# Kerodon

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### 8.1.7 Comparing $\operatorname{\mathcal{C}}$ with $\operatorname{Cospan}(\operatorname{\mathcal{C}})$

Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts. Then there is a functor from $\operatorname{\mathcal{C}}$ to the $2$-category $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ of Example 2.2.2.1, which carries each object of $\operatorname{\mathcal{C}}$ to itself and each morphism $f: X \rightarrow Y$ to the cospan $X \xrightarrow {f} Y \xleftarrow { \operatorname{id}_ Y } Y$. This observation has an $\infty$-categorical counterpart:

Construction 8.1.7.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the projection map of Notation 8.1.1.6, carrying each vertex $(f: X \rightarrow Y)$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to the vertex $Y \in \operatorname{\mathcal{C}}$. Under the bijection supplied by Proposition 8.1.3.7, we can identify $\lambda _{+}$ with a morphism of simplicial sets $\rho _{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. If $\sigma$ is an $n$-simplex of $\operatorname{\mathcal{C}}$, which we display informally as a diagram

$X_0 \xrightarrow {f_1} X_1 \xrightarrow {f_2} X_2 \rightarrow \cdots \xrightarrow {f_ n} X_ n,$

then $\rho _{+}(\sigma )$ is an $n$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ which can be depicted informally as a diagram

$\xymatrix@R =20pt@C=20pt{ X_0 \ar [dr]^-{f_1} & & X_1 \ar [dr]^{f_2} \ar [dl]^{\operatorname{id}} & & \cdots \ar [dl] \ar [dr] & & X_{n-1} \ar [dr]^{f_ n} \ar [dl]^{\operatorname{id}} & & X_ n \ar [dl]^{\operatorname{id}} \\ & X_1 \ar [dr]^{f_2} & & X_2 \ar [dl]^{\operatorname{id}} \ar [dr]^{f_3} & \cdots & X_{n-1} \ar [dl]^{\operatorname{id}} \ar [dr]^{f_ n} & & X_{n} \ar [dl]^{\operatorname{id}} & \\ & & \cdots \ar [dr]^{f_{n-1}} & & \cdots \ar [dl]^{\operatorname{id}} \ar [dr]^{f_ n} & & \cdots \ar [dl]^{\operatorname{id}} & & \\ & & & X_{n-1} \ar [dr]^{f_ n} & & X_{n} \ar [dl]^{\operatorname{id}} & & & \\ & & & & X_ n. & & & & }$

Note that $\rho _{+}$ is a monomorphism of simplicial sets.

Our goal in this section is to study the behavior of Construction 8.1.7.1 in the case where the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty$-category. In this case, we will show that $\rho _{+}$ induces an equivalence of $\operatorname{\mathcal{C}}$ with a certain restricted cospan construction (Proposition 8.1.7.6).

Construction 8.1.7.2. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. We let $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso} }(\operatorname{\mathcal{C}})$ denote the simplicial subset of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ whose $n$-simplices are diagrams

$\xymatrix@R =40pt@C=20pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl]_{\sim } & \cdots & X_{n-1,n-1} \ar [dr] \ar [dl]_{\sim } & & X_{n,n} \ar [dl]_{\sim } \\ & \cdots \ar [dr] & & \cdots \ar [dr] \ar [dl]_{\sim } & & \cdots \ar [dl]_{\sim } & \\ & & X_{0,n-1} \ar [dr] & & X_{1,n} \ar [dl]_{\sim } & & \\ & & & X_{0,n}, & & & \\ }$

where each of the leftward-directed arrows is an isomorphism in $\operatorname{\mathcal{C}}$.

Remark 8.1.7.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the simplicial set $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ of Construction 8.1.7.2 coincides with the restricted cospan construction $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ of Definition 8.1.6.1, where we take $L$ to be the collection of all morphisms of $\operatorname{\mathcal{C}}$ and $R$ to be the collection of all isomorphisms in $\operatorname{\mathcal{C}}$ (see Example 8.1.7.10 for a more general statement).

