• The characteristic embedding[1] of $X$ into $\mathcal{P}\webleft (X\webright )$ is the function
    \[ \chi _{\webleft (-\webright )}\colon X \hookrightarrow \mathcal{P}\webleft (X\webright ) \]

    defined by

    \[ \chi _{\webleft (-\webright )}\webleft (x\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{x} \]

    for each $x\in X$.

Footnotes

[1] The name “characteristic embedding” comes from the fact that there is an analogue of fully faithfulness for $\chi _{\webleft (-\webright )}$: given a set $X$, we have
\[ \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{y}\webright )=\chi _{X}\webleft (x,y\webright ), \]
for each $x,y\in X$.

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