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).

Proof. Let $\sigma $ be a thin $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$; we will show that $\sigma $ is also thin when viewed as a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ (the reverse implication follows from Remark 8.1.6.4). Choose a fully faithful functor $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is an $\infty $-category which admits pushouts and the functor $f$ preserves all pushout squares which exist in $\operatorname{\mathcal{C}}$ (see Corollary 8.3.3.17). Then $f$ induces a morphism of simplicial sets $F: \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{D}})$. The proof of Proposition 8.1.6.7 shows that every morphism $\tau _0: \Lambda ^{2}_{1} \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ can be extended to a $2$-simplex $\tau $ of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ which is thin when viewed as a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. Since $f$ preserves pushout squares, the criterion of Proposition 8.1.4.2 guarantees that $F(\tau )$ is a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{D}})$. Allowing $\tau _0$ to vary and invoking Proposition 5.4.7.9, we deduce that $F$ is a functor of $(\infty ,2)$-categories. In particular, $F(\sigma )$ is a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{D}})$. Using the criterion of Proposition 8.1.4.2 and the assumption that $f$ is fully faithful, we deduce that $\sigma $ is also thin when viewed as a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. $\square$

Proof. Let $\operatorname{\mathcal{C}}^{\simeq }$ denote the core of $\operatorname{\mathcal{C}}$ (Construction 1.3.5.4). If $f$ and $g$ are isomorphisms, then $e$ can be regarded as an edge of the simplicial set $\operatorname{Cospan}( \operatorname{\mathcal{C}}^{\simeq } )$. Since $\operatorname{\mathcal{C}}^{\simeq }$ is a Kan complex (Corollary 4.4.3.11), the simplicial set $\operatorname{Cospan}( \operatorname{\mathcal{C}}^{\simeq } )$ is also a Kan complex (Corollary 8.1.3.11), so $e$ is automatically an isomorphism when viewed as a morphism of $\operatorname{Cospan}( \operatorname{\mathcal{C}}^{\simeq } )$ (Proposition 1.4.6.10). Since the inclusion map $\operatorname{Cospan}( \operatorname{\mathcal{C}}^{\simeq } ) \hookrightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is a functor of $(\infty ,2)$-categories (Corollary 8.1.6.8), it follows that $e$ is also an isomorphism when regarded as a morphism of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ (Remark 5.4.7.5).

Now suppose that $e$ is an isomorphism in the $(\infty ,2)$-category $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$. Arguing as in the proof of Proposition 5.4.6.6, we can produce a $3$-simplex $\sigma : \Delta ^3 \rightarrow \operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$, where $\sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ is the morphism $e$, $\sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 2 \} )}$ is the identity morphism $\operatorname{id}_{X}$, and $\sigma |_{ \operatorname{N}_{\bullet }( \{ 1 < 3 \} ) }$ is the identity morphism $\operatorname{id}_{Y}$. Let us identify $\sigma $ with a diagram

in the $\infty $-category $\operatorname{\mathcal{C}}$. This diagram exhibits the identity morphism $\operatorname{id}_{X}$ as a composition of $u$ with $f$. Since $\sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 3 \} ) }$ is a thin $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$, the outer rectangular region on the left is a pushout square in $\operatorname{\mathcal{C}}$ (Corollary 8.1.6.8). It follows that the composition of $v$ with $u$ is an isomorphism in $\operatorname{\mathcal{C}}$ (since it is a pushout of the identity morphism $\operatorname{id}_{Y}$; see Corollary 7.6.3.24). Applying the two-out-of-six property to the $3$-simplex of $\operatorname{\mathcal{C}}$ given by the left edge of the diagram, we conclude that $f$ is an isomorphism in $\operatorname{\mathcal{C}}$ (see Proposition 5.4.6.5). A similar argument shows that $g$ is an isomorphism in $\operatorname{\mathcal{C}}$. $\square$