# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

### 8.1.10 Beck-Chevalley Fibrations

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty$-categories. For each object $C \in \operatorname{\mathcal{C}}$, we let $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ denote the corresponding fiber of $U$. If $U$ is a cartesian fibration, then every morphism $f: C \rightarrow B$ in $\operatorname{\mathcal{C}}$ determines a functor $f^{\ast }: \operatorname{\mathcal{E}}_{B} \rightarrow \operatorname{\mathcal{E}}_{C}$, given by contravariant transport along $f$ (Definition 5.2.2.14). If $U$ is a cocartesian fibration, then every morphism morphism $f': C' \rightarrow B$ determines a functor $f'_{!}: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{B}$, given by covariant transport along $f'$ (Definition 5.2.2.4). If both of these conditions are satisfied, then every cospan $C \xrightarrow {f} B \xleftarrow {f'} C'$ in $\operatorname{\mathcal{C}}$ determines a functor from $\operatorname{\mathcal{E}}_{C'}$ to $\operatorname{\mathcal{E}}_{C}$, given by the composition $f^{\ast } \circ f'_{!}$. Our goal in this section is to show that, under some mild assumptions, the construction $(f,f') \mapsto f^{\ast } \circ f'_{!}$ is compatible with the composition law on cospans, up to coherent homotopy. More precisely, we will show that this construction is given by contravariant transport for a certain cartesian fibration between cospan constructions (Theorem 8.1.10.3 and Remark 8.1.10.4). First, we need some terminology.

Definition 8.1.10.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which admits pushouts. We will say that an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a dual Beck-Chevalley fibration if the following conditions are satisfied:

$(1)$

The morphism $U$ is a cartesian fibration.

$(2)$

The morphism $U$ is a cocartesian fibration.

$(3)$

Suppose we are given a morphism $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{E}}$, which we display informally as a diagram

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

Assume that $f$ is $U$-cartesian, that $g'$ is $U$-cocartesian, and that $U(\sigma )$ is a pushout square in $\operatorname{\mathcal{C}}$. Then $f'$ is $U$-cartesian if and only if $g$ is $U$-cocartesian.

Example 8.1.10.2. Let $\operatorname{\mathcal{E}}= \operatorname{Mod}( \operatorname{ Ab })$ denote the category of pairs $(A,M)$, where $A$ is a commutative ring and $M$ is an $A$-module (see Example 5.0.0.2). Let $\operatorname{\mathcal{C}}$ denote the category of commutative rings and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be the forgetful functor $(A,M) \mapsto A$. Then (the nerve of) $U$ is a dual Beck-Chevalley fibration, in the sense of Definition 8.1.10.1. To see this, suppose we are given a commutative diagram $\sigma :$

$\xymatrix@R =50pt@C=50pt{ (A,M) \ar [r]^-{f} \ar [d]^{g} & (A_0, M_0) \ar [d]^{g'} \\ (A_1,M_1) \ar [r]^-{f'} & (A_{01}, M_{01}) }$

in the category $\operatorname{\mathcal{E}}$. Then:

• The morphism $f$ is $U$-cartesian if and only if the underlying map $M \rightarrow M_1$ is an isomorphism of $A$-modules.

• The morphism $g'$ is $U$-cocartesian if and only if it exhibits $M_{01}$ as obtained from $M_1$ by extending scalars along the ring homomorphism $A_1 \rightarrow A_{01}$: that is, if and only if it induces an isomorphism $A_{01} \otimes _{A_1} M_1 \rightarrow M_{01}$.

• The image of $\sigma$ is a pushout diagram in $\operatorname{\mathcal{C}}$ if and only if the induced map $A_0 \otimes _{A} A_1 \rightarrow A_{01}$ is an isomorphism.

If all three of these conditions are satisfied, then the composite map $A_0 \otimes _{A} M \rightarrow M_0 \rightarrow M_{01}$ is an isomorphism. Using the two-out-of-three property, we conclude that $f'$ is $U$-cartesian if and only if $g$ is $U$-cocartesian.

We can now formulate our main result, which we prove at the end of this section.

