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Definition Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n \geq -1$ be an integer. We say that $f$ is $n$-truncated if, for every vertex $x \in X$ having image $y = f(x)$, the induced map

\[ \pi _{m}(f): \pi _{m}(X,x) \rightarrow \pi _{m}(Y,y) \]

is injective for $m = n+1$ and bijective for $m > n+1$. If $n \leq -2$, we say that $f$ is $n$-truncated if it is $(-1)$-truncated and the map $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is surjective.