Variant 8.1.7.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $R$ be a collection of edges of $\operatorname{\mathcal{C}}$. We let $\operatorname{Cospan}^{\mathrm{all}, R}( \operatorname{\mathcal{C}})$ denote the simplicial subset $\operatorname{Cospan}^{A,R}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{C}})$, where $A$ is the collection of all edges of $\operatorname{\mathcal{C}}$. Similarly, if $L$ is a collection of edges of $\operatorname{\mathcal{C}}$, we let $\operatorname{Cospan}^{L,\mathrm{all}}(\operatorname{\mathcal{C}})$ denote the simplicial subset $\operatorname{Cospan}^{L,A}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{C}})$. Note that the simplicial set $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ of Definition 8.1.6.1 can be recovered as the intersection $\operatorname{Cospan}^{L, \mathrm{all}}(\operatorname{\mathcal{C}}) \cap \operatorname{Cospan}^{\mathrm{all}, R}(\operatorname{\mathcal{C}})$.

Proposition 8.1.7.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the simplicial set $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ is an $\infty$-category.

Proof. Let $\sigma$ be a $2$-simplex of $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$, which we identify with a diagram

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

in the $\infty$-category $\operatorname{\mathcal{C}}$ where the leftward-directed morphisms are isomorphisms. Using Corollary 7.6.3.24, we deduce that the inner region is a pushout square in $\operatorname{\mathcal{C}}$. It follows that $\sigma$ is automatically thin when regarded as a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ (Proposition 8.1.4.2), and therefore also when regarded as a $2$-simplex of $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ (Remark 8.1.6.4). To complete the proof, it will suffice to show that every diagram $\Lambda ^{2}_{1} \rightarrow \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ can be extended to a $2$-simplex of $\operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ (see Example 2.3.2.4). Using Lemma 8.1.4.4, we can restate this as follows: for every pair of morphisms $f: X_{1,1} \rightarrow X_{1,2}$ and $u: X_{1,1} \rightarrow X_{0,1}$ of $\operatorname{\mathcal{C}}$ where $u$ is an isomorphism, there exists a commutative diagram

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

where $v$ is also an isomorphism. This follows immediately from the definitions (or from Corollary 4.4.5.9). $\square$

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\rho _{+}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Cospan}( \operatorname{\mathcal{C}})$ the morphism described in Construction 8.1.7.1. Note that $\rho _{+}$ carries each object of $\operatorname{\mathcal{C}}$ to itself, and each morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ to the edge of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ given by the diagram $X \xrightarrow {f} Y \xleftarrow { \operatorname{id}_ Y }$. Since every identity morphism in $\operatorname{\mathcal{C}}$ is an isomorphism, $\rho _{+}$ factors through the $\infty$-category $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}})$. The remainder of this section is devoted to the proof of the following:

Proposition 8.1.7.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the functor $\rho _{+}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ is an equivalence of $\infty$-categories.

Example 8.1.7.7. Let $X$ be a Kan complex. Applying Proposition 8.1.7.6 (and noting that every edge of $X$ is an isomorphism), we see that $\rho _{+}: X \rightarrow \operatorname{Cospan}(X)$ is a homotopy equivalence of Kan complexes (see Corollary 8.1.3.11).

Our proof of Proposition 8.1.7.6 will require some preliminaries.

Definition 8.1.7.8. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We say that $W$ has the left two-out-of-three property if, for every $2$-simplex of $\operatorname{\mathcal{C}}$ with boundary indicated in by the diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [rr]^{h} \ar [dr]^{f} & & Z \\ & Y, \ar [ur]_{g} & }$

where $f$ belongs to $W$, $g$ belongs to $W$ if and only if $h$ belongs to $W$.

Lemma 8.1.7.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories and let $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ be a functor, corresponding to a morphism of simplicial sets $f: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. Let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which satisfy the left two-out-of-three property. Then $f$ factors through $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ if and only if $F$ satisfies the following pair of conditions:

$(1)$

For each object $X$ in $\operatorname{\mathcal{D}}$, the morphism $F$ carries every morphism of $\{ X\} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}})$ to a morphism of $\operatorname{\mathcal{C}}$ which belongs to $L$.

$(2)$

For each vertex $Y$ in $\operatorname{\mathcal{D}}$, the morphism $F$ carries every morphism of $\operatorname{Tw}(\operatorname{\mathcal{D}}) \times _{\operatorname{\mathcal{D}}} \{ Y\}$ to a morphism of $\operatorname{\mathcal{C}}$ which belongs to $R$.

Proof. For every morphism $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}$, let $\operatorname{id}_{X} \xrightarrow { e_{L} } e \xleftarrow { e_{R} } \operatorname{id}_{Y}$ be the tautological cospan in $\operatorname{Tw}(\operatorname{\mathcal{D}})$ described in Example 8.1.3.6. By virtue of Remark 8.1.6.3, the morphism $f$ factors through $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ if and only if it satisfies the following pair of conditions:

$(1')$

For every morphism $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}$, the functor $F$ carries $e_{L}$ to a morphism of $\operatorname{\mathcal{C}}$ which belongs to $L$.

