Example 1.4.3.3. Let $X$ be a topological space. Suppose we are given points $x,y \in X$ and a pair of continuous paths $f,g: [0,1] \rightarrow X$ satisfying $f(0) = x = g(0)$ and $f(1)= y = g(1)$. Then $f$ and $g$ are homotopic as morphisms of the $\infty $-category $\operatorname{Sing}_{\bullet }(X)$ (in the sense of Definition 1.4.3.1) if and only if the paths $f$ and $g$ are homotopic relative to their endpoints: that is, if and only if there exists a continuous function $H: [0,1] \times [0,1] \rightarrow X$ satisfying
\[ H(s,0) = f(s) \quad H(s,1) = g(s) \quad H(0, t) = x \quad H(1,t) = y \]
(see Exercise 1.4.3.4 for a more precise statement).