8.6.3 Dual Fibrations
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. By virtue of Corollary 8.6.2.9, $U$ admits a cartesian conjugate $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$, which is uniquely determined up to equivalence. Setting $\operatorname{\mathcal{E}}^{\vee } = \operatorname{\mathcal{E}}^{\dagger , \operatorname{op}}$ and $U^{\vee } = U^{\dagger , \operatorname{op}}$, we obtain another cocartesian fibration $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$. Our goal in this section is to give a direct characterization of the relationship between $U$ and $U^{\vee }$, which does not rely on the theory of conjugate fibrations developed in §8.6.1 and §8.6.2. To fix ideas, let us begin by considering the case $\operatorname{\mathcal{C}}= \Delta ^{0}$. In this case, the conjugate fibration $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is characterized by the requirement that the $\infty $-category $\operatorname{\mathcal{E}}^{\dagger }$ is equivalent to $\operatorname{\mathcal{E}}$. Consequently, the cocartesian fibration $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ is characterized by the requirement that $\operatorname{\mathcal{E}}^{\vee }$ is equivalent to the opposite of $\operatorname{\mathcal{E}}$. By virtue of Corollary 8.2.6.6, this is equivalent to the existence of a balanced coupling $\lambda : \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times \operatorname{\mathcal{E}}$: that is, a left fibration which satisfies the following addition conditions:
For every object $X \in \operatorname{\mathcal{E}}$, there exists an object $\widetilde{X} \in \widetilde{\operatorname{\mathcal{E}}}$ satisfying $\lambda _{+}( \widetilde{X} ) = X$ which is universal: that is, it is an initial object of the $\infty $-category $\widetilde{\operatorname{\mathcal{E}}} \times _{ \operatorname{\mathcal{E}}} \{ X \} $.
For every object $X^{\vee } \in \operatorname{\mathcal{E}}^{\vee }$, there exists an object $\widetilde{X} \in \widetilde{\operatorname{\mathcal{E}}}$ satisfying $\lambda _{-}( \widetilde{X} ) = X^{\vee }$ which is couniversal: that is, it is an initial object of the $\infty $-category $\{ X^{\vee } \} \times _{ \operatorname{\mathcal{E}}^{\vee } } \widetilde{\operatorname{\mathcal{E}}}$.
An object of $\widetilde{\operatorname{\mathcal{E}}}$ is universal if and only if it is couniversal.
We now extend the notion of balanced coupling to the relative setting.
Definition 8.6.3.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets, and let $\lambda = (\lambda _{-}, \lambda _{+}): \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ be a left fibration of simplicial sets. We will say that $\lambda $ exhibits $U^{\vee }$ as a cocartesian dual of $U$ if the following conditions are satisfied:
- $(a)$
For every vertex $C \in \operatorname{\mathcal{C}}$, the left fibration
\[ \lambda _{C}: \widetilde{\operatorname{\mathcal{E}}}_{C} \rightarrow \operatorname{\mathcal{E}}^{\vee }_{C} \times \operatorname{\mathcal{E}}_{C} \]
is a balanced coupling of $\infty $-categories.
- $(b)$
Let $\widetilde{U}: \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map $U^{\vee } \circ \lambda _{-} = U \circ \lambda _{+}$, $f: \widetilde{X} \rightarrow \widetilde{X}'$ be a $\widetilde{U}$-cocartesian edge of $\widetilde{\operatorname{\mathcal{E}}}$, and let $e: C \rightarrow C'$ be its image $\widetilde{U}(f)$ in the simplicial set $\operatorname{\mathcal{C}}$. If the object $\widetilde{X} \in \widetilde{\operatorname{\mathcal{E}}}_{C}$ is universal for the coupling $\lambda _{C}$, then the object $\widetilde{X}' \in \widetilde{\operatorname{\mathcal{E}}}_{C'}$ is universal for the coupling $\lambda _{C'}$.
We say that $U^{\vee }$ is a cocartesian dual of $U$ if there exists a left fibration $\lambda : \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ which exhibits $U^{\vee }$ as a cocartesian dual of $U$.
