What does it mean if a map corresponds to a path?

Solution 1:

The circle $S^1$ is obtained as a quotient from $I = [0,1]$ by identifying the points $0$ and $1$. You can take $p : I \to S^1, p(t) = (\cos(2\pi t), \sin(2\pi t))$ as a quotient map. Then $\gamma_f = f \circ p, \gamma_g = g \circ p$.

To obtain an assignment $f \mapsto \gamma_f$ it would be sufficient to take any map $p : I \to S^1$. But to obtain a closed path $\gamma_f$ for each $f$, we need the minimal requirement that $p$ is a closed path in $S^1$, i.e. $p(0) = p(1)$. This is the case for our above $p$. But what is the relevance of our $p$ being a quotient map?

Given a closed path $\delta : I \to X$, i.e. with $\delta(0) = \delta(1)$, we see that there exists a unique function $\bar \delta : S^1 \to X$ such that $\bar \delta \circ p = \delta$. Since $p$ is a quotient map, this map is continuous (universal property of quotient maps). Thus we get an assigment $\delta \mapsto \bar \delta $. This is easily seen to be inverse to $f \mapsto \gamma_f$ .