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11.9.1 Obsolete Discussion of Balanced Couplings

We close this section by describing an example of a balanced coupling which will play an important role in §8.6.

Proposition 11.9.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{ev}_{0}, \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the evaluation functors. Let $L$ be the collection of all morphisms $u$ in $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ such that $\operatorname{ev}_0(u)$ is an isomorphism in $\operatorname{\mathcal{C}}$, and let $R$ be the collection of all morphisms $u$ in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ such that $\operatorname{ev}_{1}(u)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then the maps $\operatorname{Cospan}( \operatorname{ev}_0 )$ and $\operatorname{Cospan}( \operatorname{ev}_1 )$ determine a balanced coupling

$\lambda : \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}^{ \mathrm{iso}, \mathrm{all} }( \operatorname{\mathcal{C}}) \times \operatorname{Fun}^{ \mathrm{all}, \mathrm{iso} }( \operatorname{\mathcal{C}}).$

The proof of Proposition 11.9.1.1 will require some preliminaries. The first step is to establish the following:

Lemma 11.9.1.2. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the morphism

$\lambda : \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}^{ \mathrm{iso}, \mathrm{all} }( \operatorname{\mathcal{C}}) \times \operatorname{Fun}^{ \mathrm{all}, \mathrm{iso} }( \operatorname{\mathcal{C}})$

is a left fibration of $\infty$-categories.

The proof of Lemma 11.9.1.2 is straightforward but somewhat tedious; we therefore defer the argument to §8.6.6, where we prove a more a general statement (Lemma 11.9.1.11). It follows from Lemma 11.9.1.2 that we can view the map

$\lambda : \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}^{ \mathrm{iso}, \mathrm{all} }( \operatorname{\mathcal{C}}) \times \operatorname{Fun}^{ \mathrm{all}, \mathrm{iso} }( \operatorname{\mathcal{C}})$

as a coupling of the $\infty$-category $\operatorname{Fun}^{ \mathrm{all}, \mathrm{iso} }( \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}^{ \mathrm{iso}, \mathrm{all} }( \operatorname{\mathcal{C}})^{\operatorname{op}}$ with itself (see Remark 8.1.6.2). To deduce Proposition 11.9.1.1, we will compare $\lambda$ with the twisted arrow coupling $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ of Example 8.2.0.2.

***

Construction 11.9.1.3. Let $Q$ be a partially ordered set and let $Q^{\operatorname{op}}$ denote the opposite partially ordered set. To avoid confusion, for each element $q \in Q$, we write $q^{\operatorname{op}}$ for the corresponding element of $Q^{\operatorname{op}}$. Let $\operatorname{Tw}(Q)$ denote the twisted arrow category of $Q$ (Example 8.1.0.5), which we identify with the partially ordered subset of $Q^{\operatorname{op}} \times Q$ consisting of those pairs $(p^{\operatorname{op}}, q)$ satisfying $p \leq q$. We then have a morphism of partially ordered sets $\xi _{Q}: \operatorname{Tw}(Q) \times [1] \rightarrow Q^{\operatorname{op}} \star Q$, given concretely by the formulae

$\xi _{Q}( p^{\operatorname{op}}, q, i ) = \begin{cases} p^{\operatorname{op}} & \text{ if } i = 0 \\ q & \text{ if } i = 1. \end{cases}$

Let $\operatorname{\mathcal{C}}$ be a simplicial set. For every nonempty finite linearly ordered set $Q$, we obtain a map

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(Q), \operatorname{Tw}(\operatorname{\mathcal{C}}) ) & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(Q^{\operatorname{op}} \star Q), \operatorname{\mathcal{C}}) \\ & \xrightarrow { \circ \xi _{Q}} & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \operatorname{Tw}(Q) \times [1]), \operatorname{\mathcal{C}}) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}( \operatorname{N}_{\bullet }(Q) ) \times \Delta ^1, \operatorname{\mathcal{C}}) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}( \operatorname{N}_{\bullet }(Q) ), \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(Q), \operatorname{Cospan}( \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) ) ). \end{eqnarray*}

This construction depends functorially on $Q$, and therefore determines a morphism of simplicial sets $\Xi : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) )$.

Remark 11.9.1.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the morphism $\Xi$ of Construction 11.9.1.3 can be described concretely on low-dimensional simplices as follows:

• On vertices, $\Xi$ is given by the formula $\Xi (f) = f$. Here we abuse notation by identifying vertices of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ and $\operatorname{Cospan}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) )$ with edges of the simplicial set $\operatorname{\mathcal{C}}$.

