ScalingStacks

Proof. We must show that the map

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

in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$ lies in $({\bf W}\cap{\bf F})_{\textup{Joyal}}$.

First of all, using the characterization ${\bf F}_{\textup{Joyal}}=\textup{rlp}(({\bf W}\cap{\bf C})_{\textup{Joyal}})$, it is easy to see

• •

  that the object ${\tt E}_{/{\tt f}(1)}\in s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$ is fibrant and

• •

  that the map ${\tt B}_{/{\tt p}({\tt f})}\rightarrow{\tt B}_{/{\tt p}({\tt f}(1))}$ is a fibration in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$.

Hence, it follows from the Reedy trick that this fiber product is in fact a homotopy pullback in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$. Thus, this map in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$ presents the map

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

in ${{\mathcal{C}\textup{at}}_{\infty}}$, which can be canonically identified with the map

$$\mathcal{E}_{/f(0)}\rightarrow\mathcal{E}_{f(1)}\underset{\mathcal{B}_{/p(f(1))}}{\times}\mathcal{B}_{/p(f(0))}$$

in ${{\mathcal{C}\textup{at}}_{\infty}}$ and therefore lies in ${\bf W}_{\textup{Joyal}}$ by assumption.

To see that it also lies in ${\bf F}_{\textup{Joyal}}$, we argue as follows. We claim that there is a Quillen adjunction

$$\alpha:\textup{Fun}([1],s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}\rightleftarrows(s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\Delta^{1}/}:\beta,$$

where

• •

  we equip $[1]$ with the Reedy category structure determined by the degree function $0\mapsto 0$ and $1\mapsto 1$,

• •

  we define

  $$\beta(\Delta^{1}\xrightarrow{{\tt y}}{\tt Y})=({\tt Y}_{/{\tt y}}\to{\tt Y}_{/{\tt y}(1)}),$$

  and

• •

  we define

  $$\alpha({\tt Z}\rightarrow{\tt W})=\left(\Delta^{1}\rightarrow\left({\tt Z}\star\Delta^{1}\coprod_{{\tt Z}\star\Delta^{\{1\}}}{\tt W}\star\Delta^{\{1\}}\right)\right).$$

It is not hard to see that indeed $\alpha\dashv\beta$, so it suffices to show that $\alpha$ is a left Quillen functor. For this, given a map

in $\textup{Fun}([1],s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}$ (reading the square vertically), observe that for this to be a (resp.​ acyclic) cofibration is precisely to require that the two relative latching maps ${\tt Z}_{1}\rightarrow{\tt Z}_{2}$ and ${\tt W}_{1}\coprod_{{\tt Z}_{1}}{\tt Z}_{2}\rightarrow{\tt W}_{2}$ are (resp.​ acyclic) cofibrations. For simplicity, let us write the composite

$$\textup{Fun}([1],s{\mathcal{S}\textup{et}})\xrightarrow{\alpha}s{\mathcal{S}\textup{et}}_{\Delta^{1}/}\rightarrow s{\mathcal{S}\textup{et}}$$

of our left adjoint with the evident forgetful functor simply as $\textup{Fun}([1],s{\mathcal{S}\textup{et}})\xrightarrow{\alpha^{\prime}}s{\mathcal{S}\textup{et}}$. Now, assuming our map ${\tt g}_{1}\rightarrow{\tt g}_{2}$ is a cofibration in $\textup{Fun}([1],s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}$, then its image $\alpha^{\prime}({\tt g}_{1})\rightarrow\alpha^{\prime}({\tt g}_{2})$ fits into the diagram

in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$ of Figure 3, in which

• •

  the front and back faces are pushouts by definition;

• •

  the quadrilateral contained in the top face is a pushout since in the composite

  $$s{\mathcal{S}\textup{et}}\xrightarrow{-\star\Delta^{\{1\}}}s{\mathcal{S}\textup{et}}_{\Delta^{\{1\}}/}\rightarrow s{\mathcal{S}\textup{et}}$$

  where the second functor is forgetful,

  • –

    the first functor commutes with colimits by [Lur09, Remark 1.2.8.2] and

  • –

    the second functor commutes with pushouts since the walking span $\textup{N}^{-1}(\Lambda^{2}_{0})\in{\mathcal{C}\textup{at}}$ has an initial object

