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Definition Let $f: X \rightarrow S$ be a morphism of simplicial sets. We say that $f$ is a left covering map if, for every pair of integers $0 \leq i < n$, every lifting problem

\[ \xymatrix { \Lambda ^{n}_{i} \ar [r] \ar@ {^{(}->}[d] & X \ar [d]^{f} \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & S } \]

admits a unique solution. We say that $f$ is a right covering map if the analogous condition holds for $0 < i \leq n$.