Subsection 2.4.8 (01L1): Isbell Duality for Sets

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The Isbell function of $X$ is the map

\[ \mathsf{I}\colon \mathcal{P}\webleft (X\webright )\to \mathsf{Sets}\webleft (X,\mathcal{P}\webleft (X\webright )\webright ) \]

defined by

\[ \mathsf{I}\webleft (U\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[x\mapsto \webleft [U,\webleft\{ x\webright\} \webright ]_{X}]\mspace {-3mu}] \]

for each $U\in \mathcal{P}\webleft (X\webright )$.

Recall from Remark 2.4.1.1.2 that we may view the powerset $\mathcal{P}\webleft (X\webright )$ of a set $X$ as the decategorification of the category of presheaves $\mathsf{PSh}\webleft (\mathcal{C}\webright )$ of a category $\mathcal{C}$. Building upon this analogy, we want to mimic the definition of the Isbell $\mathsf{Spec}$ functor, which is given on objects by

\[ \mathsf{Spec}\webleft (\mathcal{F}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Nat}\webleft (\mathcal{F},h_{\webleft (-\webright )}\webright ) \]

for each $\mathcal{F}\in \text{Obj}\webleft (\mathsf{PSh}\webleft (\mathcal{C}\webright )\webright )$. To this end, we could define

\[ \mathsf{I}\webleft (U\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,\chi _{\webleft (-\webright )}\webright ]_{X}, \]

replacing:

The Yoneda embedding $X\mapsto h_{X}$ of $\mathcal{C}$ into $\mathsf{PSh}\webleft (\mathcal{C}\webright )$ with the characteristic embedding $x\mapsto \chi _{x}$ of $X$ into $\mathcal{P}\webleft (X\webright )$ of Definition 2.5.4.1.1.
The internal Hom $\text{Nat}$ of $\mathsf{PSh}\webleft (\mathcal{C}\webright )$ with the internal Hom $\webleft [-,-\webright ]_{X}$ of $\mathcal{P}\webleft (X\webright )$ of Definition 2.4.7.1.1.

However, since $\webleft [U,\chi _{x}\webright ]_{X}$ is a subset of $U$ instead of a truth value, we get a function

\[ \mathsf{I}\colon \mathcal{P}\webleft (X\webright )\to \mathsf{Sets}\webleft (X,\mathcal{P}\webleft (X\webright )\webright ) \]

instead of a function

\[ \mathsf{I}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (X\webright ). \]

This makes some of the properties involving $\mathsf{I}$ a bit more cumbersome to state, although we still have an analogue of Isbell duality in that $\mathsf{I}_{*}\circ \mathsf{I}$ evaluates to $\text{id}_{\mathcal{P}\webleft (X\webright )}$ in the sense of Proposition 2.4.8.1.3.

The diagram

commutes, i.e. we have

\[ \mathsf{I}_{*}\webleft (\mathsf{I}\webleft (U\webright )\webright )=[\mspace {-3mu}[x\mapsto [\mspace {-3mu}[y\mapsto U]\mspace {-3mu}]]\mspace {-3mu}] \]

for each $U\in \mathcal{P}\webleft (X\webright )$.

We have

\begin{align*} \mathsf{I}_{*}\webleft (\mathsf{I}\webleft (U\webright )\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{I}_{*}\webleft ([\mspace {-3mu}[x\mapsto U^{\textsf{c}}\cup \webleft\{ x\webright\} ]\mspace {-3mu}]\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[x\mapsto \mathsf{I}\webleft (U^{\textsf{c}}\cup \webleft\{ x\webright\} \webright )]\mspace {-3mu}]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[x\mapsto [\mspace {-3mu}[y\mapsto \webleft (U^{\textsf{c}}\cup \webleft\{ x\webright\} \webright )^{\textsf{c}}\cup \webleft\{ x\webright\} ]\mspace {-3mu}]]\mspace {-3mu}]\\ & = [\mspace {-3mu}[x\mapsto [\mspace {-3mu}[y\mapsto \webleft (U\cap \webleft (X\setminus \webleft\{ x\webright\} \webright )\webright )\cup \webleft\{ x\webright\} ]\mspace {-3mu}]]\mspace {-3mu}]\\ & = [\mspace {-3mu}[x\mapsto [\mspace {-3mu}[y\mapsto \webleft (U\setminus \webleft\{ x\webright\} \webright )\cup \webleft\{ x\webright\} ]\mspace {-3mu}]]\mspace {-3mu}]\\ & = [\mspace {-3mu}[x\mapsto [\mspace {-3mu}[y\mapsto U]\mspace {-3mu}]]\mspace {-3mu}], \end{align*}

where we have used Item 2 of Proposition 2.3.11.1.2 for the fourth equality above.

The Clowder Project

2.4.8 Isbell Duality for Sets