ScalingStacks

Definition 2.1. Let $\mathcal{E}\xrightarrow{f}\mathcal{B}$ be a functor of $\infty$-categories.

1. (1)

   We say that a morphism $e_{1}\xrightarrow{\varphi}e_{2}$ in $\mathcal{E}$ is $f$-cocartesian (or simply cocartesian, if the functor $f$ is clear from the context) if it induces a pullback diagram

   in ${{\mathcal{C}\textup{at}}_{\infty}}$. In this case, we call $\varphi$ an $f$-cocartesian lift (or simply a cocartesian lift) of $f(\varphi)$ relative to $e_{1}$. We then say that the functor $f$ is a cocartesian fibration if every object of

   $$\textup{Fun}([1],\mathcal{B})\underset{s,\mathcal{B},f}{\times}\mathcal{E}$$

   admits an $f$-cocartesian lift.

2. (2)

   Dually, we say that a morphism $e_{1}\xrightarrow{\varphi}e_{2}$ in $\mathcal{E}$ is $f$-cartesian (or simply cartesian) if it induces a pullback diagram

   in ${{\mathcal{C}\textup{at}}_{\infty}}$. In this case, we call $\varphi$ an $f$-cartesian lift (or simply a cartesian lift) of $f(\varphi)$ relative to $e_{2}$. We then say that the functor $f$ is a cartesian fibration if every object of

   $$\textup{Fun}([1],\mathcal{B})\underset{t,\mathcal{B},f}{\times}\mathcal{E}$$

   admits an $f$-cartesian lift.