Proposition 8.1.9.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, let $R$ be a collection of edges of $\operatorname{\mathcal{C}}$ which admits $U$-cocartesian lifts, and let $\widetilde{R}$ denote the collection of all $U$-cocartesian edges $f$ of $\operatorname{\mathcal{E}}$ such that $U(f)$ belongs to $\overline{R}$. Then the induced map $\operatorname{Cospan}^{\mathrm{all}, \widetilde{R} }( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\mathrm{all}, R}( \operatorname{\mathcal{C}})$ is an inner fibration of simplicial sets.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Replacing $\operatorname{\mathcal{C}}$ by a full simplicial subset if necessary, we may assume that $R$ contains every degenerate edge of $\operatorname{\mathcal{C}}$. Choose integers $0 < i < n$; we wish to show that every lifting problem
8.23
\begin{equation} \begin{gathered}\label{equation:all-cocart-inner} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{f_0} \ar [d] & \operatorname{Cospan}^{\mathrm{all}, \widetilde{R}}( \operatorname{\mathcal{E}}) \ar [d] \\ \Delta ^ n \ar@ {-->}[ur] \ar [r]^-{ \overline{f} } & \operatorname{Cospan}^{\mathrm{all}, R}( \operatorname{\mathcal{C}}) } \end{gathered} \end{equation}
admits a solution. Using Proposition 8.1.3.7, we can rewrite (8.23) as a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\Lambda ^{n}_{i}) \ar [r]^-{F_0} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{Tw}(\Delta ^ n) \ar@ {-->}[ur]^{F} \ar [r]^-{ \overline{F} } & \operatorname{\mathcal{C}}, } \]
where the morphism $F$ is required to satisfy the following additional condition:
- $(\ast )$
For every pair of integers $0 \leq i \leq j \leq n$, the morphism $F$ carries $(j,j) \rightarrow (i,j)$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
Replacing $\operatorname{\mathcal{E}}$ by the fiber product $\operatorname{Tw}(\Delta ^ n) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, we can reduce to the case where $\operatorname{\mathcal{C}}= \operatorname{Tw}( \Delta ^ n )$, and $\overline{F}$ is the identity morphism. Since $U$ is an inner fibration, it follows that $\operatorname{\mathcal{E}}$ is an $\infty $-category.
Suppose first that $n \geq 3$. In this case, $F_0$ determines a commutative diagram
8.24
\begin{equation} \begin{gathered}\label{equation:all-cocart-inner2} \xymatrix@R =50pt@C=50pt{ F_0( i, i ) \ar [r] \ar [d] & F_0(i-1, i) \ar [d] \\ F_0(i, i+1) \ar [r] & F_0(i-1, i+1) } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{E}}$. Using our assumption that $f_0$ factors through $\operatorname{Cospan}^{\mathrm{all}, \widetilde{R}}(\operatorname{\mathcal{E}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{E}})$ (together with Corollary 5.1.2.4), we deduce that the horizontal maps the diagram (8.24) are $U$-cocartesian. In particular, (8.24) is a $U$-colimit diagram (Proposition 7.6.2.26). Since the image of (8.24) in $\operatorname{Tw}(\Delta ^ n)$ is a pushout square, it is a pushout diagram in $\operatorname{\mathcal{E}}$ (Corollary 7.1.6.17). Applying Proposition 8.1.4.2, we deduce that $f_0|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{E}})$, and therefore also of $\operatorname{Cospan}^{ \mathrm{all}, \widetilde{R} }(\operatorname{\mathcal{E}})$ (Remark 8.1.6.4). It follows that $f_0$ can be extended to an $n$-simplex $\sigma $ of $\operatorname{Cospan}^{\mathrm{all}, \widetilde{R} }( \operatorname{\mathcal{E}})$, which we can identify with a functor $F: \operatorname{Tw}( \Delta ^ n ) \rightarrow \operatorname{\mathcal{E}}$ satisfying condition $(\ast )$ and the identity $F|_{ \operatorname{Tw}( \Lambda ^{n}_{i} ) } = F_0$. The equality $U \circ F = \overline{F}$ is automatic, since $\operatorname{Tw}(\Delta ^ n)$ is the nerve of a partially ordered set and $\operatorname{Tw}( \Lambda ^{n}_{i} )$ contains every vertex of $\operatorname{Tw}( \Delta ^ n )$.
We now treat the case $n = 2$ (so that $i = 1$). In this case, we can identify $F_0$ with a diagram
\[ X_{0,0} \xrightarrow {r} X_{0,1} \xleftarrow {u} X_{1,1} \xrightarrow {s} X_{1,2} \xleftarrow {v} X_{2,2} \]
in the $\infty $-category $\operatorname{\mathcal{E}}$, where the morphisms $u$ and $v$ are $U$-cocartesian. Our assumption that $\overline{f}$ factors through $\operatorname{Cospan}^{\mathrm{all}, R}( \operatorname{\mathcal{C}})$ guarantees that the morphism $(2,2) \rightarrow (0,2)$ belongs to $R$. Since morphisms of $R$ admit $U$-cocartesian lifts, we can choose a $U$-cocartesian morphism $w': X_{2,2} \rightarrow X_{0,2}$ in $\operatorname{\mathcal{E}}$, where $X_{0,2}$ belongs to the fiber over the object $(0,2) \in \operatorname{\mathcal{C}}$. Since $v$ is also $U$-cocartesian, we can choose a $2$-simplex $\sigma _0$ of $\operatorname{\mathcal{E}}$ with boundary indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{2,2} \ar [rr]^{w'} \ar [dr]^{v} & & X_{0,2} \\ & X_{1,2}. \ar [ur]^{w} & } \]
Since $\operatorname{\mathcal{E}}$ is an $\infty $-category, we can choose another $2$-simplex $\sigma _1$ of $\operatorname{\mathcal{E}}$ with boundary indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{1,1} \ar [rr]^{q} \ar [dr]^{s} & & X_{0,2} \\ & X_{1,2}. \ar [ur]^{w'} & } \]
Invoking our assumption that $u$ is $U$-cocartesian, we can choose another $2$-simplex $\sigma _2$ of $\operatorname{\mathcal{E}}$ with boundary indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{1,1} \ar [rr]^{q} \ar [dr]^{u} & & X_{0,2} \\ & X_{0,1}. \ar [ur]^{t} & } \]
Using the fact that $\operatorname{\mathcal{E}}$ is an $\infty $-category, we obtain another $2$-simplex $\sigma _3$ of $\operatorname{\mathcal{E}}$ with boundary indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{0,0} \ar [rr] \ar [dr]^{r} & & X_{0,2} \\ & X_{0,1}. \ar [ur]^{t} & } \]
The $2$-simplices $\sigma _0$, $\sigma _1$, $\sigma _2$, and $\sigma _3$ determine a functor $F: \operatorname{Tw}( \Delta ^2) \rightarrow \operatorname{\mathcal{C}}$ extending $F_0$, which we display informally as a diagram
\[ \xymatrix@C =50pt@R=50pt{ X_{0,0} \ar [dr]^{r} & & X_{1,1} \ar [dl]_{u} \ar [dr]^{s} & & X_{2,2} \ar [dl]_{v} \\ & X_{0,1} \ar [dr]^{t} & & X_{1,2} \ar [dl]_{w} & \\ & & X_{0,2}. & & } \]
Since the morphism $w'$ is $U$-cocartesian, the functor $F$ satisfies condition $(\ast )$ and can therefore be viewed as a solution to the lifting problem (8.23). $\square$