• Let $e: f_0 \rightarrow f_1$ be an edge of the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$, which we identify with a $3$-simplex $\sigma$ of $\operatorname{\mathcal{C}}$ displayed informally in the diagram

$\xymatrix@C =50pt@R=50pt{ X_0 \ar [d]^{ f_0 } & X_1 \ar [l]_{g} \ar [d]^{f_1} \\ Y_0 \ar [r]^-{h} & Y_1. }$

Then $\Xi (e)$ is the cospan from $f_0$ to $f_1$ in the simplicial set $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ depicted informally in the diagram

$\xymatrix@C =50pt@R=50pt{ X_0 \ar [d]^{f_0} \ar [r]^-{\operatorname{id}} & X_0 \ar [d] & X_1 \ar [d]^{f_1} \ar [l]_{g} \\ Y_0 \ar [r]^-{h} & Y_1 & Y_1. \ar [l]_{\operatorname{id}} }$

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. It follows from Remark 11.9.1.4 that the morphism $\Xi : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) )$ of Construction 11.9.1.3 factors through the simplicial subset $\operatorname{Cospan}^{L,R}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \subseteq \operatorname{Cospan}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) )$ appearing in the statement of Proposition 11.9.1.1. Unwinding the definitions, we obtain a morphism of couplings

11.13
$$\begin{gathered}\label{equation:Xi-to-cospan} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \Xi } \ar [d] & \operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}}) \ar [d]^{ \lambda } \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [r]^-{ \rho _{-} \times \rho _{+} } & \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}}) \times \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) } \end{gathered}$$

where the vertical maps are the left fibrations of Proposition 8.1.1.11 and Lemma 11.9.1.2, and $\rho _{+}$ and $\rho _{-}$ are given by Construction 8.1.7.1 and Variant 8.1.7.14. By virtue of Remark 8.2.6.4, Proposition 11.9.1.1 is a consequence of the following more precise result:

Proposition 11.9.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the diagram (11.13) is an equivalence of couplings (in the sense of Exercise 8.2.2.7)

Proof. It follows from Proposition 8.1.7.6 that the inclusion maps

$\rho _{-}: \operatorname{\mathcal{C}}^{\operatorname{op}} \hookrightarrow \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}}) \quad \quad \rho _{+}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$

are equivalences of $\infty$-categories. By virtue of Corollary 5.1.7.16, it will suffice to show that for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism $\Xi$ induces a homotopy equivalence of Kan complexes

$\Xi _{X,Y}: \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{Cospan}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ).$

We complete the proof by observing that $\Xi _{X,Y}$ fits into a commutative diagram of Kan complexes

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( X, Y) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [d] \\ \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \ar [r]^-{ \Xi _{X,Y} } & \operatorname{Cospan}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) }$

where the left vertical map is the homotopy equivalence of Corollary 8.1.2.10, the right vertical map is the homotopy equivalence of Example 8.1.7.7, and the upper horizontal map is the homotopy equivalence of Proposition 4.6.5.10. $\square$

Corollary 11.9.1.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is an isomorphism if and only if it is universal with respect to the balanced coupling

$\lambda : \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}^{ \mathrm{iso}, \mathrm{all} }( \operatorname{\mathcal{C}}) \times \operatorname{Fun}^{ \mathrm{all}, \mathrm{iso} }( \operatorname{\mathcal{C}})$

of Proposition 11.9.1.1 (where we abuse notation by identifying $f$ with an object of the $\infty$-category $\operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) )$).

Proof. Let us abuse notation further by identifying $f$ with an object of the twisted arrow $\infty$-category $\operatorname{Tw}(\operatorname{\mathcal{C}})$, so that the comparison functor $\Xi : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) )$ of Construction 11.9.1.3 satisfies $\Xi (f) = f$ (Remark 11.9.1.4). By virtue of Proposition 11.9.1.5 and Remark 8.3.2.8, we are reduced to showing that $f$ is an isomorphism if and only if it is universal with respect to the twisted arrow coupling $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$, which follows from Example 8.2.1.5. $\square$

Example 11.9.1.7 (Path Fibrations). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, and let

