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8.1.4 Cospans in $\infty$-Categories

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, and let $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ denote the simplicial set of cospans in $\operatorname{\mathcal{C}}$ (Construction 8.1.3.1). In the special case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0 )$ is the nerve of an ordinary category $\operatorname{\mathcal{C}}_0$ which admits pushouts, Corollary 8.1.3.15 supplies an isomorphism of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ with the Duskin nerve of the $2$-category $\operatorname{Cospan}(\operatorname{\mathcal{C}}_0 )$ of Example 2.2.2.1. In particular, $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category (see Proposition 5.4.1.5). Our goal in this section is to prove an $\infty$-categorical generalization of this result.

Proposition 8.1.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category if and only if $\operatorname{\mathcal{C}}$ admits pushouts.

Our proof of Proposition 8.1.4.1 will require several steps. The main ingredient is the following characterization of thin $2$-simplices of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, which we will establish in ยง8.1.5:

Proposition 8.1.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\sigma$ be a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, which we identify with a diagram

8.6
$$\begin{gathered}\label{equation:thin-in-correspondence} \xymatrix@R =50pt@C=50pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl] & & X_{2,2} \ar [dl] \\ & X_{0,1} \ar [dr] & & X_{1,2} \ar [dl] & \\ & & X_{0,2} & & } \end{gathered}$$

in the $\infty$-category $\operatorname{\mathcal{C}}$. Then $\sigma$ is thin (in the sense of Definition 2.3.2.3) if and only if the inner region is a pushout square in the $\infty$-category $\operatorname{\mathcal{C}}$.

Corollary 8.1.4.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then every degenerate $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is thin.

Proof. Let $\sigma$ be a $1$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$, corresponding to a diagram $X \xrightarrow {f} B \xleftarrow {g} Y$ in the $\infty$-category $\operatorname{\mathcal{C}}$ where $f$ belongs to $L$ and $g$ belongs to $R$. We will show that the left-degenerate $2$-simplex $s^{1}_0(\sigma )$ is thin; a similar argument will show that the right-degenerate $2$-simplex $s^{1}_1(\sigma )$ is thin (see Remark 8.1.3.4). Unwinding the definitions, we see that $s^{1}_0(\sigma )$ corresponds to a diagram in $\operatorname{\mathcal{C}}$ of the form

$\xymatrix@R =50pt@C=50pt{ X \ar [dr]_{\operatorname{id}_ X} & & X \ar [dl]^{\operatorname{id}_ X} \ar [dr]_{f} & & Y \ar [dl]^{g} \\ & X \ar [dr]_{f} & & B \ar [dl]^{ \operatorname{id}_ B } & \\ & & B. & & }$

By virtue of Proposition 8.1.4.2, it will suffice to show that the inner region of the diagram is a pushout square in $\operatorname{\mathcal{C}}$. This follows from Corollary 7.6.3.24, since $\operatorname{id}_{B}$ and $\operatorname{id}_{X}$ are isomorphisms in $\operatorname{\mathcal{C}}$. $\square$

Lemma 8.1.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\sigma _0: \Lambda ^{2}_{1} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets, corresponding to a diagram

8.7
$$\begin{gathered}\label{equation:thin-filler-in-Corr} \xymatrix@C =50pt@R=50pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dl]^{u} \ar [dr]_{v} & & X_{2,2} \ar [dl] \\ & X_{0,1} & & X_{1,2} & } \end{gathered}$$

in the $\infty$-category $\operatorname{\mathcal{C}}$. Then any commutative diagram

8.8
$$\begin{gathered}\label{equation:thin-filler-in-Corr2} \xymatrix@R =50pt@C=50pt{ X_{1,1} \ar [r]^-{u} \ar [d]^{v} & X_{1,2} \ar [d] \\ X_{0,1} \ar [r] & X_{0,2} } \end{gathered}$$

can be obtained from an extension of $\sigma _0$ to a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. In particular, if $X_{1,2}$ and $X_{0,1}$ admit a pushout along $X_{1,1}$, then $\sigma _0$ can be extended to a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

Proof. Let us identify $\operatorname{Tw}( \Delta ^2 )$ with the simplicial set $\operatorname{N}_{\bullet }(\overline{Q})$, where $\overline{Q}$ denotes the partially ordered set $\{ (i,j) \in [2]^{\operatorname{op}} \times [2]: i \leq j \}$. Under this identification, $\operatorname{Tw}( \Lambda ^{2}_{1} )$ corresponds to the simplicial subset $\operatorname{N}_{\bullet }(Q) \subseteq \operatorname{N}_{\bullet }( \overline{Q} )$, where $Q = \overline{Q} \setminus \{ (0,2) \}$, so $\sigma _0$ determines a diagram $\tau : \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{\mathcal{C}}$.

