Definition 4.5.5.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets. We will say that $q$ is an *isofibration* if it satisfies the following condition:

- $(\ast )$
Let $B$ be a simplicial set and let $A \subseteq B$ be a simplicial subset for which the inclusion $A \hookrightarrow B$ is a categorical equivalence. Then every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]admits a solution.