$(2')$

For every morphism $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}$, the functor $F$ carries $e_{R}$ to a morphism of $\operatorname{\mathcal{C}}$ which belongs to $R$.

The implications $(1) \Rightarrow (1')$ and $(2) \Rightarrow (2')$ are immediate (if $e: X \rightarrow Y$ is any morphism of $\operatorname{\mathcal{D}}$, then $e_{L}$ is contained to the fiber $\{ X\} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}})$, and $e_{R}$ is contained in $\operatorname{Tw}(\operatorname{\mathcal{D}}) \times _{\operatorname{\mathcal{D}}} \{ Y\}$ ). We will complete the proof by showing that $(1')$ implies $(1)$; a similar argument shows that $(2')$ implies $(2)$.

Assume that condition $(1')$ is satisfied, let $X$ be an object of $\operatorname{\mathcal{D}}$, and let $u: X \rightarrow Y$ and $v: X \rightarrow Z$ be morphisms of $\operatorname{\mathcal{D}}$. Suppose we are given a morphism $g: u \rightarrow v$ in the $\infty$-category $\operatorname{\mathcal{E}}= \{ X\} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}})$; we wish to show that $F(g)$ belongs to $L$. Since $\operatorname{id}_{X}$ is initial when viewed as an object of $\operatorname{\mathcal{E}}$ (Proposition 8.1.2.1), there is a $2$-simplex of $\operatorname{\mathcal{E}}$ whose boundary is indicated in the diagram

$\xymatrix@R =50pt@C=50pt{ & \operatorname{id}_{X} \ar [dl]^{ u_{L} } \ar [dr]_{ v_{L} } & \\ u \ar [rr]^{g} & & v. }$

Assumption $(1')$ guarantees that $F(u_{L} )$ and $F(v_ L)$ belong to $L$. Since $L$ satisfies the left two-out-of-three property, it follows that $F(g)$ also belongs to $L$. $\square$

Example 8.1.7.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\sigma$ be an $n$-simplex of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, corresponding to a diagram

8.19
$$\begin{gathered}\label{equation:left-toot-consequence} \xymatrix@R =40pt@C=20pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl] & \cdots & X_{n-1,n-1} \ar [dr] \ar [dl] & & X_{n,n} \ar [dl] \\ & \cdots \ar [dr] & & \cdots \ar [dr] \ar [dl] & & \cdots \ar [dl] & \\ & & X_{0,n-1} \ar [dr] & & X_{1,n} \ar [dl] & & \\ & & & X_{0,n}. & & & \\ } \end{gathered}$$

Let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which contain all identity morphisms and have the left two-out-of-three property. Then $\sigma$ belongs to $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ if and only if each of the rightward-pointing morphisms displayed in (8.19) belong to $L$, and each of the leftward-pointing morphisms displayed in (8.19) belong to $R$.

Remark 8.1.7.11. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $R$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which has the left two-out-of-three property, and let $\operatorname{\mathcal{D}}$ be a simplicial set. Suppose we are given a pair of morphisms $f,g: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}^{ \mathrm{all}, R}( \operatorname{\mathcal{C}})$, corresponding to diagrams $F,G: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$. If $\alpha : F \rightarrow G$ is a natural transformation of diagrams, then the following conditions are equivalent:

$(a)$

For every edge $u: D' \rightarrow D$ of $\operatorname{\mathcal{D}}$, the morphism $\alpha _{u}: F(u) \rightarrow G(u)$ belongs to $R$.

$(b)$

For every degenerate edge $u: D \rightarrow D$ of $\operatorname{\mathcal{D}}$, the morphism $\alpha _{u}: F(u) \rightarrow G(u)$ belongs to $R$.

