Remark 3.5.0.2. The preceding characterization of $n$-truncated Kan complexes has a counterpart for $n$-connective Kan complexes. A Kan complex $X$ is $n$-connective if and only if it is homotopy equivalent to a Kan complex $Y$ having a single $m$-simplex for each $m < n$ (Proposition 3.5.2.9 and Remark 3.5.2.10). We will prove Proposition 3.5.0.1 by showing that, in this case, the fundamental $n$-groupoid $\pi _{\leq n}(Y)$ is *isomorphic* to an Eilenberg-MacLane space $\mathrm{K}(G,n)$, for some group $G$. See Proposition 3.5.5.16.

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