Definition 3.5.7.19. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n$ be an integer. We say that $f$ exhibits $Y$ as an $n$-truncation of $Y$ if $Y$ is $n$-truncated and $f$ is $(n+1)$-connective. We say that $Y$ is an $n$-truncation of $X$ if there exists a morphism $f: X \rightarrow Y$ which exhibits $Y$ as an $n$-truncation of $X$.
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