Theorem 8.1.10.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which admits pushouts, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a dual Beck-Chevalley fibration and let $R$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$. Then the map $\operatorname{Cospan}(U): \operatorname{Cospan}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}( \operatorname{\mathcal{C}})$ restricts to a cartesian fibration of $\infty$-categories

$V: \operatorname{Pith}( \operatorname{Cospan}^{ \mathrm{all}, R}( \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Pith}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) ).$

Moreover, a morphism of $\operatorname{Pith}( \operatorname{Cospan}^{ \mathrm{all}, R}( \operatorname{\mathcal{E}}) )$ is $V$-cartesian if and only if it corresponds to a cospan $X \xrightarrow {f} B \xleftarrow {g} Y$ in the $\infty$-category $\operatorname{\mathcal{E}}$, where $f$ is $U$-cartesian and $g$ is $U$-cocartesian.

Remark 8.1.10.4 (Contravariant Transport for Cospan Fibrations). In the situation of Theorem 8.1.10.3, let $C$ and $C'$ be objects of $\operatorname{\mathcal{C}}$, so that Proposition 8.1.7.6 supplies equivalences of $\infty$-categories

$\rho _{+}^{C}: \operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}}_ C) = V^{-1} \{ C\} \quad \quad \rho _{+}^{C'}: \operatorname{\mathcal{E}}_{C'} \hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}}_{C'}) = V^{-1} \{ C'\} .$

Let $e$ be a morphism from $C$ to $C'$ in the $(\infty ,2)$-category $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, which we identify with a pair of morphisms $C \xrightarrow {f} B \xleftarrow {f'} C'$ in the $\infty$-category $\operatorname{\mathcal{C}}$. Choose functors $f^{\ast }: \operatorname{\mathcal{E}}_{B} \rightarrow \operatorname{\mathcal{E}}_{C}$, $f'_{!}: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{B}$ and diagrams

$\xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{\mathcal{E}}_{B} \ar [r]^-{ H } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} & \Delta ^1 \times \operatorname{\mathcal{E}}_{C'} \ar [l]_{H'} \ar [d] \\ \Delta ^1 \ar [r]^-{ f } & \operatorname{\mathcal{C}}& \Delta ^{1} \ar [l]_{f'} }$

which exhibit $f^{\ast }$ and $f'_{!}$ as given by contravariant and covariant transport along $f$ and $f'$, respectively (see Definitions 5.2.2.4 and 5.2.2.14). Then the composition

\begin{eqnarray*} \operatorname{Tw}(\Delta ^1 \times \operatorname{\mathcal{E}}_{C'} ) & \simeq & \operatorname{Tw}(\Delta ^1) \times \operatorname{Tw}(\operatorname{\mathcal{E}}_{C'} ) \\ & \rightarrow & \operatorname{Tw}(\Delta ^1) \times \operatorname{\mathcal{E}}_{C'} \\ & \simeq & ( \operatorname{N}_{\bullet }( \{ (0,0) < (0,1) \} ) {\coprod }_{ \operatorname{N}_{\bullet }( \{ (0,1) \} ) } \operatorname{N}_{\bullet }( \{ (1,1) < (0,1) \} ) ) \times \operatorname{\mathcal{E}}_{C'} \\ & \xrightarrow {(H \circ (\operatorname{id}\times f'_{!}), H')} & \operatorname{\mathcal{E}}\end{eqnarray*}

can be identified with a functor $T: \Delta ^1 \times \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{E}})$ which fits into a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{\mathcal{E}}_{C'} \ar [d] \ar [r]^-{T} & \operatorname{Pith}(\operatorname{Cospan}^{\mathrm{all},W}(\operatorname{\mathcal{E}})) \ar [d]^{V} \\ \Delta ^1 \ar [r]^-{e} & \operatorname{Pith}(\operatorname{Cospan}(\operatorname{\mathcal{C}})). }$

For each object $X \in \operatorname{\mathcal{E}}_{C'}$, the characterization of $V$-cartesian morphisms given in Theorem 8.1.10.3 shows that $T|_{ \Delta ^1 \times \{ X\} }$ is a $V$-cartesian morphism of $\operatorname{Pith}(\operatorname{Cospan}^{\mathrm{all},R}(\operatorname{\mathcal{E}}))$. It follows that the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{C'} \ar [r]^-{ f'_{!}} \ar [d]^{ \rho ^{C'}_{+} } & \operatorname{\mathcal{E}}_{B} \ar [r]^-{ f^{\ast } } & \operatorname{\mathcal{E}}_{C} \ar [d]^{ \rho ^{C}_{+} } \\ V^{-1} \{ C'\} \ar [rr]^{ e^{\ast } } & & V^{-1} \{ C\} }$

commutes up to homotopy, where the functor $e^{\ast }$ is given by contravariant transport along $e$ (for the cartesian fibration $V$).

