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 $\mathcal{P}(X)$ is the function

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

defined by2

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

for each $x\in X$.

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

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

of $X$ into $\mathcal{P}(X)$ as a decategorification of the Yoneda embedding

\[ {\text{よ}}\colon \mathcal{C}^{\mathsf{op}} \hookrightarrow \mathsf{PSh}(\mathcal{C}) \]

of a category $\mathcal{C}$ into $\mathsf{PSh}(\mathcal{C})$.

Proposition 4.5.4.1.3 ▶ Properties of Characteristic Embeddings

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

  1. 1.

    Interaction With Functions. We have

Proof of Proposition 4.5.4.1.3.

Item 1: Interaction With Functions
Indeed, we have

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

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

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