Subsection 4.5.4 (01LH): The Characteristic Embedding of a Set

Table of Contents
Part 2: Sets
Chapter 4: Constructions With Sets
Section 4.5: Characteristic Functions
Subsection 4.5.4: The Characteristic Embedding of a Set (cite)

4.5.4 The Characteristic Embedding of a Set

Let $X$ be a set.

Definition 4.5.4.1.1 ▶ The Characteristic Embedding of a Set

The characteristic embedding¹ of $X$ into $\mathcal{P}\webleft (X\webright )$ is the function

\[ \chi _{\webleft (-\webright )}\colon X \hookrightarrow \mathcal{P}\webleft (X\webright ) \]

defined by²

\begin{align*} \chi _{\webleft (-\webright )}\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{x}\\ & = \left\{ x\right\} \end{align*}

for each $x\in X$.

¹The name “characteristic embedding” is justified by Corollary 4.5.5.1.2, which gives an analogue of fully faithfulness for $\chi _{\webleft (-\webright )}$.
²Here we are identifying $\mathcal{P}\webleft (X\webright )$ with $\mathsf{Sets}\webleft (X,\{ \mathsf{t},\mathsf{f}\} \webright )$ as per Item 2 of Proposition 4.5.1.1.4.

Remark 4.5.4.1.2 ▶ The Characteristic Embedding of a Set as a Decategorification of the Yoneda Embedding

Expanding upon Remark 4.5.1.1.2, Remark 4.5.2.1.2, and Remark 4.5.3.1.2, we may view the characteristic embedding

\[ \chi _{\webleft (-\webright )}\colon X\hookrightarrow \mathcal{P}\webleft (X\webright ) \]

of $X$ into $\mathcal{P}\webleft (X\webright )$ as 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 $\mathsf{PSh}\webleft (\mathcal{C}\webright )$.

Proposition 4.5.4.1.3 ▶ Properties of Characteristic Embeddings

Let $f\colon X\to Y$ be a map of sets.

1.
Interaction With Functions. We have

Proof of Proposition 4.5.4.1.3.

Item 1: Interaction With Functions

Indeed, we have

\begin{align*} \webleft [f_{!}\circ \chi _{X}\webright ]\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}\webleft (\chi _{X}\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}\webleft (\left\{ x\right\} \webright )\\ & = \left\{ f\webleft (x\webright )\right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{X'}\webleft (f\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\chi _{X'}\circ f\webright ]\webleft (x\webright ),\end{align*}

for each $x\in X$, showing the desired equality.

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