacks which I study currently. Where does this adjunction arise in the literature? It is similar to an adjunction from derived algebraic geometry (Ben-Zvi, Nadler, Prop. 3.1).

There are related conversations between Jim Dolan and Todd Trimble, which already answer my questions partially.

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# Theme of Isbell duality

Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$. If I understand correctly, this notation suggests that $\widehat{C}$ consists of "geometric" objects, whereas $\widehat{C^{\mathrm{op}}}$ consists of "algebraic" objects. Then $\mathcal{O}$ associates to some geometric object $X$ the algebraic object of all global functions on $X$, whereas $\mathrm{Spec}$ associates to some algebraic object $A$ the "affine" geometric object associated to $A$. Note both the unit $\eta_X : X \to \mathrm{Spec}(\mathcal{O}(X))$ and the counit $\varepsilon_A : A \to \mathcal{O}(\mathrm{Spec}(A))$ of this adjunction are given by evaluation. As every adjunction we get an equivalence of categories between its fixed points, i.e. those $X$ such that $\eta_X$ is an iso, and those $A$ such that $\varepsilon_A$ is an iso (Isbell-dual objects).

In general this formulation does not make sense, but it is well-known in the following special cases which are also alluded in the nlab article.

1) Algebraic geometry: There is an adjunction $\mathrm{Sch} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}} \mathrm{Ring}^{\mathrm{op}}$. Unit and counit are just evaluation. It restricts to an antiequivalence of categories between affine schemes and rings.

2) Functional analysis: There is an adjunction $\mathrm{Top} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}} {C^*\mathrm{Alg}_1}^{\mathrm{op}}$. Unit and counit are again just evaluation. It restricts to an antiequivalence of categories between compact Hausdorff spaces and commutative unital $C^*$-algebras.

Question. Are these adjunctions really special cases of Isbell duality? If not, how are they related? Is there any more general pattern?

I'm also interested in the case of $(2,1)$-categories $C$. Here $\widehat{C}$ should be the category of pseudo-functors $C^{\mathrm{op}} \to \mathrm{Gpd}$. In this setting there is an adjunction similar to 1) between stacks and cocomplete tensor categories, where $\mathcal{O} = \mathrm{Qcoh}$ associates to every stack $X$ the category of quasi-coherent modules, which may be imagined as categorified global functions on $X$. The fixed points are the tensorial stacks which I study currently. Where does this adjunction arise in the literature?

There are related conversations between Jim Dolan and Todd Trimble, which already answer my questions partially.