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8.6.5 Cocartesian Duality via Cospans

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Theorem 8.6.4.1 asserts that $U$ admits a cocartesian dual $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$, which is uniquely determined up to equivalence. In this section, we describe an alternative construction of $U^{\vee }$ due to Barwick-Glasman-Nardin ([BGN]), which uses the restricted cospan construction of ยง8.1.6.

Notation 8.6.5.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set, and let $\rho _{+}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ be the inclusion map of Construction 8.1.7.1. For every morphism of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, we let $\operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ denote the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{Cospan}(\operatorname{\mathcal{C}})} \operatorname{Cospan}(\operatorname{\mathcal{E}})$.

Suppose that $U$ is a cocartesian fibration, and let $L$ be the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$. In this case, we define $\operatorname{Cospan}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ to be the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{Cospan}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{L,\mathrm{all}}(\operatorname{\mathcal{E}})$, which we regard as a simplicial subset of $\operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$.

Remark 8.6.5.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Low-dimensional simplices of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ can be described as follows:

  • Vertices of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ can be identified with vertices of the simplicial set $\operatorname{\mathcal{E}}$.

  • Let $X$ and $Y$ be vertices of $\operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$. Then edges $e: X \rightarrow Y$ of $\operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ can be identified with pairs of edges $X \xrightarrow {f} B \xleftarrow {g} Y$ in the simplicial set $\operatorname{\mathcal{E}}$ having the property that $U(g)$ is a degenerate edge of $\operatorname{\mathcal{C}}$. If $U$ is a cocartesian fibration, then the edge $e$ belongs to $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ if and only if $f$ is $U$-cocartesian.

Example 8.6.5.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. If $\operatorname{\mathcal{C}}= \Delta ^0$, then $\operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ can be identified with the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{E}})$. If, in addition, $U$ is a cocartesian fibration, then $\operatorname{\mathcal{E}}$ is an $\infty $-category and $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ can be identified with the simplicial subset $\operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{E}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{E}})$ of Variant 8.1.7.14. In this case, Proposition 8.1.7.6 guarantees that $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ is an $\infty $-category which is equivalent to $\operatorname{\mathcal{E}}^{\operatorname{op}}$.

Remark 8.6.5.4 (Base Change). Suppose we are given a pullback diagram of simplicial sets

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

Then we have a canonical isomorphism $\operatorname{Cospan}(\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}') \simeq \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$. If $U$ is a cocartesian fibration, then $U'$ is also a cocartesian fibration, and we also obtain an isomorphism $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}') \simeq \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$. In particular, for each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ is isomorphic to the $\infty $-category $\operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{E}}_{C} )$, which is equivalent to the $\infty $-category $\operatorname{\mathcal{E}}_{C}^{\operatorname{op}}$.

Example 8.6.5.5 (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 8.6.5.1 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 8.2.6.11 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 8.2.6.10), the lower horizontal maps are equivalences of $\infty $-categories (Proposition 8.1.7.6). Applying Proposition 8.2.6.13 (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.

We can now state our main result.

Theorem 8.6.5.6. 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 8.6.5.6 at the end of this section.

Corollary 8.6.5.7 (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 $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$.

The proof of Theorem 8.6.5.6 will require some preliminaries. Our first goal is to show that if $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration, then the projection map $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is also a cocartesian fibration.

Lemma 8.6.5.8. 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 $e: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ such that $U(e)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then $\operatorname{Cospan}(U): \operatorname{Cospan}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ restricts to a cocartesian fibration of $\infty $-categories $V: \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}})$. Moreover, an edge $e: X \rightarrow Y$ of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is $V$-cocartesian if and only if it satisfies the following condition:

$(\ast )$

The edge $e$ corresponds to a cospan $X \xrightarrow {\ell } B \xleftarrow {r} Y$ in $\operatorname{\mathcal{E}}$, where $\ell $ is $U$-cocartesian and $r$ is an isomorphism.

Proof. Let $L_0$ be the collection of all morphisms in $\operatorname{\mathcal{C}}$, and let $R_0$ be the collection of all morphisms of $\operatorname{\mathcal{C}}$. Then $L_0$ and $R_0$ are pushout-compatible, in the sense of Definition 8.1.6.5 (Example 8.1.6.6). Moreover, $U$ is a Beck-Chevalley fibration relative to $(R_0, L_0)$ (Example 8.1.10.8). Since $\operatorname{Cospan}^{L_0, R_0}(\operatorname{\mathcal{C}}) = \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}})$ is an $\infty $-category, the desired result follows from (the duals of) Theorem 8.1.10.9 and Remark 8.1.10.11. $\square$

Remark 8.6.5.9. 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 $g$ of $\operatorname{\mathcal{E}}$ such that $U(g)$ is an isomorphism in $\operatorname{\mathcal{C}}$. We then have a commutative diagram of pullback squares

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

where the vertical map in the middle is a cocartesian fibration (Lemma 8.6.5.8), and the horizontal map on the lower left is an equivalence of $\infty $-categories (Proposition 8.1.7.6). It follows that the projection map $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of $\infty $-categories. Moreover, Corollary 4.5.2.29 implies that the inclusion $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is an equivalence of $\infty $-categories.

Lemma 8.6.5.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then the projection map $U^{\vee }: \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is also a cocartesian fibration. Moreover, an edge $X \rightarrow Y$ of $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ is $U^{\vee }$-cocartesian if and only if it corresponds to a cospan $X \xrightarrow {\ell } B \xleftarrow {r} Y$ in $\operatorname{\mathcal{E}}$, where $\ell $ is $U$-cocartesian and $r$ is an isomorphism in the $\infty $-category $\{ U(Y) \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.

Proof. Using Proposition 5.1.4.7 and Remark 8.6.5.4, we can reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. In particular, $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, the desired result follows by combining Remark 8.6.5.9 with Lemma 8.6.5.8. $\square$

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 8.6.5.11. 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 8.6.5.12. 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 8.6.5.8, 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 8.6.5.11 is a left fibration, which exhibits $V_{-}$ as a cocartesian dual of $V_{+}$ (in the sense of Definition 8.6.3.1).

Example 8.6.5.13. In the special case $\operatorname{\mathcal{C}}= \Delta ^0$, Proposition 8.6.5.12 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 8.2.6.9.

Proof of Theorem 8.6.5.6 from Proposition 8.6.5.12. 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 8.6.3.4 and 8.6.5.4, 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 8.6.3.4 again, we are reduced to showing that $V_{-}$ is a cocartesian dual of $V_{+}$, which follows from Proposition 8.6.5.12. $\square$

We now turn to the proof of Proposition 8.6.5.12.

Lemma 8.6.5.14. 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 8.6.5.11 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 8.6.5.8, 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 8.6.5.12. 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 8.6.5.14. 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 8.2.6.9. 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 8.2.6.14).

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). $\square$