Kerodon

Warning 9.3.4.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $i: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a left homotopy inverse $r: X \rightarrow X_0$. If $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, then $i$ is automatically a monomorphism. In general, this is not necessarily true. For example, let $(X,x)$ be a pointed Kan complex, and regard the inclusion map $i: \{ x\} \rightarrow X$ as a morphism in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces. Then $i$ has a left homotopy inverse (given by the constant map $X \rightarrow \{ x\} $). However, $i$ is a monomorphism in the $\infty $-category $\operatorname{\mathcal{S}}$ only if $x$ belongs to a contractible connected component of $X$ (Example 9.3.4.10).