Section 2. (0MWT) · ScalingStacks

2. Definitions

In this brief section, we give precise definitions of the concepts we have introduced in § 1.

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

(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)

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.

0MWE

Remark 2.2. Let $\mathcal{E}\xrightarrow{f}\mathcal{B}$ be a functor of $\infty$ -categories. The crucial consequence of a morphism $e_{1}\xrightarrow{\varphi}e_{2}$ in $\mathcal{E}$ being $f$ -cocartesian is that for any object $e\in\mathcal{E}$ , the induced diagram

is a pullback square in $\mathcal{S}$ . Dually, the crucial consequence of a morphism $e_{1}\xrightarrow{\varphi}e_{2}$ in $\mathcal{E}$ being $f$ -cartesian is that for any object $e\in\mathcal{E}$ , the induced diagram

is a pullback square in $\mathcal{S}$ . (Recall Example 1.6, and see [Lur09, Proposition 2.4.1.10].)

0MWF

Remark 2.3. A cocartesian fibration $\mathcal{E}\xrightarrow{f}\mathcal{B}$ whose fibers $f^{-1}(b)$ are all $\infty$ -groupoids is called a left fibration. These assemble into the full subcategory $\mathcal{LF}\textup{ib}(\mathcal{B})\subset\textup{co}\mathcal{C}\mathcal{F}\textup{ib}(\mathcal{B})$ . Correspondingly, the (covariant) Grothendieck construction restricts to an equivalence

from the $\infty$ -category of functors $\mathcal{B}\rightarrow\mathcal{S}$ valued in the $\infty$ -category of spaces to the $\infty$ -category $\mathcal{LF}\textup{ib}(\mathcal{B})$ of left fibrations over $\mathcal{B}$ . As a result of the fact that all morphisms in a space are equivalences, given a left fibration $\mathcal{E}\xrightarrow{f}\mathcal{B}$ , every morphism in $\mathcal{E}$ is $f$ -cocartesian.

Dually, a cartesian fibration whose fibers are all $\infty$ -groupoids is called a right fibration, the contravariant Grothendieck construction restricts to an equivalence

from the $\infty$ -category of functors $\mathcal{B}^{op}\rightarrow\mathcal{S}$ to the full subcategory $\mathcal{RF}\textup{ib}(\mathcal{B})\subset\mathcal{C}\mathcal{F}\textup{ib}(\mathcal{B})$ of right fibrations over $\mathcal{B}$ , and given a right fibration $\mathcal{E}\xrightarrow{f}\mathcal{B}$ , every morphism in $\mathcal{E}$ is $f$ -cartesian.