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Definition 9.1.5.1 (Lifting Problems in $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A lifting problem in $\operatorname{\mathcal{C}}$ is a diagram $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. In this case, a solution to the lifting problem $\sigma $ is a $3$-simplex $\overline{\sigma }: \Delta ^3 \rightarrow \operatorname{\mathcal{C}}$ for which the composition

\[ \Delta ^1 \times \Delta ^1 \xrightarrow {\alpha } \Delta ^3 \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{C}} \]

coincides with $\sigma $, where $\alpha $ denotes the map of simplicial sets given on vertices by $\alpha (i,j) = 2i+j$.