Today we introduced the idea of homotopy, and then followed it up with another dollop of abstract nonsense (congruences and quotient categories).

An easy application of the last result we proved today is that the sphere $S^n$ is not contractible. For this one applies Proposition 2.15—which states that a map $f: S^n \to Y$ can be extended to a map $g: B^{n+1} \to Y$ if and only if $f$ is nullhomotopic—with $Y = S^n$ and $f = \mathrm{id}_{S^n}$.