The implication $(a) \Rightarrow (b)$ is immediate. To prove the reverse implication, let $u: D' \rightarrow D$ be an edge of $\operatorname{\mathcal{D}}$ and let $u_{R}: \operatorname{id}_{D} \rightarrow u$ be the edge of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ described in Example 8.1.3.6. Evaluating $\alpha$ on the morphism $u_{R}$, we obtain a commutative diagram

$\xymatrix@R =50pt@C=50pt{ F( \operatorname{id}_{D} ) \ar [r]^-{ \alpha _{ \operatorname{id}_ D } } \ar [d] & G( \operatorname{id}_{D} ) \ar [d] \\ F(u) \ar [r]^-{ \alpha _{u} } & G( u ) }$

in the $\infty$-category $\operatorname{\mathcal{C}}$, where the vertical maps belong to $R$ by virtue of our assumption that $f$ and $g$ factor through $\operatorname{Cospan}^{\mathrm{all}, R}(\operatorname{\mathcal{C}})$. Applying the left two-out-of-three property, we conclude that if the upper horizontal map belongs to $R$, then the lower horizontal map also belongs to $R$.

Lemma 8.1.7.12. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which have the left two-out-of-three property, and let $\operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ spanned by those objects which correspond to diagrams $\operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{C}})$. Let $R'$ be the collection of morphisms in $\operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ which satisfy the equivalent conditions of Remark 8.1.7.11, and define $L'$ similarly. Then the isomorphism $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Cospan}( \operatorname{\mathcal{C}}) ) \simeq \operatorname{Cospan}( \operatorname{Fun}(\operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) )$ of Remark 8.1.3.9 restricts to an isomorphism of simplicial subsets $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}}) ) \simeq \operatorname{Cospan}^{L',R'}( \operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) )$.

Proof. Writing $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ as the intersection $\operatorname{Cospan}^{L, \mathrm{all}}(\operatorname{\mathcal{C}}) \cap \operatorname{Cospan}^{\mathrm{all}, R}( \operatorname{\mathcal{C}})$ (see Variant 8.1.7.4), we can reduce to the case where either $L$ or $R$ is the collection of all morphisms of $\operatorname{\mathcal{C}}$. Let us assume that $L$ is the collection of all morphisms of $\operatorname{\mathcal{C}}$, so that $L'$ is the collection of all morphisms of $\operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$. Suppose we are given another simplicial set $\operatorname{\mathcal{E}}$ and a diagram $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \times \operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$. We can identify $F$ with a morphism of simplicial sets $\operatorname{\mathcal{E}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Cospan}( \operatorname{\mathcal{C}}) )$. By virtue of Remark 8.1.6.3, this morphism factors through the simplicial subset $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$ if and only if, for every edge $u: D' \rightarrow D$ of $\operatorname{\mathcal{D}}$ and every edge $v: E' \rightarrow E$ of $\operatorname{\mathcal{E}}$, the morphism $F$ satisfies the following condition:

$(1_{u,v})$

Let $u_{R}: \operatorname{id}_{D} \rightarrow u$ and $v_{R}: \operatorname{id}_{E} \rightarrow v$ be the edges of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ and $\operatorname{Tw}(\operatorname{\mathcal{E}})$ described in Example 8.1.3.6. Then the morphism $F(u_{R}, v_{R} )$ belongs to $R$.

Identifying $F$ with a morphism $f: \operatorname{\mathcal{E}}\rightarrow \operatorname{Cospan}( \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) )$, we see that $f$ factors through $\operatorname{Cospan}( \operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) )$ if and only if it satisfies condition $(1_{u,v} )$ whenever $v$ is a degenerate edge of $\operatorname{\mathcal{E}}$. Under this assumption, $f$ factors through $\operatorname{Cospan}^{L',R'}( \operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) )$ if and only if, for every edge $u: D' \rightarrow D$ of $\operatorname{\mathcal{D}}$ and every edge $v: E' \rightarrow E$ of $\operatorname{\mathcal{E}}$, the diagram $F$ satisfies the following condition:

$(2_{u,v})$

The morphism $F( \operatorname{id}_{u}, v )$ belongs to $R$.

To complete the proof, it suffices to observe that if condition $(1_{ u, \operatorname{id}_{E} } )$ is satisfied, then condition $(1_{u,v} )$ is equivalent to condition $(2_{u,v} )$. This follows by applying the left two-out-of-three property to the upper triangle appearing in the diagram

$\xymatrix@R =50pt@C=50pt{ F( \operatorname{id}_{D}, \operatorname{id}_{E} ) \ar [r]^-{ F( u_{R}, \operatorname{id}) } \ar [d]^{ F( \operatorname{id}, v_ R) } \ar [dr]^{ F( u_{R}, v_{R} ) } & F(u, \operatorname{id}_ E) \ar [d]^{ F( \operatorname{id}_ u, v_ R) } \\ F( \operatorname{id}_ D, v) \ar [r]^-{ F( u_ R, \operatorname{id}_ v) } & F( u, v ). }$
$\square$

Lemma 8.1.7.13. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\operatorname{\mathcal{D}}$ be a simplicial set, and suppose we are given a pair of diagrams $f,g: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}})$, corresponding to diagrams $F,G: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The diagrams $f$ and $g$ are isomorphic when viewed as objects of the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) )$.