Example 8.6.3.2. Let $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}^{\vee }$ be $\infty $-categories, and let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^0$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \Delta ^0$ denote the projection maps. Then a left fibration $\widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{ \Delta ^0 } \operatorname{\mathcal{E}}= \operatorname{\mathcal{E}}^{\vee } \times \operatorname{\mathcal{E}}$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ if and only if it is a balanced coupling. In particular, $U^{\vee }$ is a cocartesian dual of $U$ if and only if $\operatorname{\mathcal{E}}^{\vee }$ is equivalent to the opposite $\infty $-category $\operatorname{\mathcal{E}}^{\operatorname{op}}$ (Corollary 8.2.6.6).
Moreover, it suffices to verify condition $(b)$ in the special case where $\operatorname{\mathcal{C}}_0 = \Delta ^1$ is the standard $1$-simplex.
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets, and let $\lambda : \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ be a left fibration. Condition $(a)$ of Definition 8.6.3.1 guarantees that, for each vertex $C \in \operatorname{\mathcal{C}}$, the coupling $\lambda _{C}: \widetilde{\operatorname{\mathcal{E}}}_{C} \rightarrow \operatorname{\mathcal{E}}^{\vee }_{C} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}_{C}$ is representable by an equivalence of $\infty $-categories $G_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow (\operatorname{\mathcal{E}}^{\vee }_{C})^{\operatorname{op}}$ (Theorem 8.2.6.5). Heuristically, one can think of condition $(b)$ as requiring that the equivalence $G_{C}$ depends functorially on $C$. We can articulate this heuristic more precisely as follows:
Proposition 8.6.3.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \Delta ^1$ be cocartesian fibrations of $\infty $-categories, and let $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ and $F^{\vee }: \operatorname{\mathcal{E}}^{\vee }_0 \rightarrow \operatorname{\mathcal{E}}^{\vee }_{1}$ be functors given by covariant transport along the nondegenerate edge of $\Delta ^1$. Let $\lambda = (\lambda _{-}, \lambda _{+}): \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{ \Delta ^1 } \operatorname{\mathcal{E}}$ be a left fibration, and suppose that the associated couplings
\[ \lambda _0: \widetilde{\operatorname{\mathcal{E}}}_0 \rightarrow \operatorname{\mathcal{E}}_{0}^{\vee } \times \operatorname{\mathcal{E}}_{0} \quad \quad \lambda _{1}: \widetilde{\operatorname{\mathcal{E}}}_{1} \rightarrow \operatorname{\mathcal{E}}_{1}^{\vee } \times \operatorname{\mathcal{E}}_{1} \]
are representable by functors $G_0: \operatorname{\mathcal{E}}_0 \rightarrow (\operatorname{\mathcal{E}}_0^{\vee })^{\operatorname{op}}$ and $G_1: \operatorname{\mathcal{E}}_1 \rightarrow (\operatorname{\mathcal{E}}_1^{\vee })^{\operatorname{op}}$, respectively. If $\lambda $ satisfies condition $(b)$ of Definition 8.6.3.1, then the diagram of $\infty $-categories
8.79
\begin{equation} \begin{gathered}\label{equation:balanced-coupling-family-over-edge1} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{0} \ar [r]^-{F} \ar [d]^{ G_0 } & \operatorname{\mathcal{E}}_{1} \ar [d]^{ G_1 } \\ (\operatorname{\mathcal{E}}^{\vee }_{0})^{\operatorname{op}} \ar [r]^-{ (F^{\vee } )^{\operatorname{op}} } & ( \operatorname{\mathcal{E}}^{\vee }_{1})^{\operatorname{op}} } \end{gathered} \end{equation}
commutes up to isomorphism.
Proof.
