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8.6.3 Uniqueness of Conjugate Fibrations

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. It follows from Corollary 8.6.2.4 that $U$ admits a cartesian conjugate $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$. When $\operatorname{\mathcal{C}}$ is an $\infty $-category, Proposition 8.6.2.8 guarantees that $U^{\dagger }$ is unique up to equivalence: any cartesian conjugate of $U$ is equivalent to the cartesian fibration

\[ V: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \]

of Construction 8.6.2.2. In this section, we will establish a more general uniqueness result, which applies even when $\operatorname{\mathcal{C}}$ is not an $\infty $-category. Our proof will use an alternative construction of the conjugate fibration due to Barwick-Glasman-Nardin ([BGN]), which involves the restricted cospan construction of ยง8.1.6.

Notation 8.6.3.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\rho _{-}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ be the comparison map of Variant 8.1.7.14. For every cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, we let $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ denote the fiber product $\operatorname{Cospan}(\operatorname{\mathcal{E}})^{\mathrm{all},R}(\operatorname{\mathcal{E}}) \times _{\operatorname{Cospan}(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}^{\operatorname{op}}$, where $R$ is the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$ (see Definition 8.1.6.1).

Remark 8.6.3.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Low-dimensional simplices of the simplicial set $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ can be described as follows:

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

  • Let $X$ and $Y$ be vertices of $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$. 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 $g$ is $U$-cocartesian and $U(f)$ is a degenerate edge of $\operatorname{\mathcal{C}}$.

Example 8.6.3.3. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category, so the projection map $\operatorname{\mathcal{E}}\rightarrow \Delta ^{0}$ is a cocartesian fibration of simplicial sets. Then the simplicial set $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\Delta ^0)$ of Notation 8.6.3.1 can be identified with the simplicial set $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}})$ of Construction 8.1.7.2. In particular, $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\Delta ^0)$ is an $\infty $-category (Proposition 8.1.7.5), and Proposition 8.1.7.6 supplies an equivalence of $\infty $-categories $\rho _{+}: \operatorname{\mathcal{E}}\rightarrow \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\Delta ^0)$.

Remark 8.6.3.4 (Base Change). Suppose we are given a pullback diagram of simplicial sets

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

where $U$ and $U'$ are cocartesian fibrations. Then we have a canonical isomorphism of simplicial sets

\[ \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}') \simeq \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{\mathcal{C}}'^{\operatorname{op}}. \]

In particular, for each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{Cospan}_{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \{ C\} $ is isomorphic to the $\infty $-category $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}_{C} )$, which is equivalent to the $\infty $-category $\operatorname{\mathcal{E}}_{C}$.

We can now state a preliminary version of our main result:

Proposition 8.6.3.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then the projection map $V: \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian fibration, which is a cartesian conjugate of $U$.

The proof of Proposition 8.6.3.5 will require some preliminaries. Our first step is to show that the projection map $V$ is a cartesian fibration.

Lemma 8.6.3.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories, let $R$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$, and let $L$ 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 cartesian fibration of $\infty $-categories $V: \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{C}})$. Moreover, an edge $e: X \rightarrow Y$ of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is $V$-ccartesian 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 an isomorphism and $r$ is $U$-cocartesian.

Proof. Let $L_0$ be the collection of all isomorphisms 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 Theorem 8.1.10.9 and Remark 8.1.10.11. $\square$

Remark 8.6.3.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories, let $R$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$, and let $L$ 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

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

where the vertical map in the middle is a cartesian fibration (Lemma 8.6.3.6), 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 $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian fibration of $\infty $-categories. Moreover, Corollary 4.5.2.29 implies that the inclusion $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is an equivalence of $\infty $-categories.

Lemma 8.6.3.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then the projection map $V: \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is also a cartesian fibration. Moreover, an edge $X \rightarrow Y$ of $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ is $V$-cartesian if and only if it corresponds to a cospan $X \xrightarrow {\ell } B \xleftarrow {r} Y$ in $\operatorname{\mathcal{E}}$, where $r$ is $U$-cocartesian and $\ell $ is an isomorphism in the $\infty $-category $\{ U(X) \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.

