The definitions in Definition 2.4.1.1.1 are decategorifications of co/presheaves, representable co/presheaves, $\textup{Hom}$ profunctors, and the Yoneda embedding:[1]
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A function
\[ f\colon X\to \{ \mathsf{t},\mathsf{f}\} \]
is a decategorification of a presheaf
\[ \mathcal{F}\colon \mathcal{C}^{\mathsf{op}}\to \mathsf{Sets}, \]with the characteristic functions $\chi _{U}$ of the subsets of $X$ being the primordial examples (and, in fact, all examples) of these.
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The characteristic function
\[ \chi _{x}\colon X\to \{ \mathsf{t},\mathsf{f}\} \]
of an element $x$ of $X$ is a decategorification of the representable presheaf
\[ h_{X}\colon \mathcal{C}^{\mathsf{op}} \to \mathsf{Sets} \]of an object $x$ of a category $\mathcal{C}$.
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The characteristic relation
\[ \chi _{X}\webleft (-_{1},-_{2}\webright )\colon X\times X\to \{ \mathsf{t},\mathsf{f}\} \]
of $X$ is a decategorification of the $\textup{Hom}$ profunctor
\[ \textup{Hom}_{\mathcal{C}}\webleft (-_{1},-_{2}\webright )\colon \mathcal{C}^{\mathsf{op}}\times \mathcal{C}\to \mathsf{Sets} \]of a category $\mathcal{C}$.
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The characteristic embedding
\[ \chi _{\webleft (-\webright )}\colon X\hookrightarrow \mathcal{P}\webleft (X\webright ) \]
of $X$ into $\mathcal{P}\webleft (X\webright )$ is a decategorification of the Yoneda embedding
\[ {\text{よ}}\colon \mathcal{C}^{\mathsf{op}} \hookrightarrow \mathsf{PSh}\webleft (\mathcal{C}\webright ) \]of a category $\mathcal{C}$ into $\mathsf{PSh}\webleft (\mathcal{C}\webright )$.
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There is also a direct parallel between unions and colimits:
- An element of $\mathcal{P}\webleft (X\webright )$ is a union of elements of $X$, viewed as one-point subsets $\webleft\{ x\webright\} \in \mathcal{P}\webleft (A\webright )$.
- An object of $\mathsf{PSh}\webleft (\mathcal{C}\webright )$ is a colimit of objects of $\mathcal{C}$, viewed as representable presheaves $h_{X}\in \text{Obj}\webleft (\mathsf{PSh}\webleft (\mathcal{C}\webright )\webright )$.