$(2)$

The diagrams $F$ and $G$ are isomorphic when viewed as objects of the $\infty$-category $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$.

Proof. Let $\operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ spanned by those functors $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ which correspond to diagrams $\operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso} }(\operatorname{\mathcal{C}})$. Lemma 8.1.7.12, identifies $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) )$ with the $\infty$-category $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) )$. We are therefore reduced to proving that $F$ and $G$ are isomorphic when viewed as objects of the $\infty$-category $\operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ if and only if they are isomorphic when viewed as objects of the $\infty$-category $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) )$. This is a special case of Corollary 8.1.6.12. $\square$

Proof of Proposition 8.1.7.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. We wish to show that the comparison map $\rho _{+}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}})$ of Construction 8.1.7.1. Let $\operatorname{\mathcal{D}}$ be a simplicial set; we will show that composition with $\rho _{+}$ induces a bijection $\theta : \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) )^{\simeq } )$.

Let $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}$ denote the projection map, and let $W$ be the collection of all edges $e$ of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ such that $\lambda _{+}(e)$ is a degenerate edge of $\operatorname{\mathcal{D}}$. Let $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}})[W^{-1}], \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ spanned by those diagrams $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ which carry each edge of $W$ to an isomorphism in $\operatorname{\mathcal{C}}$ (Notation 6.3.1.1). Using Lemmas 8.1.7.9 and 8.1.7.13, we can identify $\theta$ with the map $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}})[W^{-1}], \operatorname{\mathcal{C}})^{\simeq } )$ given by composition $\lambda _{+}$. To complete the proof, it will suffice to show that $\lambda _{+}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ with respect to $W$, in the sense of Definition 6.3.1.9. This follows from Corollary 6.3.6.4, since the morphism $\lambda _{+}$ is universally localizing (Corollary 8.1.2.4). $\square$

Variant 8.1.7.14. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the projection map $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ determines a morphism of simplicial sets $\rho _{-}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. If $\sigma$ is an $n$-simplex of $\operatorname{\mathcal{C}}^{\operatorname{op}}$, which we display informally as a diagram

$X_0 \xleftarrow {f_1} X_1 \xleftarrow {f_2} X_2 \leftarrow \cdots \xleftarrow {f_ n} X_ n,$

then $\rho _{-}(\sigma )$ is an $n$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ which is depicted informally by the diagram

$\xymatrix@R =20pt@C=20pt{ X_0 \ar [dr]^-{\operatorname{id}} & & X_1 \ar [dr]^{\operatorname{id}} \ar [dl]^{f_1} & & \cdots \ar [dl] \ar [dr] & & X_{n-1} \ar [dr]^{\operatorname{id}} \ar [dl]^{f_{n-1}} & & X_ n \ar [dl]^{f_ n} \\ & X_0 \ar [dr]^{\operatorname{id}} & & X_1 \ar [dl]^{f_1} \ar [dr]^{\operatorname{id}} & \cdots & X_{n-2} \ar [dl]^{f_{n-2}} \ar [dr]^{\operatorname{id}} & & X_{n-1} \ar [dl]^{f_{n-1}} & \\ & & \cdots \ar [dr]^{\operatorname{id}} & & \cdots \ar [dl]^{f_1} \ar [dr]^{\operatorname{id}} & & \cdots \ar [dl]^{f_2} & & \\ & & & X_{0} \ar [dr]^{\operatorname{id}} & & X_{1} \ar [dl]^{f_1} & & & \\ & & & & X_0. & & & & }$

If $\operatorname{\mathcal{C}}$ is an $\infty$-category, then $\rho _{-}$ restricts an equivalence $\operatorname{\mathcal{C}}^{\operatorname{op}} \hookrightarrow \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}})$, where $\operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{C}})$ is the simplicial subset $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{C}})$ where $L$ is the collection of isomorphisms in $\operatorname{\mathcal{C}}$, and $R$ is the collection of all morphisms of $\operatorname{\mathcal{C}}$.