Proof. Using Proposition 5.1.4.8 and Remark 8.6.3.4, 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 8.6.3.7 with Lemma 8.6.3.6. $\square$

Construction 8.6.3.9. Let $\operatorname{\mathcal{C}}$ be a simplicial set, and suppose we are given a pair of morphisms $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$. Let $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ be a morphism of simplicial sets for which the diagram

8.76
\begin{equation} \begin{gathered}\label{equation:compare-dagger-with-conjugacy} \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{ U } \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}

is commutative. Let $\lambda _{+}: \operatorname{Tw}( \operatorname{\mathcal{E}}^{\dagger } ) \rightarrow \operatorname{\mathcal{E}}^{\dagger }$ be the projection map of Notation 8.1.1.6 and let $\iota : \operatorname{Tw}(\operatorname{\mathcal{C}}^{\operatorname{op}}) \xrightarrow {\sim } \operatorname{Tw}(\operatorname{\mathcal{C}})$ be the isomorphism described in Remark 8.1.1.7. Then we can extend (8.76) to a commutative diagram

8.77
\begin{equation} \begin{gathered}\label{equation:compare-dagger-with-conjugacy2} \xymatrix@C =50pt@R=50pt{ \operatorname{Tw}(\operatorname{\mathcal{E}}^{\dagger }) \ar [r]^-{(\lambda _{+}, \iota \circ \operatorname{Tw}(U^{\dagger }))} \ar [d]^{ \operatorname{Tw}(U^{\dagger }) } & \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{ U } \\ \operatorname{Tw}( \operatorname{\mathcal{C}}^{\operatorname{op}} ) \ar [r]^-{ \iota } & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}

Using Proposition 8.1.3.7, we can identify the outer rectangle with a diagram

8.78
\begin{equation} \begin{gathered}\label{equation:compare-dagger-with-conjugacy3} \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \ar [r] \ar [d]^{U^{\dagger }} & \operatorname{Cospan}( \operatorname{\mathcal{E}}) \ar [d]^{ \operatorname{Cospan}(U) } \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \ar [r]^-{\rho _{-}} & \operatorname{Cospan}( \operatorname{\mathcal{C}}), } \end{gathered} \end{equation}

which we can identify with a comparison map $\Psi : \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{E}}) \times _{\operatorname{Cospan}(\operatorname{\mathcal{C}})} \operatorname{\mathcal{C}}^{\operatorname{op}}$.

Remark 8.6.3.10. In the situation of Construction 8.6.3.9, the morphism $\Psi $ can be described explicitly on low-dimensional simplices as follows:

  • If $X$ is a vertex of $\operatorname{\mathcal{E}}^{\dagger }$ having image $C = U^{\dagger }(X)$, then $\Psi (X)$ is the vertex of $\operatorname{Cospan}( \operatorname{\mathcal{E}})$ corresponding to the vertex $T( X, \operatorname{id}_{ C } ) \in \operatorname{\mathcal{E}}$.

  • Let $X$ and $Y$ be vertices of $\operatorname{\mathcal{E}}^{\dagger }$, having images $C= U^{\dagger }(X)$ and $D = U^{\dagger }(Y)$. Let $f: X \rightarrow Y$ be an edge of $\operatorname{\mathcal{E}}^{\dagger }$, and let us identify $U^{\dagger }(f)$ with an edge $e: D \rightarrow C$ in the simplicial set $\operatorname{\mathcal{C}}$. Then $\Psi ( f): \Psi ( X) \rightarrow \Psi ( Y)$ is the edge of $\operatorname{Cospan}(\operatorname{\mathcal{E}})$ corresponding to the pair of edges $T(X, \operatorname{id}_{C} ) \xrightarrow { T( f , e_{R} ) } T( Y, e ) \xleftarrow { T(\operatorname{id}_{Y}, e_{L} ) } T(Y, \operatorname{id}_{D} )$ in $\operatorname{\mathcal{E}}$; here $e_{L}: \operatorname{id}_{D} \rightarrow e$ and $e_{R}: \operatorname{id}_{C} \rightarrow e$ denote the edges of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ described in Example 8.1.3.6.

We will deduce Proposition 8.6.3.5 from the following more precise result:

Proposition 8.6.3.11. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. Suppose we are given a morphism $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ for which the diagram

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{ U } \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}} \]

is commutative. The following conditions are equivalent:

$(a)$

The morphism $T$ exhibits $U^{\dagger }$ as a cartesian conjugate of $U$, in the sense of Definition 8.6.1.1.

$(b)$

The comparison map $\Psi $ of Construction 8.6.3.9 factors through the simplicial set $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ of Notation 8.6.3.1, and the map $\operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ is an equivalence of cartesian fibrations over $\operatorname{\mathcal{C}}$.

Proof. Using Proposition 5.1.7.15, we see that $(b)$ is equivalent to the following three conditions:

$(b_0)$

The map $\Psi $ factors through the simplicial subset $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{E}}) \times _{\operatorname{Cospan}(\operatorname{\mathcal{C}})} \operatorname{\mathcal{C}}^{\operatorname{op}}$.

$(b_1)$

Let $V: \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be the cartesian fibration of Lemma 8.6.3.8. Then $\Psi $ carries $U^{\dagger }$-cartesian edges of $\operatorname{\mathcal{E}}^{\dagger }$ to $V$-cartesian edges of $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$.