  (although really we have only rewritten this pushout to improve readability), and the dotted arrow is then the induced map;

• •

  the left face is a pushout by inspection;

• •

  all maps labeled as cofibrations are such

  • –

    by inspection,

  • –

    by the assumption that ${\tt g}_{1}\rightarrow{\tt g}_{2}$ is a cofibration in $\textup{Fun}([1],s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}$,

  • –

    because ${\bf C}_{\textup{Joyal}}\subset s{\mathcal{S}\textup{et}}$ is closed under pushouts, or

  • –

    because ${\bf C}_{\textup{Joyal}}\subset s{\mathcal{S}\textup{et}}$ is closed under composition;

  and

• •

  the maps labeled with the symbol $\stackrel{{\scriptstyle?}}{{\approx}}$ are weak equivalences in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$ if ${\tt g}_{1}\rightarrow{\tt g}_{2}$ is additionally a weak equivalence in $\textup{Fun}([1],s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}$

  • –

    by the assumption that ${\tt g}_{1}\rightarrow{\tt g}_{2}$ is an acyclic cofibration in $\textup{Fun}([1],s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}$,

  • –

    because $({\bf W}\cap{\bf C})_{\textup{Joyal}}\subset s{\mathcal{S}\textup{et}}$ is closed under pushouts, or

  • –

    because $({\bf W}\cap{\bf C})_{\textup{Joyal}}\subset s{\mathcal{S}\textup{et}}$ is closed under composition.

Now, because the left and back faces are both pushouts, then the composite rectangle which they form is also a pushout. But this is the same as the composite rectangle formed by the front and right faces. As the front face is a pushout, it follows that the right face is also a pushout as well. Thus, the functor

$$\textup{Fun}([1],s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}\xrightarrow{\alpha^{\prime}}s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$$

preserves both cofibrations and acyclic cofibrations, since these are each closed under pushout in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$. But the cofibrations and acyclic cofibrations in $(s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\Delta^{1}/}$ are created by the forgetful functor $s{\mathcal{S}\textup{et}}_{\Delta^{1}/}\rightarrow s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$, and so the functor

$$\textup{Fun}([1],s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}\xrightarrow{\alpha}(s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\Delta^{1}/}$$

is indeed a left Quillen functor, as claimed.

We now return to our given composite

$$\Delta^{1}\xrightarrow{{\tt f}}{\tt E}\stackrel{{\scriptstyle{\tt p}}}{{\twoheadrightarrow}}{\tt B}$$

in $s{\mathcal{S}\textup{et}}_{\textup{Joyal}}$. This can be considered as defining a fibration in $(s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\Delta^{1}/}$, and hence applying our right Quillen functor

$$(s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\Delta^{1}/}\xrightarrow{\beta}\textup{Fun}([1],s{\mathcal{S}\textup{et}}_{\textup{Joyal}})_{\textup{Reedy}}$$

yields another fibration. In particular, the resulting relative matching map

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

at the object $1\in[1]$ must lie in ${\bf F}_{\textup{Joyal}}\subset s{\mathcal{S}\textup{et}}$, as desired. ∎