Corollary 8.1.10.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which admits pushouts, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a dual Beck-Chevalley fibration, let $L$ be the collection of all $U$-cartesian morphisms of $\operatorname{\mathcal{C}}$, and let $R$ be the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{C}}$. Then $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is an $(\infty ,2)$-category, and $U$ induces a right fibration of $\infty$-categories $\operatorname{Pith}(\operatorname{Cospan}^{L,R}( \operatorname{\mathcal{E}})) \rightarrow \operatorname{Pith}(\operatorname{Cospan}(\operatorname{\mathcal{C}}))$.

Proof. The collections $L$ and $R$ are closed under composition (Corollary 5.1.2.4) and pushout-compatible by virtue of our assumption that $U$ is a Beck-Chevalley fibration. Using Proposition 8.1.6.7, we see that $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is an $(\infty ,2)$-category. Moreover, the pith of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ can be identified with the subcategory of $\operatorname{Pith}( \operatorname{Cospan}^{ \mathrm{all}, R}(\operatorname{\mathcal{E}}) )$ spanned by those morphisms which are cartesian with respect to the fibration $V: \operatorname{Pith}( \operatorname{Cospan}^{ \mathrm{all}, R}( \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Pith}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) )$ of Theorem 8.1.10.3. The desired result now follows from Corollary 5.1.4.15. $\square$

For later use, we will prove a more general form of Theorem 8.1.10.3, where we place some restrictions on the cospans under consideration. This will allow us to loosen the requirements of Definition 8.1.10.1.

Definition 8.1.10.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which are pushout-compatible (Definition 8.1.6.5). We will say that an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a dual Beck-Chevalley fibration relative to $(L,R)$ if the following conditions are satisfied:

$(1)$

Every morphism of $\operatorname{\mathcal{C}}$ which belongs to $L$ admits $U$-cartesian lifts (Definition 8.1.9.5).

$(2)$

Every morphism of $\operatorname{\mathcal{C}}$ which belongs to $R$ admits $U$-cocartesian lifts.

$(3)$

Suppose we are given a morphism $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{E}}$, which we display informally as a diagram

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

Assume that $f$ is $U$-cartesian, that $g$ is $U$-cocartesian, that $U(f)$ belongs to $L$, that $U(g)$ belongs to $R$, and that $U(\sigma )$ is a pushout square in $\operatorname{\mathcal{C}}$. Then $f'$ is $U$-cartesian if and only if $g$ is $U$-cocartesian.

Example 8.1.10.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which admits pushouts and let $A$ denote the collection of all morphisms of $\operatorname{\mathcal{C}}$. Then an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a dual Beck-Chevalley fibration (in the sense of Definition 8.1.10.6) if and only if if it is a dual Beck-Chevalley fibration relative to $(A,A)$ (in the sense of Definition 8.1.10.6).

Example 8.1.10.8. In the situation of Definition 8.1.10.6, suppose that $L$ is the collection of all isomorphisms in $\operatorname{\mathcal{C}}$. Then condition $(1)$ is equivalent to the requirement that $U$ is an isofibration (Example 8.1.9.8), and condition $(3)$ is automatic. Similarly, if $R$ is the collection of all isomorphisms in $\operatorname{\mathcal{C}}$, then condition $(2)$ is the requirement that $U$ is an isofibration, and condition $(3)$ is automatic.

Theorem 8.1.10.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which are closed under composition and pushout-compatible. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a dual Beck-Chevalley fibration with respect to $(L,R)$ and define $\widetilde{L}$ and $\widetilde{R}$ as in Proposition 8.1.9.10. Then the functor $V: \operatorname{Pith}( \operatorname{Cospan}^{\widetilde{L}, \widetilde{R}}(\operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}}) )$ of Remark 8.1.9.11 is a cartesian fibration of $\infty$-categories. Moreover, a morphism $e$ of $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is $V$-cartesian if and only if it satisfies the following condition:

$(\ast )$

The morphism $e$ corresponds to a cospan $X \xrightarrow {f} B \xleftarrow {g} Y$ in the $\infty$-category $\operatorname{\mathcal{E}}$, where $f$ is $U$-cartesian and $g$ is $U$-cocartesian.