$(b_2)$

For each vertex $C \in \operatorname{\mathcal{C}}$, the morphism $\Psi $ restricts to an equivalence of $\infty $-categories

\[ \Psi _{C}: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{Cospan}^{\dagger }( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} = \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}_{C} ). \]

For every edge $e: D \rightarrow C$ of $\operatorname{\mathcal{C}}$, let $e_{L}: \operatorname{id}_{D} \rightarrow e$ and $e_{R}: \operatorname{id}_{C} \rightarrow e$ denote the edges of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ described in Example 8.1.3.6. Using Remark 8.6.3.10, we can rewrite condition $(b_0)$ as follows:

$(b'_0)$

Let $Y$ be a vertex of $\operatorname{\mathcal{E}}^{\dagger }$ having image $D = U^{\dagger }(Y)$ in $\operatorname{\mathcal{C}}$, and let $e: C \rightarrow D$ be an edge of $\operatorname{\mathcal{C}}$. Then $T( \operatorname{id}_{Y}, e_{L} ): T(Y, \operatorname{id}_{D} ) \rightarrow T( Y, e )$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.

Similarly, by combining Remark 8.6.3.10 with the characterization of $V$-cartesian edges supplied by Lemma 8.6.3.8, we can rewrite condition $(b_1)$ as follows:

$(b'_1)$

Let $f: X \rightarrow Y$ be a $U^{\dagger }$-cartesian edge of $\operatorname{\mathcal{E}}^{\dagger }$, and let us identify $U^{\dagger }(f)$ with an edge $e: D \rightarrow C$ of $\operatorname{\mathcal{C}}$. Then $T(f, e_{R} ): T(X, \operatorname{id}_{C}) \rightarrow T(Y,e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$.

Unwinding the definitions, we observe that for each vertex $C \in \operatorname{\mathcal{C}}$, the functor $\Psi _{C}$ factors as a composition

\[ \operatorname{\mathcal{E}}^{\dagger }_{C} \xrightarrow { T_{C} } \operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}_{C} ) \simeq \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} , \]

where the second map is the equivalence of Proposition 8.1.7.6. We can therefore rewrite $(b_2)$ as follows:

$(b'_2)$

For each vertex $C \in \operatorname{\mathcal{C}}$, the morphism $T$ restricts to an equivalence of $\infty $-categories $T_{C}: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$.

The equivalence of $(a)$ and $(b)$ now follows from Proposition 8.6.1.13. $\square$

Example 8.6.3.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Applying Construction 8.6.3.9 to the evaluation functor

\[ \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}\quad \quad (C, f_ C, u: C \rightarrow C') \mapsto f_ C(u), \]

we we obtain a comparison functor

\[ \Psi : \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\dagger }( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}). \]

It follows from Propositions 8.6.3.11 and 8.6.2.3 that this functor is an equivalence of $\infty $-categories.

Proof of Proposition 8.6.3.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Using Corollary 8.6.2.4, we can choose a cartesian fibration $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and a morphism $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ which exhibits $U^{\dagger }$ as a cartesian conjugate of $U$. Applying Proposition 8.6.3.11, we see that Construction 8.6.3.9 supplies a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}^{\dagger } \ar [rr]^-{\Psi } \ar [dr]_{U^{\dagger }} & & \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \ar [dl]^{V} \\ & \operatorname{\mathcal{C}}^{\operatorname{op}}, & } \]

where the horizontal map is an equivalence of cartesian fibrations over $\operatorname{\mathcal{C}}^{\operatorname{op}}$. It follows that $V$ is also a cartesian conjugate of $U$. $\square$

Warning 8.6.3.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Proposition 8.6.3.5 guarantees the existence of a morphism

\[ T: \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}} \]

which exhibits the projection map $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ as a cartesian conjugate of $U$. Beware that the construction of $T$ requires making some auxiliary choices. For example, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then we can construct the datum $T$ by choosing a homotopy inverse to the equivalence $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\dagger }( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ of Example 8.6.3.12.

Corollary 8.6.3.14 (Uniqueness). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then $U$ admits a cartesian conjugate, which is uniquely determined up equivalence.

Proof. Combining Propositions 8.6.3.11 and 8.6.3.5, we see that a cartesian fibration $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is conjugate to $U$ if and only if it is equivalent to the projection map $\operatorname{Cospan}^{\dagger }( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$. $\square$

We close this section with a simple application of Proposition 8.6.3.5.

