$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. The following conditions are equivalent:


The morphism $f$ is a Kan fibration (Definition


For every square diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A_{} \ar [d]^{i} \ar [r] & X_{} \ar [d]^{f} \\ B_{} \ar [r] \ar@ {-->}[ur] & S_{} } \]

where $i$ is anodyne, there exists a dotted arrow rendering the diagram commutative.

The implication $(b) \Rightarrow (a)$ is immediate from the definitions (since the horn inclusions $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ are anodyne for $n > 0$). The reverse implication follows from the weak saturation of the collection of morphisms which are weakly left orthogonal to $f$ (Proposition