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

$$\lim\left(\hbox to163.1pt{\vbox to87.5pt{\pgfpicture\makeatletter\hbox{\hskip 81.54857pt\lower-43.81284pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{}{}{{}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{{}}{{}}{{}}{{}}{{}}}{{{\lx@inpgf@ignorespaces}}}{{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-81.54857pt}{-37.52953pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -112.84 -51.93)} \pgfsys@hbox{58}\lxSVG@closescope }}}{{{\lx@inpgf@ignorespaces{}}}{{}}{{}}{{}}{{}}{{}}}} \lxSVG@closescope }}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{\lx@inpgf@ignorespaces}}{}{}{}{{{}{}}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.39998pt} \lx@inpgf@ignorespaces{}{}{}{}{{}}{}{}{{}}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 91.09 38.78 L 91.09 12.18}{fill:none} {{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}}}{{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{0.0}{-1.0}{1.0}{0.0}{65.8333pt}{8.60081pt}\lxSVG@begingroup@{transform=matrix(0.0 -1.0 1.0 0.0 91.09 11.9)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@begingroup@{stroke-linejoin=round} \lxSVG@drawpath@unclipped{M -2.88 3.32 C -2.35 1.33 -1.18 0.39 0 0 C -1.18 -0.39 -2.35 -1.33 -2.88 -3.32}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{68.18607pt}{17.20837pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 94.35 23.81)} \pgfsys@hbox{58}\lxSVG@closescope }}} \lxSVG@closescope }}} \lxSVG@closescope {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{\lx@inpgf@ignorespaces}}{}{}{}{{{}{}}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.39998pt} \lx@inpgf@ignorespaces{}{}{}{}{}{{}}{}{}{{}}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 36.42 1.29 L 73.76 1.29}{fill:none} {{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}}}{{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{52.0629pt}{0.93pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 72.04 1.29)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@begingroup@{stroke-linejoin=round} \lxSVG@drawpath@unclipped{M -2.88 3.32 C -2.35 1.33 -1.18 0.39 0 0 C -1.18 -0.39 -2.35 -1.33 -2.88 -3.32}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{53.5028pt}{0.93pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 74.03 1.29)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@begingroup@{stroke-linejoin=round} \lxSVG@drawpath@unclipped{M -2.88 3.32 C -2.35 1.33 -1.18 0.39 0 0 C -1.18 -0.39 -2.35 -1.33 -2.88 -3.32}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{34.18051pt}{-4.43666pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 47.3 -6.14)} \pgfsys@hbox{58}\lxSVG@closescope }}} \lxSVG@closescope }}} \lxSVG@closescope {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{\lx@inpgf@ignorespaces}}{}{}{}{{{}{}}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.39998pt} \lx@inpgf@ignorespaces{}{}{}{}{}{{}}{}{}{{}}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 0 -10.97 L 0 -34.77}{fill:none} {{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}}}{{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{-23.8899pt}\lxSVG@begingroup@{transform=matrix(0.0 -1.0 1.0 0.0 0 -33.06)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@begingroup@{stroke-linejoin=round} \lxSVG@drawpath@unclipped{M -2.88 3.32 C -2.35 1.33 -1.18 0.39 0 0 C -1.18 -0.39 -2.35 -1.33 -2.88 -3.32}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{-25.3298pt}\lxSVG@begingroup@{transform=matrix(0.0 -1.0 1.0 0.0 0 -35.05)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@begingroup@{stroke-linejoin=round} \lxSVG@drawpath@unclipped{M -2.88 3.32 C -2.35 1.33 -1.18 0.39 0 0 C -1.18 -0.39 -2.35 -1.33 -2.88 -3.32}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{2.35277pt}{-18.97476pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 3.26 -26.26)} \pgfsys@hbox{58}\lxSVG@closescope }}} \lxSVG@closescope }}} \lxSVG@closescope {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{\lx@inpgf@ignorespaces}}{}{}{}{{{}{}}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.39998pt} \lx@inpgf@ignorespaces{}{}{}{}{{}}{}{}{{}}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -69.07 -48.47 L -36.97 -48.47}{fill:none} {{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}}}{{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-26.51804pt}{-35.02953pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -36.69 -48.47)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@begingroup@{stroke-linejoin=round} \lxSVG@drawpath@unclipped{M -2.88 3.32 C -2.35 1.33 -1.18 0.39 0 0 C -1.18 -0.39 -2.35 -1.33 -2.88 -3.32}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-39.95552pt}{-41.66006pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -55.29 -57.65)} \pgfsys@hbox{58}\lxSVG@closescope }}} \lxSVG@closescope }}} \lxSVG@closescope \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}\right)$$

