Yoneda Lemma

The Yoneda Lemma syas that the set of morphisms from a representable Presheaf $Y$ into an arbitrary Presheaf $X$ is in natural isomorphism with the set X

Definition

Let $\mathcal{C}$ be a locally-small category.

$$\left({\int }_{Y:\mathcal{C}}\mathcal{C}[Y,X]\to FY\right)\cong FX$$

Proof

$$\begin{array}{rl}(\Rightarrow )& x:FX\\ & Y:\mathcal{C}\\ & f:\mathcal{C}[Y,X]\\ & \text{Have }Ff:\mathcal{C}[X,Y]\\ & \text{Then }(Ff)x:FY\\ (\Leftarrow )& x:FY\\ & X:\mathcal{C}\\ & f:\mathcal{C}[Y,X]\\ & \text{Then }{\alpha }_X{1}_X:FX\\ (\cong )& \text{Follows immediately from naturality.}\end{array}$$

Corollary

$$A\cong B⟹{\int }_X\mathcal{C}[A,X]\cong \mathcal{C}[B,X]$$

Resources

https://proofwiki.org/wiki/Yoneda_Lemma
https://www.youtube.com/watch?v=mLRgKPwyg4Y

VisualMath

https://www.youtube.com/watch?v=MphWHbArqnI
Great for intuition.

Understanding Yoneda

Richard Southwell

https://www.youtube.com/watch?v=h64yZs8ThtQ

https://math.mit.edu/~rud/TeXthings/yoneda%20lemma%20rapport(1).pdf

Emily Riehl

Good introduction to Yoneda Representation.
https://www.youtube.com/watch?v=SsgEvrDFJsM
Also riehl2016-category-theory.