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

Remark By construction, the collection of anodyne morphisms is weakly saturated. In particular:

  • Every isomorphism of simplicial sets is anodyne.

  • If $i: A_{} \rightarrow B_{}$ and $j: B_{} \rightarrow C_{}$ are anodyne morphisms of simplicial sets, then the composition $j \circ i$ is anodyne.

  • For every pushout diagram of simplicial sets

    \[ \xymatrix@R =50pt@C=50pt{ A_{} \ar [d]_{i} \ar [r] & A'_{} \ar [d]^{i'} \\ B_{} \ar [r] & B'_{}, } \]

    if $i$ is anodyne, then $i'$ is also anodyne.

  • Suppose there exists a commutative diagram of simplicial sets

    \[ \xymatrix@R =50pt@C=50pt{ A_{} \ar [d]^{i} \ar [r] & A'_{} \ar [d]^{i'} \ar [r] & A_{} \ar [d]^{i} \\ B_{} \ar [r] & B'_{} \ar [r] & B_{}, } \]

    where the horizontal compositions are the identity. If $i'$ is anodyne, then $i$ is anodyne.