Set $Q_0 = Q \setminus \{ (0,0), (2,2) \}$ and $\tau _0 = \tau |_{ \operatorname{N}_{\bullet }( Q_0 ) }$. Lemma 8.1.4.4 is equivalent to the assertion that the restriction map $\operatorname{\mathcal{C}}_{ \tau / } \rightarrow \operatorname{\mathcal{C}}_{ \tau _0 / }$ is surjective on vertices. To prove this, it will suffice to show that the inclusion map $\operatorname{N}_{\bullet }(Q_0) \hookrightarrow \operatorname{N}_{\bullet }(Q)$ is right anodyne (Corollary 4.3.6.13), or equivalently that it is right cofinal (Proposition 7.2.1.3). This is a special case of Corollary 7.2.3.7, since the inclusion map $Q_0 \hookrightarrow Q$ has a left adjoint (given by $(0,0) \mapsto (0,1)$ and $(2,2) \mapsto (1,2)$). $\square$

Remark 8.1.4.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and suppose we are given a morphism of simplicial sets $\varphi : \operatorname{Tw}( \Delta ^2 ) \rightarrow \operatorname{\mathcal{C}}$, which we display as a diagram

$\xymatrix@R =50pt@C=50pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dl] \ar [dr] & & X_{2,2} \ar [dl] \\ & X_{0,1} \ar [dr] & & X_{1,2} \ar [dl] & \\ & & X_{0,2}. & & }$

The proof of Lemma 8.1.4.4 shows that $\varphi$ is a colimit diagram if and only if the inner region is a pushout square.

We now study the problem of filling outer horns in simplicial sets of the form $\operatorname{Cospan}(\operatorname{\mathcal{C}})$.

Lemma 8.1.4.6. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets. Suppose we are given an integer $n \geq 3$ and a lifting problem

8.9
$$\begin{gathered}\label{equation:relative-outer-horn-filling} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{Cospan}(\operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{Cospan}(U) } \\ \Delta ^ n \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur] & \operatorname{Cospan}(\operatorname{\mathcal{D}}), } \end{gathered}$$

where $\sigma _0$ corresponds to a morphism of simplicial sets $F_0: \operatorname{Tw}( \Lambda ^{n}_{0} ) \rightarrow \operatorname{\mathcal{C}}$ with the following properties:

$(a)$

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

$(b)$

The morphism $F_0$ carries the edge $(1,n) \rightarrow (0,n)$ to a $U$-cartesian edge of $\operatorname{\mathcal{C}}$.

Then the lifting problem (8.9) admits a solution.

Proof. Using Proposition 8.1.3.7, we can rewrite (8.9) as a lifting problem

8.10
$$\begin{gathered}\label{equation:relative-outer-horn-filling2} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\Lambda ^{n}_{0}) \ar [r]^-{ F_0 } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ U } \\ \operatorname{Tw}(\Delta ^ n) \ar [r]^-{ \overline{F} } \ar@ {-->}[ur] & \operatorname{\mathcal{D}}. } \end{gathered}$$

Let $P$ denote the set of all ordered pairs $(i,j)$, where $i$ and $j$ are integers satisfying $0 \leq i \leq j \leq n$. We regard $P$ as a partially ordered set by identifying it with its image in the product $[n]^{\operatorname{op}} \times [n]$ (so that $(i,j) \leq (i',j')$ if and only if $i' \leq i$ and $j \leq j'$). In what follows, we will identify $\operatorname{Tw}( \Delta ^ n )$ with the nerve $\operatorname{N}_{\bullet }(P)$; under this identification, $\operatorname{Tw}( \Lambda ^{n}_{0} )$ corresponds to a simplicial subset $K_0 \subseteq \operatorname{N}_{\bullet }(P)$.

