Corollary 4.2.4.6. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then:
- $(1)$
The morphism $q$ is a left covering map if and only if, for every square diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]where $i$ is left anodyne, there exists a unique dotted arrow rendering the diagram commutative.
- $(2)$
The morphism $q$ is a right covering map if and only if, for every square diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]where $i$ is right anodyne, there exists unique a dotted arrow rendering the diagram commutative.