The Clowder Project

Recall from Remark 4.4.1.1.2 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

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

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 4.5.4.1.1.

• •

  The internal Hom $\operatorname {\mathrm{Nat}}$ of $\mathsf{PSh}\webleft (\mathcal{C}\webright )$ with the internal Hom $\webleft [-,-\webright ]_{X}$ of $\mathcal{P}\webleft (X\webright )$ of Definition 4.4.7.1.1.

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

instead of a function

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 $\operatorname {\mathrm{id}}_{\mathcal{P}\webleft (X\webright )}$ in the sense of Proposition 4.4.8.1.3.