Let $S = \{ (i_0, j_0) < (i_1, j_1) < \cdots < (i_ d, j_ d) \}$ be a nonempty linearly ordered subset of $P$, so that we have inequalities $0 \leq i_ d \leq i_{d-1} \leq \cdots \leq i_0 \leq j_0 \leq j_{1} \leq \cdots \leq j_ d \leq n$. In this case, we write $\tau _{S}$ for the corresponding nondegenerate $d$-simplex of $\operatorname{N}_{\bullet }(P)$. We will say that $S$ is basic if $\tau _ S$ is contained in $K_0$. Equivalently, $S$ is basic if the set $\{ i_0, i_1, \cdots , i_ d, j_0, j_1, \cdots , j_ d \}$ does not contain $\{ 1 < 2 < \cdots < n \}$. If $S$ is not basic, we let $\mathrm{pr}(S)$ denote the largest integer $j$ such that $S$ contains the pair $(i,j)$ for some $i \neq 0$. If no such integer exists, we define $\mathrm{pr}(S) = 0$. We will refer to $\mathrm{pr}(S)$ as the priority of $S$. We say that $S$ is prioritized if it is not basic and contains the pair $(0, \mathrm{pr}(S) )$.

Let $\{ S_1, S_2, \cdots , S_ m \}$ be an enumeration of the collection of all prioritized linearly ordered subsets of $P$ which satisfies the following conditions:

• The sequence of priorities $\mathrm{pr}(S_1), \mathrm{pr}(S_2), \cdots , \mathrm{pr}( S_ m)$ is nondecreasing. That is, if $1 \leq k \leq \ell \leq m$, then we have $\mathrm{pr}( S_{k} ) \leq \mathrm{pr}(S_{\ell })$.

• If $\mathrm{pr}( S_{k} ) = \mathrm{pr}( S_{\ell } )$ for $k \leq \ell$, then $| S_{k} | \leq | S_{\ell } |$.

For $1 \leq \ell \leq m$, let $\tau _{\ell } \subseteq \operatorname{N}_{\bullet }(P)$ denote the simplex $\tau _{S_{\ell }}$ and let $K_{\ell } \subseteq \operatorname{N}_{\bullet }(P)$ denote the union of $K_0$ with the simplices $\{ \tau _1, \tau _2, \cdots , \tau _{\ell } \}$, so that we have inclusion maps

$K_0 \hookrightarrow K_1 \hookrightarrow K_{2} \hookrightarrow \cdots \hookrightarrow K_{m}.$

We claim that $K_ m = \operatorname{N}_{\bullet }(P)$: that is, $K_ m$ contains $\tau _{S}$ for every nonempty linearly ordered subset $S \subseteq P$. If $S$ is basic, there is nothing to prove. We may therefore assume that $S$ has priority $p$ for some integer $p \geq 0$. The union $S \cup \{ (0,p) \}$ is then a prioritized linearly ordered subset of $P$, and therefore coincides with $S_{\ell }$ for some $1 \leq \ell \leq m$. In this case, we have $\tau _{S} \subseteq \tau _{\ell } \subseteq K_{\ell } \subseteq K_ m$.

We will complete the proof by constructing a compatible sequence of maps $F_{\ell }: K_{\ell } \rightarrow \operatorname{\mathcal{C}}$ extending $F_0$ and satisfying $U \circ F_{\ell } = \overline{F}|_{ K_{\ell } }$. Fix an integer $1 \leq \ell \leq m$, and suppose that $F_{\ell -1}$ has already been constructed. Write $S_{\ell } = \{ (i_0, j_0) < (i_1, j_1) < \cdots < (i_ d, j_ d) \}$, so that the simplex $\tau _{\ell }$ has dimension $d$. Let $p$ be the priority of $S_{\ell }$. Since $S_{\ell }$ is prioritized, it contains $(0,p)$; we can therefore write $(0,p) = (i_{ d' }, j_{d'} )$ for some integer $0 \leq d' \leq d$. Let $L \subseteq \Delta ^ d$ denote the inverse image of $K_{\ell -1}$ under the map $\tau _{\ell }: \Delta ^{d} \rightarrow \operatorname{N}_{\bullet }(P)$. We will show that $L$ coincides with the horn $\Lambda ^{d}_{d'}$, so that the pullback diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ L \ar [r] \ar [d] & K_{\ell -1} \ar [d] \\ \Delta ^{d} \ar [r]^-{ \tau _{\ell } } & K_{\ell }. }$

is also a pushout square (Lemma 3.1.2.11). This can be stated more concretely as follows:

