Warning 9.2.4.2. Let $f: X_0 \rightarrow X$ be a morphism of Kan complexes. The assertion that $f$ is a monomorphism can be given two different interpretations:

- $(1)$
The map $f$ is a monomorphism in the ordinary category of $\operatorname{Set_{\Delta }}$ of simplicial sets.

- $(2)$
The map $f$ is a monomorphism in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces.

Beware that these conditions are unrelated to one another. Condition $(2)$ is homotopy invariant: it is the requirement that $f$ restricts to a homotopy equivalence of $X_0$ with a summand of $X$ (Example 9.2.4.10). Condition $(1)$ is very far from being homotopy invariant: we can always arrange that it is satisfied by replacing $X$ by a homotopy equivalent Kan complex (see Exercise 3.1.7.11).