Construction 8.6.3.15. Let $e$ denote the nondegenerate edge of $\Delta ^1$, viewed as an object of the $\infty $-category $\operatorname{Tw}(\Delta ^1)$. For every simplicial set $\operatorname{\mathcal{C}}$, we let

\[ \Xi : \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \]

denote the morphism of simplicial sets which corresponds, under the bijection of Proposition 8.1.3.7, to the composite map

\begin{eqnarray*} \operatorname{Tw}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) & \simeq & \{ e \} \times \operatorname{Tw}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \\ & \hookrightarrow & \operatorname{Tw}(\Delta ^1) \times \operatorname{Tw}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Tw}( \Delta ^1 \times \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \\ & \xrightarrow { \operatorname{Tw}(\operatorname{ev}) } & \operatorname{Tw}( \operatorname{\mathcal{C}}). \end{eqnarray*}

Remark 8.6.3.16. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let

\[ \lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \quad \quad \lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}} \]

be the projection maps of Notation 8.1.1.6. Then the morphism $\Xi $ of Construction 8.6.3.15 fits into a commutative diagram

8.79
\begin{equation} \begin{gathered}\label{equation:compatibilities-of-Xi} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{ \rho _{-} } & \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \ar [l]_{\operatorname{ev}_0} \ar [r]^-{ \operatorname{ev}_{1} } \ar [d]^{\Xi } & \operatorname{\mathcal{C}}\ar [d]^{\rho _{+}} \\ \operatorname{Cospan}(\operatorname{\mathcal{C}}^{\operatorname{op}}) & \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \ar [l]_{ \operatorname{Cospan}(\lambda _{-}) } \ar [r]^-{ \operatorname{Cospan}(\lambda _{+} ) } & \operatorname{Cospan}(\operatorname{\mathcal{C}}), } \end{gathered} \end{equation}

where $\rho _{+}$ and $\rho _{-}$ are the embeddings of Construction 8.1.7.1 and Variant 8.1.7.14. Moreover, the composition of $\Xi $ with the diagonal map $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}})$ is the unit map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{C}}) )$ of Construction 8.1.3.5.

In the situation of Remark 8.6.3.16, suppose that the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category. Then the projection map $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cocartesian fibration of $\infty $-categories, and a morphism $u$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is $\lambda _{-}$-cocartesian if and only if $\lambda _{+}(u)$ is an isomorphism in $\operatorname{\mathcal{C}}$ (Corollary 8.1.1.14). Using the commutativity of the diagram (8.79), we see that the morphism $\Xi $ factors through the simplicial subset $\operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} ) \subseteq \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{C}}) )$ of Notation 8.6.3.1. In this case, we have the following:

Proposition 8.6.3.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the morphism $\Xi : \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} )$ is an equivalence of $\infty $-categories.

Proof. Let $V: \operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ be the projection map. It follows from Lemma 8.6.3.8 (and Corollary 8.1.1.14) that $V$ is a cartesian fibration, and that a morphism in $\operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} )$ is $V$-cartesian if and only if its image under the composite map

\[ \operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} ) \hookrightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \xrightarrow { \operatorname{Cospan}( \lambda _{+} ) } \operatorname{Cospan}(\operatorname{\mathcal{C}}) \]

is contained in $\operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{C}})$. Recall that the evaluation functor $\operatorname{ev}_0: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is also a cartesian fibration, and that a morphism $f$ of $\operatorname{Fun}( \Delta ^1,\operatorname{\mathcal{C}})$ is $\operatorname{ev}_0$-cartesian if and only if $\operatorname{ev}_1(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$ (Corollary 5.3.7.3). Invoking Remark 8.6.3.16, we obtain a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \ar [rr]^{ \Xi } \ar [dr]^{ \operatorname{ev}_0 } & & \operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} ) \ar [dl]_{V} \\ & \operatorname{\mathcal{C}}, & } \]

where $\Xi $ carries $\operatorname{ev}_0$-cartesian morphisms of the $\infty $-category $\operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}})$ to $V$-cartesian morphisms of the $\infty $-category $\operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{C}}^{\operatorname{op}} )$. By virtue of Proposition 5.1.7.15, it will suffice to show that for each object $X \in \operatorname{\mathcal{C}}$, the functor $\Xi $ restricts to an equivalence of fibers

\[ \Xi _{X}: \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \rightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}} \operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} ) \simeq \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) ). \]

We conclude by observing that $\Xi _{X}$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [d] \ar [r] & \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d]^{\rho _{+}} \\ \{ X\} \otimes _{\operatorname{\mathcal{C}}} \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \ar [r]^-{ \Xi _{X} } & \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) ) } \]

where the left vertical map is the equivalence of Corollary 4.6.4.18, the right vertical map is the equivalence of Proposition 8.1.7.6, and the upper horizontal map is the equivalence of Proposition 8.1.2.9. $\square$

Corollary 8.6.3.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the evaluation map $\operatorname{ev}_0: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a cartesian conjugate of the cocartesian fibration $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.