This will be old news to anyone who does algebraic topology, but oddly enough I can’t seem to find it all written in one place anywhere, and in particular I can’t find the bit about $\mathsf{hPairTop}$ at all.

In algebraic topology you (for example) associate every topological space $X$ with a group, like $\pi_1(X, x_0)$ or $H_5(X)$. All of these operations turn out to be functors. This isn’t surprising, because as far as I’m concerned the definition of a functor is “any time you take one type of object and naturally make another object”.

The surprise is that these objects also respect homotopy in a nice way; proving this is a fair amount of the “setup …