Kerodon

Warning 1.4.4.6. In the situation of Example 1.4.4.5, the concatenation $g \star f$ is not the only path which is a composition of $f$ and $g$ in the $\infty $-category $\operatorname{Sing}_{\bullet }(X)$. Any path in $X$ which is homotopic to $g \star f$ (with endpoints fixed) has the same property, by virtue of Proposition 1.4.4.2 (and Example 1.4.3.3). For example, we can replace $g \star f$ by a reparametrization, such as the path

When viewing $\operatorname{Sing}_{\bullet }(X)$ as an $\infty $-category, all of these paths have an equal claim to be regarded as “the” composition of $f$ and $g$.