Remark 2.4.8.1.2 (01L3): Motivation for the Isbell Function

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

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.