Proof. Let us say that a morphism $e$ of $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is special if it satisfies condition $(\ast )$. We first show that every special morphism $e$ of $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is $V$-cartesian. Suppose we are given an integer $n \geq 2$ and a lifting problem

8.27
\begin{equation} \begin{gathered}\label{equation:cospan-cartesian-fibration} \xymatrix@C =50pt@R=50pt{ \Lambda ^{n}_{n} \ar [r]^-{h_0} \ar [d] & \operatorname{Pith}(\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})) \ar [d]^{U'} \\ \Delta ^{n} \ar@ {-->}[ur]^{h} \ar [r]^-{ \overline{h} } & \operatorname{Pith}(\operatorname{Cospan}^{L, R}( \operatorname{\mathcal{C}}), } \end{gathered} \end{equation}

where the composition

$\Delta ^{1} \simeq \operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \Lambda ^{n}_{n} \xrightarrow {h_0} \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$

coincides with the edge $e$; we wish to show that (8.27) admits a solution. Let us identify $\overline{h}$ with a diagram $\overline{F}: \operatorname{Tw}( \Delta ^ n ) \rightarrow \operatorname{\mathcal{E}}$ and $h_0$ with a diagram $H_0: \operatorname{Tw}( \Lambda ^{n}_{n} ) \rightarrow \operatorname{\mathcal{C}}$ satisfying $U \circ H_0 = \overline{H}|_{ \operatorname{Tw}( \Lambda ^{n}_{n} )}$. We first treat the case $n=2$. In this case, we can identify $F_0$ with a pair of cospans

$X_{0,0} \xrightarrow {f'} X_{0,2} \xleftarrow {g'} X_{2,2} \quad \quad X_{1,1} \xrightarrow {f} X_{1,2} \xleftarrow {g} X_{2,2}$

in the $\infty$-category $\operatorname{\mathcal{E}}$, where $f,f' \in \widetilde{L}$ and $g,g' \in \widetilde{R}$. Since $g$ is $U$-cocartesian, we can lift $\overline{H}( (2,2) \rightarrow (1,2) \rightarrow (0,2) )$ to a $2$-simplex $\sigma _0$ of $\operatorname{\mathcal{E}}$ whose boundary we display in the diagram

$\xymatrix@R =50pt@C=50pt{ X_{2,2} \ar [rr]^{g'} \ar [dr]^{g} & & X_{0,2} \\ & X_{1,2}. \ar [ur]^{g''} & }$

Since $g$ and $g'$ are $U$-cocartesian by assumption, Corollary 5.1.2.4 guarantees that $g''$ is also $U$-cocartesian. Since $U$ is an inner fibration, we can lift $\overline{H}( (1,1) \rightarrow (1,2) \rightarrow (0,2) )$ to a $2$-simplex $\sigma _1$ of $\operatorname{\mathcal{E}}$, whose boundary we display in the diagram

$\xymatrix@R =50pt@C=50pt{ X_{1,1} \ar [rr]^{s} \ar [dr]^{f} & & X_{0,2} \\ & X_{1,2}. \ar [ur]^{g''} & }$

Since morphisms of $R$ admit $U$-cocartesian lifts, we can lift $\overline{H}( (1,1) \rightarrow (0,1) \rightarrow (0,2) )$ to a $2$-simplex $\sigma _2$ of $\operatorname{\mathcal{E}}$ displayed in the diagram

$\xymatrix@R =50pt@C=50pt{ X_{1,1} \ar [rr]^{s} \ar [dr]^{g'''} & & X_{0,2} \\ & X_{0,1}, \ar [ur]^{f''} & }$

where $g'''$ is $U$-cocartesian. Applying condition $(3)$ of Definition 8.1.10.6 to the diagram

$\xymatrix@R =50pt@C=50pt{ X_{1,1} \ar [r]^-{f} \ar [d]^{g'''} & X_{1,2} \ar [d]^{ g'' } \\ X_{0,1} \ar [r]^-{f''} & X_{0,2}, }$

we deduce that the morphism $f''$ is $U$-cartesian. We can therefore lift $\overline{H}( (0,0) \rightarrow (0,1) \rightarrow (0,2) )$ to a $2$-simplex $\sigma _3$ of $\operatorname{\mathcal{E}}$ which we display as a diagram

$\xymatrix@R =50pt@C=50pt{ X_{0,0} \ar [rr]^{f'} \ar [dr]^{f'''} & & X_{0,2} \\ & X_{0,1}, \ar [ur]^{f''} & }$

The $2$-simplices $\sigma _0$, $\sigma _1$, $\sigma _2$, and $\sigma _3$ can then be amalgamed into a functor $H: \operatorname{Tw}( \Delta ^2 ) \rightarrow \operatorname{\mathcal{E}}$ which we display informally as a diagram

$\xymatrix@C =50pt@R=50pt{ X_{0,0} \ar [dr]^{f'''} & & X_{1,1} \ar [dr]^{f} \ar [dl]_{g'''} & & X_{2,2} \ar [dl]_{g} \\ & X_{0,1} \ar [dr]^{f''} & & X_{1,2} \ar [dl]^{g''} & \\ & & X_{0,2}, & & }$

which is a solution to the lifting problem (8.27).