$\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$

denote the functors given by evaluation at the vertices $0,1 \in \Delta ^1$. Then $\operatorname{ev}_0$ is a cartesian fibration, and $\operatorname{ev}_1$ is a cocartesian fibration (Example 5.3.7.4). Let $L$ denote the collection of $\operatorname{ev}_1$-cartesian morphisms of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (that is, the collection of morphisms $f$ such that $\operatorname{ev}_0(f)$ is an isomorphism of $\operatorname{\mathcal{C}}$), and let $R$ denote the collection of $\operatorname{ev}_0$-cocartesian morphisms of $\operatorname{\mathcal{C}}$ (that is, the collection of morphisms $f$ such that $\operatorname{ev}_1(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$). Applying the construction of Notation 8.6.3.1 to the cocartesian fibration $\operatorname{ev}_1$, we obtain an $\infty$-category $\operatorname{Cospan}^{\operatorname{CCart}}( \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}})$. The morphism $\Xi$ of Construction 11.9.1.3 determines a morphism $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}^{\operatorname{CCart}}( \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}})$ which fits into a commutative diagram

$\xymatrix@R =50pt@C=40pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{\Xi } \ar [d] & \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})/\operatorname{\mathcal{C}}) \ar [d] \ar [r] & \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \ar [d]^{\operatorname{ev}_0, \operatorname{ev}_1} \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [r] & \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{C}}) \times \operatorname{\mathcal{C}}\ar [r] & \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}}) \times \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}). }$

Here the right half of the diagram is a pullback square, the vertical maps are left fibrations (Proposition 8.1.1.11 and Lemma 11.9.1.2), the lower horizontal maps are equivalences of $\infty$-categories (Proposition 8.1.7.6). Applying Proposition 11.9.1.5 (and Corollary 4.5.2.29), we deduce that the $\Xi : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})/\operatorname{\mathcal{C}})$ is an equivalence of $\infty$-categories.

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Our goal below is to show that the induced map $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian conjugate of $U$.

To show that $U^{\vee }: \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian dual of $U$, we will need an auxiliary construction.

Notation 11.9.1.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories, let $L$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$, and let $R$ denote the collection of all morphisms $f$ of $\operatorname{\mathcal{E}}$ such that $U(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$. We let

$\operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) = \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}})$

denote the relative exponential of Construction 4.5.9.1. Evaluation at the vertices $0,1 \in \Delta ^1$ determines evaluation functors $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$. Let $\widetilde{L}$ denote the collection of all morphisms $f$ of $\operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ such that $\operatorname{ev}_0(f)$ is $U$-cocartesian, and let $\widetilde{R}$ denote the collection of all morphisms $f$ of $\operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ such that $\operatorname{ev}_1(f)$ is an isomorphism. The evaluation maps $\operatorname{ev}_0$ and $\operatorname{ev}_1$ then induce a functor

$V: \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Cospan}^{L, R}( \operatorname{\mathcal{E}}) \times _{ \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}).$

We will prove the following:

Proposition 11.9.1.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories, let $V_{-}: \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Lemma 8.6.3.6, and let $V_{+}: \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Remark 8.1.9.3. Then the functor

$V: \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Cospan}^{L, R}( \operatorname{\mathcal{E}}) \times _{ \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}})$

of Notation 11.9.1.8 is a left fibration, which exhibits $V_{-}$ as a cocartesian dual of $V_{+}$ (in the sense of Definition 8.6.4.1).

Example 11.9.1.10. In the special case $\operatorname{\mathcal{C}}= \Delta ^0$, Proposition 11.9.1.9 reduces to the assertion that the map

$\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{E}}) \times \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}})$

is a balanced coupling of $\infty$-categories, which is the content of Proposition 11.9.1.1.

Proof of Theorem 11.9.1.12 from Proposition 11.9.1.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $U^{\vee }: \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the projection map; we wish to show that $U^{\vee }$ is a cocartesian dual of $U$. Using Corollary 5.6.7.3, we can choose a pullback diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}', }$

where $U'$ is a cocartesian fibration of $\infty$-categories. Using Remarks 8.6.4.4 and 8.6.3.4, we can replace $U$ by $U'$ and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is an $\infty$-category. In this case, we have commutative diagrams

$\xymatrix@R =50pt@C=50pt{ \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \ar [d]^{U^{\vee }} \ar [r] & \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}}) \ar [d]^{V_{-}} & \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}}) \ar [d]^{ V_{+} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso} }(\operatorname{\mathcal{C}}) & \operatorname{\mathcal{C}}\ar [r] & \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}), }$

where the vertical maps are cocartesian fibrations, the diagram on the left is a pullback square, and the diagram on the right is a categorial pullback square (the horizontal maps are equivalences of $\infty$-categories by virtue of Proposition 8.1.7.6). Using Remark 8.6.4.4 again, we are reduced to showing that $V_{-}$ is a cocartesian dual of $V_{+}$, which follows from Proposition 11.9.1.9. $\square$

We now turn to the proof of Proposition 11.9.1.9.

