Definition 1.5.5.8. Let $X$ be a simplicial set. We say that $X$ is a contractible Kan complex if the projection map $X \rightarrow \Delta ^0$ is a trivial Kan fibration (Definition 1.5.5.1). In other words, $X$ is a contractible Kan complex if every map $\sigma _0: \operatorname{\partial \Delta }^ n \rightarrow X$ can be extended to an $n$-simplex of $X$.
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