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Notation 9.1.5.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often denote a lifting problem $\sigma $ in $\operatorname{\mathcal{C}}$ by a diagram

9.8
\begin{equation} \begin{gathered}\label{equation:lifting-property-in-infinity2} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r]^-{u_0} & X \ar [d]^{g} \\ B \ar [r]^-{ \overline{u} } \ar@ {-->}[ur] & Y. } \end{gathered} \end{equation}

Here the dotted arrow in the diagram does not indicate part of the data supplied by lifting problem $\sigma $; instead, it indicates part of the data of a hypothetical solution.

Stated more concretely, the lifting problem $\sigma $ is given by the following data:

  • Four objects of $\operatorname{\mathcal{C}}$, which are indicated by $A$, $B$, $X$, and $Y$ in the diagram (9.8).

  • Five morphisms of $\operatorname{\mathcal{C}}$, which we will denote by $f: A \rightarrow B$, $g: X \rightarrow Y$, $u_0: A \rightarrow X$, $\overline{u}: B \rightarrow Y$, and $\overline{u}_0: A \rightarrow Y$. Here the first four of these morphisms are indicated as outer edges of the diagram (9.8), while the fifth is left implicit.

  • A pair of $2$-simplices $\tau _{1}$ and $\tau _{2}$ of $\operatorname{\mathcal{C}}$, whose boundaries are indicated in the diagrams

    \[ \xymatrix@R =50pt@C=50pt{ A \ar [dr]_{\overline{u}_0} \ar [r]^-{u_0} & X \ar [d]^{g} & A \ar [d]^{f} \ar [dr]^{\overline{u}_0} & \\ & Y & B \ar [r]^-{\overline{u}} & Y. } \]

    In other words, $\tau _1$ and $\tau _2$ exhibit the morphism $\overline{u}_0$ as a composition $\overline{u} \circ f$ and a composition $g \circ u_0$, respectively.

A solution to the lifting problem $\sigma $ is given by the following additional data:

  • A morphism $u: B \rightarrow X$ (indicated by the dotted arrow in the diagram (9.8).

  • A pair of $2$-simplices $\tau _0$ and $\tau _3$ of $\operatorname{\mathcal{C}}$, whose boundaries are indicated in the diagrams

    \[ \xymatrix@R =50pt@C=50pt{ & X \ar [d]^{g} & A \ar [d]^{f} \ar [r]^-{u_0} & X \\ B \ar [ur]^{u} \ar [r]^-{\overline{u}} & Y & B. \ar [ur]_{u} & } \]

    In other words, $\tau _0$ exhibits $\overline{u}$ as a composition $g \circ u$, and $\tau _3$ exhibits $u_0$ as a composition $u \circ f$.

  • A $3$-simplex of $\operatorname{\mathcal{C}}$ having boundary $(\tau _0, \tau _1, \tau _2, \tau _3)$.