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8.1.9 Cospan Fibrations

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. Beware that the induced map $\operatorname{Cospan}(U): \operatorname{Cospan}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ is usually not an inner fibration. For example, in the special case $\operatorname{\mathcal{C}}= \Delta ^0$, the morphism $U$ is an inner fibration if and only if $\operatorname{\mathcal{E}}$ is an $\infty $-category. In this case, the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{E}})$ is usually not an $\infty $-category (unless $\operatorname{\mathcal{E}}$ is a Kan complex). However, it contains an $\infty $-category $\operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso} }(\operatorname{\mathcal{E}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{E}})$ (Construction 8.1.7.2), which is canonically equivalent to the $\infty $-category $\operatorname{\mathcal{E}}$ (Proposition 8.1.7.6). In this section, we describe a generalization which applies to any simplicial set $\operatorname{\mathcal{C}}$. Our main result can be stated as follows:

Proposition 8.1.9.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $W$ denote the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$. Then the induced map $\operatorname{Cospan}^{\mathrm{all},W}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ is an inner fibration of simplicial sets.

We will give the proof of Proposition 8.1.9.1 at the end of this section.

Remark 8.1.9.2. In the situation of Proposition 8.1.9.1, let $C$ be a vertex of $\operatorname{\mathcal{C}}$ and let $\operatorname{\mathcal{E}}_{C}$ denote the fiber $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Note that a morphism $u$ in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ is an isomorphism if and only if it is $U$-cocartesian when viewed as a morphism in $\operatorname{\mathcal{E}}$ (Proposition 5.1.4.11). It follows that the fiber $\{ C\} \times _{ \operatorname{Cospan}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{\mathrm{all},W}(\operatorname{\mathcal{E}})$ can be identified with the $\infty $-category $\operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}_{C} )$. In particular, Proposition 8.1.7.6 supplies an equivalence of $\infty $-categories

\[ \rho _{+}: \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \{ C\} \times _{ \operatorname{Cospan}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{ \mathrm{all}, W}( \operatorname{\mathcal{E}}). \]

Remark 8.1.9.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let $W$ be the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$. Then we have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Cospan}^{ \mathrm{all},W}( \operatorname{\mathcal{E}}) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Cospan}( \operatorname{\mathcal{C}}), } \]

where the horizontal maps on the left are equivalences of $\infty $-categories (Proposition 8.1.7.6), and the right half of the diagram is a pullback square (Proposition 5.1.1.8). It follows that from Proposition 8.1.9.1 that map $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}})$ is an inner fibration of $\infty $-categories. In fact, it is even an isofibration: this follows easily from the description of isomorphisms in the $\infty $-categories $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ and $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}})$ supplied by Corollary 8.1.6.10 (together with the fact that $U$ is an isofibration; see Proposition 5.1.4.8). Applying Theorem 5.1.6.1 to the right side of the diagram, we conclude that the map $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ is also a cocartesian fibration of simplicial sets. Moreover, Corollary 4.5.2.29 guarantees that the induced map

\[ \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}}) \simeq \operatorname{\mathcal{C}}\times _{ \operatorname{Cospan}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{\mathrm{all},W}(\operatorname{\mathcal{E}}) \]

is an equivalence of $\infty $-categories.

Remark 8.1.9.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let $W$ be the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$. It follows from Proposition 8.1.9.1 that $U$ also induces an inner fibration $\operatorname{Cospan}^{W, \mathrm{all}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$, whose fiber over an object $C \in \operatorname{\mathcal{C}}$ is equivalent to the opposite of the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ (see Variant 8.1.7.14). This construction will play an important role in ยง8.6.

For later use, it will be convenient to have a generalization of Proposition 8.1.9.1, where we impose some additional constraints on the cospans that we consider.

Definition 8.1.9.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $\overline{f}: \overline{X} \rightarrow \overline{Y}$ be an edge of $\operatorname{\mathcal{C}}$. We say that $\overline{f}$ admits $U$-cartesian lifts if, for every vertex $Y \in \operatorname{\mathcal{E}}$ satisfying $U(Y) = \overline{Y}$, there is a $U$-cartesian edge $f: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$. We say that $\overline{f}$ admits $U$-cocartesian lifts if, for every vertex $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = \overline{X}$, there is a $U$-cocartesian edge $f: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$.

Remark 8.1.9.6. In the situation of Definition 8.1.9.5, the edge $\overline{f}$ admits $U$-cocartesian lifts if and only if it it admits $U^{\operatorname{op}}$-cartesian lifts, when regarded as an edge of the opposite simplicial set $\operatorname{\mathcal{E}}^{\operatorname{op}}$.

Example 8.1.9.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. Then $U$ is a cocartesian fibration if and only if every edge of $\operatorname{\mathcal{C}}$ admits $U$-cocartesian lifts. Similarly, $U$ is a cartesian fibration if and only if every edge of $\operatorname{\mathcal{C}}$ admits $U$-cartesian lifts.

Example 8.1.9.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. The following conditions are equivalent:

  • The morphism $U$ is an isofibration of $\infty $-categories.

  • Every isomorphism of $\operatorname{\mathcal{C}}$ admits $U$-cocartesian lifts.

  • Every isomorphism of $\operatorname{\mathcal{C}}$ admits $U$-cartesian lifts.

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.