We now treat the case $n \geq 3$. By virtue of Lemma 8.1.4.6, it will suffice to show that the following conditions are satisfied:

$(a)$

The functor $F_0$ carries the edge $(n,n) \rightarrow (n-1,n)$ of $\operatorname{Tw}( \Lambda ^{n}_{n} )$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.

$(b)$

The functor $F_0$ carries the edge $(0, n-1) \rightarrow (0,n)$ of $\operatorname{Tw}( \Lambda ^{n}_{n} )$ to a $U$-cartesian edge of $\operatorname{\mathcal{E}}$.

Assertion $(a)$ follows immediately from our requirement that $f_0$ factors through $\operatorname{Cospan}^{\widetilde{L}, \widetilde{R}}(\operatorname{\mathcal{E}})$. To prove $(b)$, we observe that $F_0$ determines a commutative diagram $\tau :$

$\xymatrix@C =50pt@R=50pt{ F_0(n-1,n-1) \ar [r]^-{f} \ar [d]^{g} & F_0( n-1, n) \ar [d]^{g'} \\ F_0( 0, n-1 ) \ar [r]^-{f'} & F_0( 0, n) }$

in the $\infty$-category $\operatorname{\mathcal{E}}$, where $f \in \widetilde{L}$ and $g \in \widetilde{R}$. Our assumption that $e$ is special guarantees that $f$ is $U$-cartesian, and our assumption that $\overline{h}$ factors through the pith of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ guarantees that $U(\tau )$ is a pushout diagram in $\operatorname{\mathcal{C}}$. Applying Corollary 5.1.2.4 to the diagram

$\xymatrix@R =50pt@C=50pt{ F_0(n,n) \ar [dr] \ar [rr] & & F_0(0,n) \\ & F_0(n-1, n), \ar [ur]^{g'} & }$

we see that $g'$ is $U$-cocartesian. Condition $(3)$ of Definition 8.1.10.6 then guarantees that $f'$ is $U$-cartesian, as desired. This completes the proof that every special morphism of $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is $V$-cartesian.

It follows from Remark 8.1.9.11 that $V$ is an inner fibration of $\infty$-categories. To show that $V$ is a cartesian fibration, it will suffice to show that for every object $Y \in \operatorname{\mathcal{E}}$ and every morphism $\overline{e}: \overline{X} \rightarrow U(Y)$ in the $\infty$-category $\operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$, there exists a special morphism $e: X \rightarrow Y$ in $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}}) )$ satisfying $V(e) = \overline{e}$. Let us identify $\overline{e}$ with a cospan $\overline{X} \xrightarrow { \overline{f} } \overline{B} \xleftarrow { \overline{g} } U(Y)$ in the $\infty$-category $\operatorname{\mathcal{C}}$. Then $\overline{g}$ belongs to $R$, and can therefore be lifted to a $U$-cocartesian morphism $g: Y \rightarrow B$ in the $\infty$-category $\operatorname{\mathcal{E}}$. Since $\overline{f}$ belongs to $L$, it can be lifted to a $U$-cartesian morphism $f: X \rightarrow B$ in the $\infty$-category $\operatorname{\mathcal{E}}$. The cospan $X \xrightarrow {f} B \xleftarrow {g} Y$ then determines a special morphism $e: X \rightarrow Y$ of $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}}) )$ satisfying $V(e) = \overline{e}$.