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

$$\lim\left(\hbox to183.95pt{\vbox to87.96pt{\pgfpicture\makeatletter\hbox{\hskip 91.97398pt\lower-43.97923pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{}{}{{}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{{}}{{}}{{}}{{}}{{}}}{{{\lx@inpgf@ignorespaces}}}{{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-91.97398pt}{-37.8195pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -127.26 -52.33)} \pgfsys@hbox{58}\lxSVG@closescope }}}{{{\lx@inpgf@ignorespaces{}}}{{}}{{}}{{}}{{}}{{}}}} \lxSVG@closescope }}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{\lx@inpgf@ignorespaces}}{}{}{}{{{}{}}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.39998pt} \lx@inpgf@ignorespaces{}{}{}{}{{}}{}{}{{}}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 41.9 0 L 82.73 0}{fill:none} {{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}}}{{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{59.98895pt}{0.0pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 83.01 0)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@begingroup@{stroke-linejoin=round} \lxSVG@drawpath@unclipped{M -2.88 3.32 C -2.35 1.33 -1.18 0.39 0 0 C -1.18 -0.39 -2.35 -1.33 -2.88 -3.32}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{43.72485pt}{-6.65831pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 60.5 -9.21)} \pgfsys@hbox{58}\lxSVG@closescope }}} \lxSVG@closescope }}} \lxSVG@closescope {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{\lx@inpgf@ignorespaces}}{}{}{}{{{}{}}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.39998pt} \lx@inpgf@ignorespaces{}{}{}{}{{}}{}{}{{}}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -2.19 -12.26 L -2.19 -36.06}{fill:none} {{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}}}{{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{0.0}{-1.0}{1.0}{0.0}{-1.58507pt}{-26.2598pt}\lxSVG@begingroup@{transform=matrix(0.0 -1.0 1.0 0.0 -2.19 -36.34)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@begingroup@{stroke-linejoin=round} \lxSVG@drawpath@unclipped{M -2.88 3.32 C -2.35 1.33 -1.18 0.39 0 0 C -1.18 -0.39 -2.35 -1.33 -2.88 -3.32}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{0.7677pt}{-19.40974pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 1.06 -26.86)} \pgfsys@hbox{58}\lxSVG@closescope }}} \lxSVG@closescope }}} \lxSVG@closescope { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.39998pt} \lx@inpgf@ignorespaces{}{}{}{}{{}}{}{}{}{{}}\lxSVG@stroke\lxSVG@drawpath@unclipped{M 101.14 11.89 L 101.14 34.62}{fill:none} {{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{0.0}{-1.0}{1.0}{0.0}{73.09724pt}{8.39302pt}\lxSVG@begingroup@{transform=matrix(0.0 -1.0 1.0 0.0 101.14 11.61)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@begingroup@{stroke-linejoin=round} \lxSVG@drawpath@unclipped{M -2.88 3.32 C -2.35 1.33 -1.18 0.39 0 0 C -1.18 -0.39 -2.35 -1.33 -2.88 -3.32}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}}}{{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{0.0}{1.0}{-1.0}{0.0}{73.09724pt}{25.01987pt}\lxSVG@begingroup@{transform=matrix(0.0 1.0 -1.0 0.0 101.14 34.62)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linejoin=miter} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@drawpath@unclipped{M 0 2.71 C 0.95 2.71 1.72 2.1 1.72 1.36 C 1.72 0.61 0.95 0 0 0}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}}}\lx@inpgf@ignorespaces \lxSVG@closescope { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.39998pt} \lx@inpgf@ignorespaces{}{}{}{}{{}}{}{}{}{{}}\lxSVG@stroke\lxSVG@drawpath@unclipped{M -77.14 -48.87 L -47.03 -48.87}{fill:none} {{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{-1.0}{0.0}{0.0}{1.0}{-55.75073pt}{-35.3195pt}\lxSVG@begingroup@{transform=matrix(-1.0 0.0 0.0 1.0 -77.14 -48.87)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linejoin=miter} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@drawpath@unclipped{M 0 2.71 C 0.95 2.71 1.72 2.1 1.72 1.36 C 1.72 0.61 0.95 0 0 0}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}}}{{}{{}}{}{}{{}}{{{\lx@inpgf@ignorespaces}}{{{\lx@inpgf@ignorespaces}}{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-33.79065pt}{-35.3195pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -46.76 -48.87)} \lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke-dasharray=none,stroke-dashoffset=0.0pt} \lxSVG@begingroup@{stroke-linecap=round} \lxSVG@begingroup@{stroke-linejoin=round} \lxSVG@drawpath@unclipped{M -2.88 3.32 C -2.35 1.33 -1.18 0.39 0 0 C -1.18 -0.39 -2.35 -1.33 -2.88 -3.32}{fill:none} \lxSVG@closescope \lxSVG@closescope }}{{\lx@inpgf@ignorespaces}}}}\lx@inpgf@ignorespaces \lxSVG@closescope \lxSVG@closescope {\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}\right)$$

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. ∎