ScalingStacks

Definition 4.1.1. (T.5.2.8.1) A commutative square in an $\infty$-category $\mathcal{C}$ is a map $q\colon\Delta^{1}\times\Delta^{1}\to\mathcal{C}$, which we write somewhat informally as

suppressing the homotopies. The space of lifts for $q$ is defined as follows. Restricting to the diagonal $\Delta^{1}\to\Delta^{1}\times\Delta^{1}$, we get a morphism $h\colon A\to Y$ in $\mathcal{C}$, which can be viewed as an object $\overline{Y}$ in the $\infty$-category $\mathcal{C}_{A/}$. The diagram $q$ can be encoded as a pair of objects $B,X\in\mathcal{C}_{A//\overline{Y}}$ and the space of lifts for $q$ is given as the mapping space

$$L\left(q\right)=\operatorname{Map}_{\mathcal{C}_{A//\overline{Y}}}\left(\overline{B},\overline{X}\right).$$