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 7.6.3.23). 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 7.1.5.15). 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

\[ X_{0,0} \xrightarrow {r} X_{0,1} \xleftarrow {u} X_{1,1} \xrightarrow {s} X_{1,2} \leftarrow {v} X_{2,2} \]

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). $\square$

Proposition 8.1.9.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories, let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which are pushout-compatible, and assume that morphisms of $R$ admit $U$-cocartesian lifts. Let $\widetilde{L}$ denote the collection of all morphisms $f$ of $\operatorname{\mathcal{E}}$ such that $U(f) \in L$, and let $\widetilde{R}$ denote the collection of all $U$-cocartesian morphisms $f$ of $\operatorname{\mathcal{E}}$ such that $U(f) \in R$. Then the collections $\widetilde{L}$ and $\widetilde{R}$ are also pushout-compatible.

Proof. Let $f: X \rightarrow X_1$ be a morphism of $\operatorname{\mathcal{E}}$ which belongs to $\widetilde{L}$, and let $g': X \rightarrow X_0$ be a morphism of $\operatorname{\mathcal{E}}$ which belongs to $\widetilde{R}$. We wish to show that there exists a pushout diagram

8.25
\begin{equation} \begin{gathered}\label{equation:dual-Beck-Chevalley-nonsense} \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{g'} \ar [d]^{f} & X_0 \ar [d]^{f'} \\ X_1 \ar [r]^-{g} & X_{01} } \end{gathered} \end{equation}

in the $\infty $-category $\operatorname{\mathcal{E}}$, where $f'$ belongs to $\widetilde{L}$ and $g$ belongs to $\widetilde{R}$. Since $L$ and $R$ are pushout compatible, there exists a pushout diagram

8.26
\begin{equation} \begin{gathered}\label{equation:dual-Beck-Chevalley-nonsense2} \xymatrix@R =50pt@C=50pt{ U(X) \ar [r]^-{ U(g') } \ar [d]^{ U(f) } & U(X_0) \ar [d]^{ \overline{f}' } \ar [d] \\ U(X_1) \ar [r]^-{ \overline{g}} & \overline{X}_{01} } \end{gathered} \end{equation}

in the $\infty $-category $\operatorname{\mathcal{C}}$, where $\overline{f}'$ belongs to $L$ and $\overline{g}$ belongs to $R$. Our assumption on $U$ guarantees that $\overline{g}$ can be lifted to a $U$-cocartesian morphism $g: X_0 \rightarrow X_{01}$ of $\operatorname{\mathcal{E}}$. Since $U$ is an inner fibration, the lower left half of (8.26) can be lifted to a $2$-simplex $\sigma $ of $\operatorname{\mathcal{E}}$ which we display as a diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [dr]^{ h } & \\ X_1 \ar [r]^-{g} & X_{01}. } \]

Since $g'$ is $U$-cocartesian, we can then lift the upper right half of (8.26) to a $2$-simplex $\tau $ of $\operatorname{\mathcal{E}}$ which we display as a diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{g'} \ar [dr]^{ h } & X_0 \ar [d]^{f'} \\ & X_{01}. } \]

Amalgamating $\sigma $ and $\tau $, we obtain a diagram of the form (8.25), where $f' \in \widetilde{L}$ and $g \in \widetilde{R}$. We will complete the proof by showing that this diagram is a pushout square in the $\infty $-category $\operatorname{\mathcal{E}}$. Since (8.26) is a pushout square in $\operatorname{\mathcal{C}}$, it will suffice to show that 8.25 is a $U$-pushout square (Corollary 7.1.5.16). This is a special case of Proposition 7.6.3.23, since the horizontal morphisms appearing in the diagram are $U$-cocartesian. $\square$

Remark 8.1.9.11. In the situation of Proposition 8.1.9.10, suppose that the collections $L$ and $R$ are closed under composition. Then $\widetilde{L}$ and $\widetilde{R}$ are also closed under composition (see Corollary 5.1.2.4). Applying Proposition 8.1.6.7, we deduce that the simplicial sets $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ and $\operatorname{Cospan}^{\widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}})$ are $(\infty ,2)$-categories. Moreover, it follows from the proof of Proposition 8.1.9.10 that for every pushout diagram $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{g'} \ar [d]^{f} & X_0 \ar [d]^{f'} \\ X_1 \ar [r]^-{g} & X_{01} } \]

in $\operatorname{\mathcal{E}}$ where $f$ belongs to $\widetilde{L}$ and $g$ belongs to $\widetilde{R}$, the image $U(\sigma )$ is a pushout diagram in $\operatorname{\mathcal{C}}$. Combining this observation with Corollary 8.1.6.8 and Proposition 8.1.4.2, we see that a $2$-simplex of $\operatorname{Cospan}^{\widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}})$ is thin if and only if its image in $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is thin. In particular:

  • The induced map $\overline{V}: \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is a functor of $(\infty ,2)$-categories.

  • The functor $\overline{V}$ is an inner fibration (since it is a pullback of the inner fibration $\operatorname{Cospan}^{\mathrm{all}, \widetilde{R} }( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\mathrm{all}, R}( \operatorname{\mathcal{C}})$ of Proposition 8.1.9.9).

  • The underlying functor $V: \operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$ is also an inner fibration (since it is a pullback of $\overline{V}$).