Subsection 2.4.2 (006K): The Yoneda Lemma for Sets

Preferences

Font:

PDF Style:

Here's a breakdown of the differences between each PDF style:

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

Let $X$ be a set and let $U\subset X$ be a subset of $X$.

We have

\[ \chi _{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{U}\webright )=\chi _{U}\webleft (x\webright ) \]

for each $x\in X$, giving an equality of functions

\[ \chi _{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{\webleft (-\webright )},\chi _{U}\webright )=\chi _{U}. \]

Proof of Proposition 2.4.2.1.1.

Clear.

The characteristic embedding is fully faithful, i.e., we have

\[ \chi _{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{y}\webright )=\chi _{X}\webleft (x,y\webright ) \]

for each $x,y\in X$.

Proof of Corollary 2.4.2.1.2.

This follows from Proposition 2.4.2.1.1.

Noticed something off, or have any comments? Feel free to reach out!

You can also use the contact form below: