Theorem 2.4.5.11 (Homotopy Extension Lifting Property). Let $X_{\bullet }$ be a simplicial set. The following conditions are equivalent:
- $(1)$
The simplicial set $X_{\bullet }$ is a Kan complex.
- $(2)$
The inclusion of simplicial sets $\{ 0\} \hookrightarrow \Delta ^1$ induces a trivial Kan fibration $\operatorname{Fun}( \Delta ^1, X_{\bullet } ) \rightarrow \operatorname{Fun}( \{ 0\} , X_{\bullet } ) \simeq X_{\bullet }$.
- $(3)$
The inclusion of simplicial sets $\{ 1\} \hookrightarrow \Delta ^1$ induces a trivial Kan fibration $\operatorname{Fun}( \Delta ^1, X_{\bullet } ) \rightarrow \operatorname{Fun}( \{ 1\} , X_{\bullet } ) \simeq X_{\bullet }$.