We now complete the proof of Theorem 8.1.10.3 by showing that every $V$-cartesian morphism $e: X \rightarrow Y$ of $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) )$ is special. Let $\overline{e}$ denote the image of $e$ in $\operatorname{Pith}(\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}))$. Arguing as above, we can lift $\overline{e}$ to a special morphism $e': X' \rightarrow Y$ of $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) )$. Then $e'$ is also $V$-cartesian. Applying Remark 5.1.3.8, we can choose a $2$-simplex $\sigma$ of $\operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ which exhibits $e$ as the composition of $e'$ with an isomorphism in the $\infty$-category $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) )$. Let us identify $\sigma$ with a diagram

$\xymatrix@C =50pt@R=50pt{ X \ar [dr]^{u} & & X' \ar [dr]^{f} \ar [dl]_{v} & & Y \ar [dl] \\ & X'' \ar [dr]^{f'} & & B' \ar [dl]^{w} & \\ & & B & & }$

in the $\infty$-category $\operatorname{\mathcal{E}}$. Corollary 8.1.6.10 implies that $u$ and $v$ are isomorphisms in $\operatorname{\mathcal{E}}$. Since the inner region is a pushout diagram in $\operatorname{\mathcal{E}}$, it follows that $w$ is also an isomorphism (Corollary 7.6.3.20). Our assumption that $e'$ is special guarantees that $f$ is $U$-cartesian. Applying Corollary 5.1.2.5, we deduce that $f'$ is $U$-cartesian. It follows that any composition of $f'$ with $u$ is $U$-cartesian (Corollary 5.1.2.4), so that the morphism $e$ is also special. $\square$

Proof of Theorem 8.1.10.3. Apply Theorem 8.1.10.9 in the special case $L = A = R$, where $A$ is the collection of all morphisms in the $\infty$-category $\operatorname{\mathcal{C}}$ (Example 8.1.10.7). $\square$

Remark 8.1.10.10. In the situation of Theorem 8.1.10.9, the morphism

$\overline{V}: \operatorname{Cospan}^{\widetilde{L}, \widetilde{R}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$

is a locally cartesian fibration. To prove this, we observe that Remark 8.1.9.11 guarantees that $\overline{V}$ is an inner fibration of $(\infty ,2)$-categories and that the diagram

$\xymatrix@C =50pt@R=50pt{ \operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) ) \ar [d]^{V} \ar [r] & \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) \ar [d]^{ \overline{V} } \\ \operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) ) \ar [r] & \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) }$

is a pullback square. Since every morphism of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is contained in the pith $\operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$, the desired result follows from Theorem 8.1.10.9 (see Remark 5.1.5.6).

Remark 8.1.10.11. In the situation of Theorem 8.1.10.9, suppose that $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is an $\infty$-category (this condition is satisfied, for example, if either $L$ or $R$ consists of isomorphisms; see Proposition 8.1.7.5). Then every $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is thin. Applying Remark 8.1.9.11, we deduce that every $2$-simplex of $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is thin, so that $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is also an $\infty$-category (Example 2.3.2.4). In this case, Theorem 8.1.10.9 asserts that the projection map $V: \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is a cartesian fibration.

We also have the following variant of Corollary 8.1.10.5:

Corollary 8.1.10.12. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which are closed under composition and pushout-compatible. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a dual Beck-Chevalley fibration with respect to $(L,R)$, let $\widetilde{L}$ denote the collection of all $U$-cartesian 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:

$(1)$

The simplicial set $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is an $(\infty ,2)$-category.

$(2)$

The morphism $U$ induces a right fibration $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$.

$(3)$

If $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is an $\infty$-category, then $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is an $\infty$-category, and $U$ induces a right fibration $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$.

Example 8.1.10.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty$-categories and let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which are closed under composition and pushout-compatible. Assume that the following conditions are satisfied:

$(0)$

Let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{E}}$. If $U(f)$ belongs to $L$, then $f$ is $U$-cartesian. If $U(f)$ belongs to $R$, then $f$ is $U$-cocartesian.

$(1)$

For every object $Y \in \operatorname{\mathcal{E}}$ and every morphism $\overline{f}: \overline{X} \rightarrow U(Y)$ of $\operatorname{\mathcal{C}}$ which belongs to $L$, there exists a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$.

$(2)$

For every object $X \in \operatorname{\mathcal{E}}$ and every morphism $\overline{f}: U(X) \rightarrow \overline{Y}$ of $\operatorname{\mathcal{C}}$ which belongs to $R$, there exists a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$.

Then $U$ is a dual Beck-Chevalley fibration relative to $(L,R)$. Applying Remark 8.1.10.10, we deduce that the projection map

$V: \operatorname{Cospan}(\operatorname{\mathcal{E}}) \times _{\operatorname{Cospan}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$

is a locally cartesian fibration. Corollary 8.1.10.12 guarantees that each fiber of $V$ is a Kan complex, so that $V$ is a right fibration (Corollary 5.1.5.12).