Kerodon

Warning 4.5.3.15. Let $f: A \rightarrow B$ be a morphism of simplicial sets. By virtue of Corollary 3.3.7.7, the morphism $f$ is anodyne if and only if it is both a monomorphism and a weak homotopy equivalence. Beware that the analogous assertion for inner anodyne morphisms is false. If $f$ is inner anodyne, then it is both a monomorphism (Remark 1.5.6.5) and a categorical equivalence (Corollary 4.5.3.14). However, the converse fails: a monomorphism $A \hookrightarrow B$ which is a categorical equivalence need not be inner anodyne. For example, an inner anodyne morphism of simplicial sets is automatically bijective on vertices (Exercise 1.5.6.6). However, there can be other obstructions as well: see Example 4.5.3.16.