$(\ast )$

Let $(i,j)$ be an element of $S_{\ell }$, and set $S' = S_{\ell } \setminus \{ (i,j) \}$. Then the simplex $\tau _{S'}$ is contained in $K_{\ell -1}$ if and only if $(i,j) \neq (0,p)$.

We first prove $(\ast )$ in the case where $(i,j) \neq (0,p)$; in this case, we wish to show that $\tau _{S'}$ is contained in $K_{\ell -1}$. If $S'$ is basic, then $\tau _{S'}$ is contained in $K_0$ and there is nothing to prove. Let us therefore assume that $S'$ is not basic. Let $p' = \mathrm{pr}(S')$ denote the priority of $S'$. Then the union $S' \cup \{ (0,p') \}$ is a prioritized subset of $P$, and therefore has the form $S_{k}$ for some $1 \leq k \leq m$. By construction, we have $\mathrm{pr}(S_ k) = p' \leq p = \mathrm{pr}( S_{\ell } )$. Moreover, if $p' = p$, then our assumption $(i,j) \neq (0,p)$ guarantees that $S_ k = S'$, so that $| S_{k} | < | S_{\ell } |$. It follows that $k < \ell$, so that we have $\tau _{S'} \subseteq \tau _ k \subseteq K_{k} \subseteq K_{\ell -1}$.

We now prove $(\ast )$ in the case $(i,j) = (0,p)$; in this case, we wish to show that $\tau _{S'}$ is not contained in $K_{\ell -1}$. Assume otherwise. Then, since $S'$ is not basic, it is contained in $S_{k}$ for some $k < \ell$. The inequalities

$p = \mathrm{pr}(S') \leq \mathrm{pr}( S_ k ) \leq \mathrm{pr}( S_{\ell } ) = p.$

ensure that $S_{k}$ has priority $p$. Since $S_{k}$ is prioritized, it contains $(0,p)$, and therefore contains the union $S_{\ell } = S' \cup \{ (0,p) \}$. The inequality $|S_ k | \leq | S_{\ell } |$ then forces $k = \ell$, contradicting our assumption that $k < \ell$. This completes the proof of $(\ast )$.

Let $\rho _0$ denote the composite map $\Lambda ^{d}_{d'} = L \xrightarrow { \tau _{\ell } } K_{\ell -1} \xrightarrow { F_{\ell -1} } \operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that the lifting problem

8.11
$$\begin{gathered}\label{equation:relative-outer-horn-filling3} \xymatrix@R =50pt@C=50pt{ \Lambda ^{d}_{d'} \ar [r]^-{ \rho _0 } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \Delta ^{d} \ar@ {-->}[ur] \ar [r]^-{ \overline{F} \circ \tau _{\ell } } & \operatorname{\mathcal{D}}} \end{gathered}$$

admits a solution. We consider three cases:

• If $0 < d' < d$, then the lifting problem (8.11) admits a solution by virtue of our assumption that $U$ is an inner fibration of simplicial sets.

• Suppose that $d' = 0$: that is, the pair $(0,p)$ is the smallest element of $S_{\ell }$. Then $S_{\ell }$ does not contain any pairs $(i,j)$ with $i \neq 0$, so we have $p = 0$. Since the set $S_{\ell }$ is not basic, we must have $S_{\ell } = \{ (0,0) < (0,1) < \cdots < (0,n-1) < (0,n) \}$. In this case, the lifting problem (8.11) admits a solution by virtue of assumption $(a)$.

