Kerodon

Proof. We will show that $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ satisfies each condition of Definition 5.4.1.1:

$(1)$

Let $\sigma _0: \Lambda ^{2}_{1} \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets; we wish to show that $\sigma _0$ can be extended to a thin $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$. Let us identify $\sigma _0$ with a diagram

8.17

\begin{equation} \begin{gathered}\label{equation:correspondence-infinity-two-strong} \xymatrix@C =50pt@R=50pt{ X_{0,0} \ar [dr]^{u} & & X_{1,1} \ar [dl]_{u'} \ar [dr]^{v} & & X_{2,2} \ar [dl]_{v'} \\ & X_{0,1} & & X_{1,2} & } \end{gathered} \end{equation}

in the $\infty $-category $\operatorname{\mathcal{C}}$, where the morphisms $u$ and $v$ belong to $L$ and the morphisms $u'$ and $v'$ belong to $R$. Combining our assumption that $L$ and $R$ are pushout-compatible with Lemma 8.1.4.4, we can enlarge (8.17) to a commutative diagram

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

where $w$ belongs to $L$, $w'$ belongs to $R$, and the lower square is a pushout. By virtue of Proposition 8.1.4.2, this extension can be viewed as a thin $2$-simplex of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. Using our assumption that $L$ and $R$ are closed under composition, we see that $\sigma $ belongs to the simplicial subset $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$, and is therefore also thin when regarded as a $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ (Remark 8.1.6.4).

$(2)$

Every degenerate $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is thin; this follows from Corollary 8.1.4.3 and Remark 8.1.6.4.

$(3)$

Let $n \geq 3$ and let $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}})$ be a morphism of simplicial sets with the property that the $2$-simplex $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0< 1 < n\} ) }$ is left-degenerate. Using Lemma 8.1.4.7, we can extend $\sigma _0$ to an $n$-simplex $\sigma $ of $\operatorname{Cospan}( \operatorname{\mathcal{C}})$. Since every edge of $\Delta ^ n$ is contained in $\Lambda ^{n}_{0}$, the extension $\sigma $ is automatically contained in $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ (Remark 8.1.6.4).

$(4)$

Let $n \geq 3$ and let $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}})$ be a morphism of simplicial sets with the property that the $2$-simplex $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0< n-1 < n\} ) }$ is right-degenerate. Then $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$; this follows by applying $(3)$ to the opposite simplicial set $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})^{\operatorname{op}} \simeq \operatorname{Cospan}^{R,L}(\operatorname{\mathcal{C}})$ (see Remark 8.1.6.2).