Example Let $X$ be a topological space, and regard the simplicial set $\operatorname{Sing}_{\bullet }(X)$ as an $\infty $-category. Then:

  • The objects of $\operatorname{Sing}_{\bullet }(X)$ are the points of $X$.

  • The morphisms of $\operatorname{Sing}_{\bullet }(X)$ are continuous paths $f: [0,1] \rightarrow X$. The source of a morphism $f$ is the point $f(0)$, and the target is the point $f(1)$.

  • For every point $x \in X$, the identity morphism $\operatorname{id}_{x}$ is the constant path $[0,1] \rightarrow X$ taking the value $x$.