We kick off this lecture by defining a rather pathetic little functor called $\pi_0$. This functor sucks (as far as gleaming interesting information about a topological space goes) since it has the misfortune of taking values in the category $\mathsf{Sets}$, and basically the only thing you can do with a set is count it—the only obstruction to two sets being isomorphic is that they should have the same cardinality.

But fear not: $\pi_0$ has a big brother called $\pi_1$ (in fact, an infinite family of siblings $\pi_n$ for $n \ge 0$.) These functors are much more interesting: $\pi_1$ takes values in $\mathsf{Groups}$ (and there are many obstructions to two groups being isomorphic), and $\pi_n$ for $n \ge 2$ takes values in $\mathsf{Ab}$. We will study $\pi_1$ next lecture—for the higher $\pi_n$'s you'll have to wait until Algebraic Topology II. 🤓