ScalingStacks

Proof. Suppose we are given any edge ${\tt f}\in{\tt B}_{1}$. Let us write $\delta_{0}({\tt f})={\tt b}_{2}\in{\tt B}_{0}$ and $\delta_{1}({\tt f})={\tt b}_{1}\in{\tt B}_{0}$. Suppose further that we are given any vertex ${\tt e}_{2}\in{\tt E}_{0}$ with ${\tt p}({\tt e}_{2})={\tt b}_{2}$. Then we must find a JL-${\tt p}$-cartesian lift of the edge ${\tt f}\in{\tt B}_{1}$ relative to the vertex ${\tt e}_{2}\in{\tt E}_{0}$.

Let us respectively write $e_{1}\xrightarrow{\tilde{f}}e_{2}$ and $b_{1}\xrightarrow{f}b_{2}$ for the morphisms in $\mathcal{E}$ and $\mathcal{B}$ presented by the maps $\Delta^{1}\xrightarrow{\tilde{{\tt f}}}{\tt E}$ and $\Delta^{1}\xrightarrow{{\tt f}}{\tt B}$ in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$ (for any choice of edge $\tilde{{\tt f}}\in{\tt E}_{1}$). Then, according to Theorem 3.3, for an edge $\tilde{{\tt f}}\in{\tt E}_{1}$ to be ${\tt p}$-cartesian, it suffices to verify that the morphism $\tilde{f}$ in $\mathcal{E}$ is $p$-cartesian.

Now, the given data define a vertex

$$({\tt f},{\tt e}_{2})\in\left(\underline{\hom}(\Delta^{1},{\tt B})\underset{\textup{ev}_{1},{\tt B},{\tt p}}{\times}{\tt E}\right)_{0}$$

of the fiber product in $s{\mathcal{S}\textup{et}}$. Moreover, by [Lur09, Proposition 1.2.7.3(1)] and the Reedy trick, this fiber product is a homotopy pullback in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$. Hence, this vertex defines an object

$$(f,b_{2})\in\textup{Fun}([1],\mathcal{B})\underset{t,\mathcal{B},p}{\times}\mathcal{E}$$

of the pullback in ${{\mathcal{C}\textup{at}}_{\infty}}$.

Next, it is easy to check that the map

$$\underline{\hom}(\Delta^{1},{\tt E})\xrightarrow{\underline{\hom}(\Delta^{1},{\tt p})}\underline{\hom}(\Delta^{1},{\tt B})$$

lies in ${\bf F}_{\textup{Joyal}}\subset s{\mathcal{S}\textup{et}}$, simply by using

• •

  the fact that ${\bf F}_{\textup{Joyal}}=\textup{rlp}(({\bf W}\cap{\bf C})_{\textup{Joyal}})$,

• •

  the adjunction $-\times\Delta^{1}:s{\mathcal{S}\textup{et}}\rightleftarrows s{\mathcal{S}\textup{et}}:\underline{\hom}_{s{\mathcal{S}\textup{et}}}(\Delta^{1},-)$, and

• •

  the fact that $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}\xrightarrow{-\times\Delta^{1}}s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$ is a left Quillen functor and hence in particular preserves acyclic cofibrations.

Moreover, the map

$$\underline{\hom}(\Delta^{1},{\tt E})\xrightarrow{\textup{ev}_{1}}{\tt E}$$

also lies in ${\bf F}_{\textup{Joyal}}\subset s{\mathcal{S}\textup{et}}$, since by applying the dual of [Lur09, Corollary 2.4.7.12] to the map ${\tt E}\xrightarrow{\textup{id}_{{\tt E}}}{\tt E}$ in $s{\mathcal{S}\textup{et}}^{f}_{\textup{Joyal}}$ we see that it is a JL-cocartesian fibration, which by [Lur09, Remark 2.0.0.5] implies that it is in particular a fibration in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$. Thus, our conditions that $\delta_{0}(\tilde{{\tt f}})={\tt e}_{2}$ and that ${\tt p}(\tilde{{\tt f}})={\tt f}$ translate into the single condition that $\tilde{{\tt f}}\in{\tt E}_{1}$ define a vertex of the limit

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in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$.

Now, we claim that the above limit is in fact a homotopy limit in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$. For this, we appeal to a more elaborate version of the Reedy trick: we endow the category

$$\textup{N}^{-1}(\textup{sd}^{2}(\Delta^{1}))=(\bullet\rightarrow\bullet\leftarrow\bullet\rightarrow\bullet\leftarrow\bullet)\in{\mathcal{C}\textup{at}}$$

with the Reedy structure determined by the degree function described by the picture $(0\rightarrow 1\leftarrow 2\rightarrow 1\leftarrow 0)$. Using [Hir03, Proposition 15.10.2(1)], it is easy to see that this Reedy category has cofibrant constants, and hence by [Hir03, Theorem 15.10.8(1)] we obtain a Quillen adjunction

$$\textup{const}:s{\mathcal{S}\textup{et}}_{\textup{Joyal}}\rightleftarrows\textup{Fun}(\textup{N}^{-1}(\textup{sd}^{2}(\Delta^{1})),s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}:\lim.$$

Then, to see that the diagram in the above limit defines a fibrant object of $\textup{Fun}(\textup{N}^{-1}(\textup{sd}^{2}(\Delta^{1})),s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}$, since it is objectwise fibrant, it only remains to check that the latching map

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

lies in ${\bf F}_{\textup{Joyal}}\subset s{\mathcal{S}\textup{et}}$. For this, we use the characterization ${\bf F}_{\textup{Joyal}}=\textup{rlp}(({\bf W}\cap{\bf C})_{\textup{Joyal}})$: given any solid commutative square

in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$, unwinding the definitions we see that a dotted lift is equivalent to a dotted lift in the diagram

in $s{\mathcal{S}\textup{et}}$. But it is easy to see that the left map lies in $({\bf W}\cap{\bf C})_{\textup{Joyal}}$, while the right map lies in ${\bf F}_{\textup{Joyal}}$ by assumption. Hence the diagram in the above limit defines a fibrant object of $\textup{Fun}(\textup{N}^{-1}(\textup{sd}^{2}(\Delta^{1})),s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}$, and so the above limit is a homotopy limit in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$ and thus presents the limit

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in ${{\mathcal{C}\textup{at}}_{\infty}}$.

Now, we have assumed that $p$ is a cartesian fibration, so in particular there must exist a $p$-cartesian lift $\tilde{f}$ of $f$ relative to the object $e_{2}$. This defines an object of the above limit in ${{\mathcal{C}\textup{at}}_{\infty}}$, which must therefore be represented by a vertex of the above homotopy limit in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$: this selects the desired JL-${\tt p}$-cartesian lift $\tilde{{\tt f}}\in{\tt E}_{1}$ of ${\tt f}\in{\tt B}_{1}$ relative to ${\tt e}_{2}\in{\tt E}_{0}$. So the map ${\tt E}\stackrel{{\scriptstyle{\tt p}}}{{\twoheadrightarrow}}{\tt B}$ in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$ is indeed a JL-cartesian fibration, as claimed. ∎