Lemma 11.9.1.11. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories. Then the functor

$V: \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Cospan}^{L, R}( \operatorname{\mathcal{E}}) \times _{ \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}})$

of Notation 11.9.1.8 is a left fibration.

Proof. Let $\pi : \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}$ denote the functor given by projection onto the second factor. Let $L'$ denote the collection of all $\pi$-cocartesian morphisms of $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$: that is, morphisms $(u,v)$ where $u$ is a $U$-cocartesian morphism in $\operatorname{\mathcal{E}}$. Let $R'$ denote the collection of all morphisms $(u,v)$ of $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ where $v$ is an isomorphism in $\operatorname{\mathcal{E}}$. It follows from Proposition 8.1.9.10 (and Example 8.1.6.6) that the pair $(L',R')$ is pushout-compatible, in the sense of Definition 8.1.6.5. Moreover, we have a canonical isomorphism of simplicial sets

$\operatorname{Cospan}^{L',R'}(\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}) \simeq \operatorname{Cospan}^{L,R}( \operatorname{\mathcal{E}}) \times _{ \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})} \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}}).$

Using Lemma 8.6.3.6, we see that $\pi$ induces a cocartesian fibration $\operatorname{Cospan}^{L',R'}(\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}})$, where the target is an $\infty$-category (Proposition 8.1.7.5). It follows that the simplicial set $\operatorname{Cospan}^{L',R'}(\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}})$ is also an $\infty$-category.

Let $\operatorname{ev}_0, \operatorname{ev}_{1}: \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$ denote functors given by evaluation at $0,1 \in \Delta ^1$, so that $\operatorname{ev}_0$ and $\operatorname{ev}_1$ determine a functor

$\operatorname{ev}: \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}.$

By construction we have $\widetilde{L} = \operatorname{ev}^{-1}( L' )$ and $\widetilde{R} = \operatorname{ev}^{-1}( R' )$. Note that $\operatorname{ev}$ is a pullback of the map $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) = \operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Moreover, we can identify $V$ with the map

$\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Cospan}^{ L', R'}( \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}})$

induced by $\operatorname{ev}$. By virtue of Example 8.1.10.13, to show that $V$ is a left fibration, it will suffice to verify the following:

$(0)$

Every element of $\widetilde{L}$ is $\operatorname{ev}$-cocartesian, and every element of $\widetilde{R}$ is $\operatorname{ev}$-cartesian. This follows from Lemma 5.3.7.1.

$(1)$

Fix an object $C \in \operatorname{\mathcal{C}}$ and a morphism $f: X \rightarrow Y$ in the $\infty$-category $\operatorname{\mathcal{E}}_{C}$, which we identify with an object of the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. Let $(u,v): (X',Y') \rightarrow (X,Y)$ be a morphism of $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ which belongs to $R'$ (so that $v$ is an isomorphism in $\operatorname{\mathcal{E}}$). Then we can write $(u,v) = \operatorname{ev}(w)$ for some morphism $w: f' \rightarrow f$ in the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ (the morphism $w$ then belongs to $\widetilde{R}$ and is therefore automatically $\operatorname{ev}$-cartesian). To prove this, we note that $U(u) = U(v)$ determines an edge $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. Replacing $\operatorname{\mathcal{E}}$ by the fiber product $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, we can reduce to the situation where $\operatorname{\mathcal{C}}= \Delta ^1$ is a standard simplex. In this case, we are reduced to the problem of constructing a diagram $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{E}}$ whose boundary is indicated in the simplex

$\xymatrix@R =50pt@C=50pt{ X' \ar@ {-->}[r]^{f'} \ar [d]^{u} & Y' \ar [d]^{v} \\ X \ar [r]^-{f} & Y, }$

which is possible by virtue of our assumption that $v$ is an isomorphism.

