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Example 1.3.4.5. Let $X$ be a topological space and suppose we are given continuous paths $f,g: [0,1] \rightarrow X$ which are composable in the sense that $f(1) = g(0)$, and let $g \star f: [0,1] \rightarrow X$ denote the path obtained by concatenating $f$ and $g$, given concretely by the formula

\[ (g \star f)(t) = \begin{cases} f(2t) & \text{ if } 0 \leq t \leq 1/2 \\ g(2t-1) & \text{ if } 1/2 \leq t \leq 1. \end{cases} \]

Then $g \star f$ is a composition of $f$ and $g$ in the $\infty $-category $\operatorname{Sing}_{\bullet }(X)$. More precisely, the continuous map

\[ \sigma : | \Delta ^2 | \rightarrow X \quad \quad \sigma ( t_0, t_1, t_2) = \begin{cases} f( t_1 + 2 t_2 ) & \text{ if } t_0 \geq t_2 \\ g( t_2 - t_0 ) & \text{ if } t_0 \leq t_2. \end{cases} \]

can be regarded as a $2$-simplex of $\operatorname{Sing}_{\bullet }(X)$ which witnesses $g \star f$ as a composition of $f$ and $g$.