ScalingStacks

Definition 3.2. Let ${\tt E}\xrightarrow{{\tt p}}{\tt B}$ be an inner fibration in $s{\mathcal{S}\textup{et}}$. Following [Lur09, Definition 2.4.1.1], we will say that an edge $\Delta^{1}\xrightarrow{{\tt f}}{\tt E}$ is JL-${\tt p}$-cartesian (or simply JL-cartesian) if the induced map

$${\tt E}_{/{\tt f}}\rightarrow{\tt E}_{/{\tt f}(1)}\underset{{\tt B}_{/{\tt p}({\tt f}(1))}}{\times}{\tt B}_{/{\tt p}({\tt f})}$$

lies in $({\bf W}\cap{\bf F})_{\textup{Joyal}}\subset s{\mathcal{S}\textup{et}}$.²² 2 The map ${\tt E}_{/{\tt f}}\rightarrow{\tt E}_{/{\tt f}(1)}$ should be thought of simply as “postcomposition with ${\tt f}$”: the restriction map ${\tt E}_{/{\tt f}}\rightarrow{\tt E}_{/{\tt f}(0)}$ lies in $({\bf W}\cap{\bf F})_{\textup{Joyal}}$. Similarly for the map ${\tt B}_{/{\tt p}({\tt f})}\rightarrow{\tt B}_{/{\tt p}({\tt f}(1))}$. In this case, we will refer to the edge ${\tt f}\in{\tt E}_{1}$ as a JL-${\tt p}$-cartesian lift (or simply a JL-cartesian lift) of the edge ${\tt p}({\tt f})\in{\tt B}_{1}$ relative to the vertex ${\tt f}(1)\in{\tt E}_{0}$. Following [Lur09, Definition 2.4.2.1], we then say that the morphism ${\tt f}$ is a JL-cartesian fibration if every vertex of

$$\underline{\hom}(\Delta^{1},{\tt B})\underset{\textup{ev}_{1},{\tt B},{\tt f}}{\times}{\tt E}$$

admits an ${\tt f}$-cartesian lift.