Corollary 1.5.5.12. Let $X$ be a simplicial set. The following conditions are equivalent:
- $(1)$
The simplicial set $X$ is a contractible Kan complex.
- $(2)$
For every monomorphism of simplicial sets $i: A \hookrightarrow B$ and every map of simplicial sets $f_0: A \rightarrow X$, there exists a map $f: B \rightarrow X$ such that $f_0 = f \circ i$.