Category-Theory

The Yoneda Lemma The Yoneda lemma has been being complainted as “hard to understand” for a long time. Indeed it is not very easy, so I was trying to come up with a way to understand it more intuitively. Everyone knows the analogy by Ravi Vakil: You work at a particle accelerator. You want to understand some particle. All you can do are throw other particles at it and see what happens. If you understand how your mystery particle responds to all possible test particles at all possible test energies, then you know everything there is to know about your mystery particle. ...

Stone–Čech compactification One of the proof of the Stone–Čech compactification is to consider the continuous functions from a given topological space $X$ to $[0, 1]$. Crux 1: constructing $[0, 1]^{C}$ Let $C$ be the space of all continuous functions from $X$ to $[0, 1]$, consider $[0, 1]^{C}$, there is a natural map from $X$ to $[0, 1]^{C}$: for each $x \in X$, define $\phi(x) = f \mapsto f(x)$. With product topology $\phi$ is continuous. Also by Tychonnof’s theorem $[0, 1]^{C}$ is compact. Now the closure of the image is obviously compact Hausdorff, and it satisfies a nice property: any continuous map $f : X \to [0, 1]$ extends uniquely to a continuous map $\overline{\phi(X)} \to [0,1]$: for any $\phi(x)$, it has to map to $f(x)$! ...