$(2)$

Fix an object $C \in \operatorname{\mathcal{C}}$ and a morphism $f: X \rightarrow Y$ as above, and let $(u,v): (X,Y) \rightarrow (X',Y')$ be a morphism of $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ which belongs to $L'$ (so that $u$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$). Then we can write $(u,v) = \operatorname{ev}(w)$, for some morphism $w: f \rightarrow f'$ in the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ (the morphism $w$ then belongs to $\widetilde{L}$ and is therefore automatically $\operatorname{ev}$-cocartesian). This follows from Proposition 5.3.7.2 (or by a direct argument similar to the proof of $(1)$).

$\square$

Proof of Proposition 11.9.1.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories, and let

$V: \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Cospan}^{L, R}( \operatorname{\mathcal{E}}) \times _{ \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}})$

be the left fibration of Lemma 11.9.1.11. We wish to show that the left fibraiton $V$ exhibits $V_{-}: \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ as a cocartesian dual of $V_{+}: \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$. For each object $C \in \operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ denote the corresponding fiber of $U$. Then we have a canonical isomorphism

$\{ C\} \times _{ \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) } \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) ) \simeq \operatorname{Cospan}^{ \widetilde{L}_{C}, \widetilde{R}_{C} }( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}_{C} ) ),$

where $\widetilde{L}_{C}$ denotes the collection of all morphisms of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}_{C} )$ for which the image in $\operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{E}}_{C} )$ is an isomorphism, and $\widetilde{R}_{C}$ denotes the collection of morphisms $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}_{C} )$ for which the image in $\operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{E}}_{C} )$ is an isomorphism. The left fibration $V$ then restricts to a coupling of $\infty$-categories

$V_{C}: \operatorname{Cospan}^{ \widetilde{L}_{C}, \widetilde{R}_{C} }( \operatorname{\mathcal{E}}_{C} ) \rightarrow \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{E}}_ C) \times \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}_{C} )$

which is balanced by virtue of Proposition 11.9.1.1. Moreover, if $f$ is an object of the $\infty$-category $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) )$ satisfying $V(f) = C$, then $f$ is universal (with respect to the coupling $V_{C}$) if and only if it is an isomorphism when regarded as a morphism in the $\infty$-category $\operatorname{\mathcal{E}}_{C}$ (Corollary 11.9.1.6).

Let $u: f \rightarrow g$ be a morphism in the $\infty$-category $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{Fun}( \operatorname{\mathcal{C}}\times \Delta ^1 / \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) )$, having image $\overline{u}: C \rightarrow D$ in the $\infty$-category $\operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$. Assume that the image of $u$ in $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is $V_{-}$-cocartesian and that the image of $u$ in $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}})$ is $V_{+}$-cocartesian. To complete the proof, we must show that if $f$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{E}}_{C}$, then $g$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{E}}_{D}$. To prove this, let us identify $u$ with a commutative diagram

$\xymatrix@R =50pt@C=50pt{ X_{-} \ar [d]^{f} \ar [r]^-{s_{-}} & B_{-} \ar [d]^{ h } & Y_{-} \ar [l]_{t_{-}} \ar [d]^{g} \\ X_{+} \ar [r]^-{ s_{+} } & B_{+} & Y_{+} \ar [l]_{t_{+}} }$

in the $\infty$-category $\operatorname{\mathcal{E}}$, where $s_{-}$ is $U$-cocartesian and $t_{+}$ is an isomorphism. Since the image of $u$ in $\operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso} }(\operatorname{\mathcal{E}})$ is $V_{+}$-cocartesian, the morphism $s_{+}$ is $U$-cocartesian. Applying Corollary 5.1.2.4, we deduce that the morphism $h$ is $U$-cocartesian. Since the image of $u$ in $\operatorname{Cospan}^{ L,R}(\operatorname{\mathcal{E}})$ is $V_{-}$-cocartesian, the morphism $t_{-}$ is an isomorphism. Applying Corollary 5.1.2.5, we deduce that $g$ is $U$-cocartesian when regarded as a morphism of $\operatorname{\mathcal{E}}$, and is therefore an isomorphism in the $\infty$-category $\operatorname{\mathcal{E}}_{D}$ (Example 5.1.3.6). $\square$

Theorem 11.9.1.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then the projection map $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is also a cocartesian fibration, which is a cocartesian dual of $U$.

We will give the proof of Theorem 11.9.1.12 at the end of this section.

Corollary 11.9.1.13 (The Dual of a Path Fibration). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the projection map $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ of Notation 8.1.1.6 is a cocartesian dual of the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$.