Construction 9.2.7.1. Suppose we are given a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur] & Y. } \]
in an $\infty $-category $\operatorname{\mathcal{C}}$, given by a morphism $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. We let $\operatorname{Sol}( \sigma )$ denote the simplicial set $ \{ \sigma \} \times _{ \operatorname{Fun}( Q, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}})$, where $Q \subset \Delta ^3$ is the simplicial subset described in Remark 9.2.5.2. We will refer to $\operatorname{Sol}( \sigma )$ as the space of solutions to the lifting problem $\sigma $.