• Suppose that $d' = d$: that is, the pair $(0,p)$ is the largest element of $S_{\ell }$. Our assumption that $S_{\ell }$ is not basic then guarantees that $p =n$ and $(1,n) \in S_{\ell }$: that is, we have $S_{\ell } = \{ (i_0, j_0) < (i_1, j_1) < \cdots < (1,n) < (0,n) \}$. In this case, the lifting problem (8.11) admits a solution by virtue of assumption $(b)$.

$\square$

Specializing Lemma 8.1.4.6 to the case $\operatorname{\mathcal{D}}= \Delta ^0$, we obtain the following:

Lemma 8.1.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets, which we identify with a diagram $X: \operatorname{Tw}( \Lambda ^{n}_{0} ) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. Assume that $n \geq 3$ and that the morphisms $X(0,0) \rightarrow X(0,1)$ and $X(1,n) \rightarrow X(0,n)$ are isomorphisms in $\operatorname{\mathcal{C}}$. Then $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$.

Proof of Proposition 8.1.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. By virtue of Lemma 8.1.4.7 and Corollary 8.1.4.3, the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ satisfies conditions $(2)$ and $(3)$ of Definition 5.4.1.1. Since $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is isomorphic to $\operatorname{Cospan}(\operatorname{\mathcal{C}})^{\operatorname{op}}$ (Remark 8.1.3.4), it also satisfies condition $(4)$ of Definition 5.4.1.1. It follows that $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category if and only if it satisfies the following condition:

$(\ast )$

Every morphism of simplicial sets $\Lambda ^{2}_{1} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ can be extended to a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$.

Using Lemma 8.1.4.4, we can rewrite condition $(\ast )$ as follows:

$(\ast ')$

For every diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [dr] & & Y \ar [dl] \ar [dr] & & Z \ar [dl] \\ & B & & C & }$

in the $\infty$-category $\operatorname{\mathcal{C}}$, there exists a pushout of $B$ and $C$ along $Y$.

It is clear that if the $\infty$-category $\operatorname{\mathcal{C}}$ admits pushouts, then it satisfies condition $(\ast ')$. The converse follows by applying condition $(\ast ')$ to diagrams of the form

$\xymatrix@R =50pt@C=50pt{ X \ar [dr]^{\operatorname{id}_{X}} & & Y \ar [dl] \ar [dr] & & Z \ar [dl]_{\operatorname{id}_{Z}} \\ & X & & Z. & }$
$\square$

Corollary 8.1.4.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ admit pushouts. The following conditions are equivalent:

$(1)$

The functor $F$ carries pushout squares in $\operatorname{\mathcal{C}}$ to pushout squares in $\operatorname{\mathcal{D}}$.

$(2)$

The induced map $\operatorname{Cospan}(F): \operatorname{Cospan}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{D}})$ is a functor of $(\infty ,2)$-categories, in the sense of Definition 5.4.7.1.

Proof. The implication $(1) \Rightarrow (2)$ follows immediately from the criterion of Proposition 8.1.4.2. For the converse implication, suppose that $\operatorname{Cospan}(F): \operatorname{Cospan}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{D}})$ is a functor of $2$-categories, and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a pushout square, which we display as a diagram

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

Let $\rho : \operatorname{Tw}( \Delta ^2 ) \rightarrow \Delta ^1 \times \Delta ^1$ denote the morphism of simplicial sets given on vertices by the formula $\rho (i,j) = (\max (0, 1-i) , \max (0,j-1) )$. Then $\sigma \circ \rho$ can be identified with a $2$-simplex $\tau$ of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, corresponding to a diagram

$\xymatrix@R =50pt@C=50pt{ X_0 \ar [dr]_{\operatorname{id}} & & X \ar [dl] \ar [dr] & & X_1 \ar [dl]^{\operatorname{id}} \\ & X_0 \ar [dr] & & X_1 \ar [dl] \\ & & X_{01} & & }$

in the $\infty$-category $\operatorname{\mathcal{C}}$. It follows from the criterion of Proposition 8.1.4.2 that $\tau$ is a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. If $\operatorname{Cospan}(F)$ is a functor of $(\infty ,2)$-categories, then it carries $\tau$ to a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{D}})$. Applying the criterion of Proposition 8.1.4.2 again, we conclude that $F(\sigma )$ is a pushout square in $\operatorname{\mathcal{D}}$. $\square$