# Kerodon

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### 8.1.6 Restricted Cospans

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, and let $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ be the simplicial set introduced in Construction 8.1.3.1. By definition, the edges of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ correspond to cospans in the $\infty$-category $\operatorname{\mathcal{C}}$: that is, pairs of morphisms $X \xrightarrow {f} B \xleftarrow {g} Y$ having a common target. In practice, it will sometimes be useful to consider a variant of this construction, where we place additional restrictions on the morphisms $f$ and $g$.

Definition 8.1.6.1 (Restricted Cospans). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $L$ and $R$ be collections of edges of $\operatorname{\mathcal{C}}$. We let $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ denote the simplicial subset of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ whose $n$-simplices are given by diagrams $X: \operatorname{Tw}(\Delta ^ n) \rightarrow \operatorname{\mathcal{C}}$ which satisfy the following condition:

• For every pair of integers $0 \leq i \leq j \leq n$, the edge $X_{i,i} \rightarrow X_{i,j}$ belongs to $L$ and the edge $X_{j,j} \rightarrow X_{i,j}$ belongs to $R$.

Remark 8.1.6.2 (Symmetry). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $L$ and $R$ be collections of edges of $\operatorname{\mathcal{C}}$. Then the isomorphism $\operatorname{Cospan}(\operatorname{\mathcal{C}}) \xrightarrow {\sim } \operatorname{Cospan}( \operatorname{\mathcal{C}})^{\operatorname{op}}$ of Remark 8.1.3.4 restricts to an isomorphism of simplicial subsets $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) \xrightarrow {\sim } \operatorname{Cospan}^{R,L}(\operatorname{\mathcal{C}})^{\operatorname{op}}$.

Remark 8.1.6.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $L$ and $R$ be collections of edges of $\operatorname{\mathcal{C}}$. Suppose we are given a morphism of simplicial sets $f: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$, corresponding to a morphism $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ (see Proposition 8.1.3.7). For every edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}$, let $e_{L}: \operatorname{id}_{X} \rightarrow e$ and $e_{R}: \operatorname{id}_{Y} \rightarrow e$ be the edges of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ described in Example 8.1.3.6. Then $f$ factors through the simplicial subset $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ if and only if the edge $F(e_{L})$ belongs to $L$ and the edge $F(e_{R} )$ belongs to $R$, for every edge $e$ of $\operatorname{\mathcal{D}}$.

Remark 8.1.6.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $L$ and $R$ be collections of edges of $\operatorname{\mathcal{C}}$, and let $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ denote the restricted cospan construction of Definition 8.1.6.1. Note that a morphism of simplicial sets $K \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ factors through $\operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}})$ if and only if its restriction to the $1$-skeleton $\operatorname{sk}_1(K)$ factors through $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$. In particular, if $\sigma$ is a $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ which is thin when viewed as a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, then it is also thin when viewed as a $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$. For a partial converse, see Corollary 8.1.6.8.

We now formulate a criterion which guarantees that the simplicial set $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category.

Definition 8.1.6.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$. We will say that $L$ and $R$ are pushout-compatible if, for every morphism $f_0: X \rightarrow X_0$ of $\operatorname{\mathcal{C}}$ which belongs to $L$ and every morphism $f_1: X \rightarrow X_1$ of $\operatorname{\mathcal{C}}$ which belongs to $R$, there exists a pushout diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f_0} \ar [d]^{ f_1 } & X_0 \ar [d]^{ f'_1 } \\ X_{1} \ar [r]^-{ f'_0 } & X_{01} }$

where $f'_0$ belongs to $L$ and $f'_1$ belongs to $R$.

Example 8.1.6.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $L$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $R$ be the collection of all isomorphisms in $\operatorname{\mathcal{C}}$. Assume that $L$ is stable under isomorphism (that is, if $f$ if a morphism of $\operatorname{\mathcal{C}}$ which is isomorphic to an element of $L$ in the $\infty$-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, then $f$ also belongs to $L$). Then $L$ and $R$ are pushout-compatible.

Proposition 8.1.6.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $L$ and $R$ denote collections of morphisms of $\operatorname{\mathcal{C}}$ which are closed under composition. If $L$ and $R$ are pushout-compatible (in the sense of Definition 8.1.6.5), then the simplicial set $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category.

Proof. We will show that $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ satisfies each condition of Definition 5.5.1.3:

$(1)$

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).

$\square$

Corollary 8.1.6.8. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $L$ and $R$ denote collections of morphisms of $\operatorname{\mathcal{C}}$ which are closed under composition. If $L$ and $R$ are pushout-compatible, then a $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is thin if and only if it is thin when viewed as a $2$-simplex of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$.

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 5.5.7.9, 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 $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. $\square$

Remark 8.1.6.9. In the situation of Proposition 8.1.6.7, let $\sigma$ be an $n$-simplex of the $(\infty ,2)$-category $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$, corresponding to a diagram

8.18
\begin{equation} \begin{gathered}\label{pith-of-cospan} \xymatrix@R =40pt@C=20pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl] & \cdots & X_{n-1,n-1} \ar [dr] \ar [dl] & & X_{n,n} \ar [dl] \\ & \cdots \ar [dr] & & \cdots \ar [dr] \ar [dl] & & \cdots \ar [dl] & \\ & & X_{0,n-1} \ar [dr] & & X_{1,n} \ar [dl] & & \\ & & & X_{0,n} & & & \\ } \end{gathered} \end{equation}

in the $\infty$-category $\operatorname{\mathcal{C}}$. Then $\sigma$ is contained in the pith $\operatorname{Pith}( \operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}}) )$ if and only if each of the rectangular regions in the diagram (8.18) is a pushout square in the $\infty$-category $\operatorname{\mathcal{C}}$. This follows from the thinness criterion of Corollary 8.1.6.8 (together with Proposition 7.6.3.21).

Corollary 8.1.6.10 (Invertible Cospans). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which contain all isomorphisms and are closed under composition. Assume that $L$ and $R$ are pushout-compatible and let $e: X \rightarrow Y$ be a morphism in the $(\infty ,2)$-category $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$, which we identify with a diagram $X \xrightarrow {f} B \xleftarrow {g} Y$ in the $\infty$-category $\operatorname{\mathcal{C}}$. Then $e$ is an isomorphism in $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ if and only if $f$ and $g$ are isomorphisms in $\operatorname{\mathcal{C}}$.

Proof. Let $\operatorname{\mathcal{C}}^{\simeq }$ denote the core of $\operatorname{\mathcal{C}}$ (Construction 1.2.5.4). 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 1.3.6.10). 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 5.5.7.5).

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 5.5.6.6, 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

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

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 7.6.3.20). 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 5.5.6.5). A similar argument shows that $g$ is an isomorphism in $\operatorname{\mathcal{C}}$. $\square$

Exercise 8.1.6.11. Show that, if the conditions of Corollary 8.1.6.10 are satisfied, then the diagram $Y \xrightarrow {g} B \xleftarrow {f} X$ is a homotopy inverse of $e$, when regarded as a morphism from $Y$ to $X$ in the $(\infty ,2)$-category $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$.

Corollary 8.1.6.12. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing objects $X$ and $Y$. Let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which are pushout-compatible, contain all isomorphisms of $\operatorname{\mathcal{C}}$, and are closed under composition. Then $X$ and $Y$ are isomorphic as objects of the $\infty$-category $\operatorname{\mathcal{C}}$ if and only if they are isomorphic as objects of the $(\infty ,2)$-category $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$.