Let $\widetilde{U}$ denote the composite map
\[ \widetilde{\operatorname{\mathcal{E}}} \xrightarrow {\lambda } \operatorname{\mathcal{E}}^{\vee } \times _{ \Delta ^1 } \operatorname{\mathcal{E}}\rightarrow \Delta ^1. \]
Using Proposition 5.1.4.13, we see that $\lambda $ is a cocartesian fibration, and that an edge $e$ of $\widetilde{\operatorname{\mathcal{E}}}$ is $\widetilde{U}$-cocartesian if and only if $\lambda _{+}(e)$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ and $\lambda _{-}(e)$ is a $U^{\vee }$-cocartesian edge of $\operatorname{\mathcal{E}}^{\vee }$. Let $\widetilde{F}: \widetilde{\operatorname{\mathcal{E}}}_0 \rightarrow \widetilde{\operatorname{\mathcal{E}}}_1$ be given by covariant transport along the nondegenerate edge of $\Delta ^1$. Using Remark 5.2.8.5, we see that the diagram of $\infty $-categories
8.80
\begin{equation} \begin{gathered}\label{equation:balanced-coupling-family-over-edge2} \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{E}}}_0 \ar [r]^-{ \widetilde{F} } \ar [d]^{ \lambda _0 } & \widetilde{\operatorname{\mathcal{E}}}_1 \ar [d]^{ \lambda _1} \\ \operatorname{\mathcal{E}}_0^{\vee } \times \operatorname{\mathcal{E}}_0 \ar [r]^-{ F^{\vee } \times F} & \operatorname{\mathcal{E}}_1^{\vee } \times \operatorname{\mathcal{E}}_1 } \end{gathered} \end{equation}
commutes up to isomorphism. Since $\lambda _1$ is an isofibration, we can replace $\widetilde{F}$ by an isomorphic functor to arrange that the diagram (8.80) is strictly commutative (see Corollary 4.4.5.6). Condition $(b)$ of Definition 8.6.3.1 guarantees that the functor $\widetilde{F}$ carries universal objects of $\widetilde{\operatorname{\mathcal{E}}}_0$ (for the coupling $\lambda _0$) to universal objects of $\widetilde{\operatorname{\mathcal{E}}}_1$ (for the coupling $\lambda _1$). The commutativity of the diagram (8.79) now follows from Corollary 8.2.4.4.
$\square$
Corollary 8.6.3.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets having homotopy transport representations
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}, \operatorname{hTr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}, \]
and let $\operatorname{hTr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}^{\operatorname{op}}$ denote the functor $C \mapsto \operatorname{hTr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}(C)^{\operatorname{op}} = (\operatorname{\mathcal{E}}^{\vee }_{C})^{\operatorname{op}}$. Let $\lambda : \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ be a left fibration such that, for each vertex $C \in \operatorname{\mathcal{C}}$, the coupling $\lambda _{C}: \widetilde{\operatorname{\mathcal{E}}}_ C \rightarrow \operatorname{\mathcal{E}}^{\vee }_{C} \times \operatorname{\mathcal{E}}_{C}$ is representable by a functor $G_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow ( \operatorname{\mathcal{E}}^{\vee }_{C})^{\operatorname{op}}$. if $\lambda $ satisfies condition $(b)$ of Definition 8.6.3.1, then the construction $C \mapsto [G_ C]$ determines a natural transformation of functors $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \rightarrow \operatorname{hTr}_{ \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}}^{\operatorname{op}}$.
Corollary 8.6.3.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets having homotopy transport representations
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}, \operatorname{hTr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}. \]
Let $\lambda : \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ be a left fibration which exhibits $U^{\vee }$ as a cocartesian dual of $U$. Then $\lambda $ induces an isomorphism of functors $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \xrightarrow {\sim } \operatorname{hTr}_{ \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}}^{\operatorname{op}}$, which carries each vertex $C \in \operatorname{\mathcal{C}}$ to (the isomorphism class of) a functor which represents the balanced coupling $\lambda _{C}: \widetilde{\operatorname{\mathcal{E}}}_ C \rightarrow \operatorname{\mathcal{E}}^{\vee }_{C} \times \operatorname{\mathcal{E}}_{C}$.
Proof.
Combine Corollary 8.6.3.8 with Theorem 8.2.6.5.
$\square$
Corollary 8.6.3.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets having homotopy transport representations
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}, \operatorname{hTr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}. \]
If $U^{\vee }$ is a cocartesian dual of $U$, then $\operatorname{hTr}_{\operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}}$ is isomorphic to the functor
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}^{\operatorname{op}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}} \quad \quad C \mapsto \operatorname{\mathcal{E}}_{C}^{\operatorname{op}}. \]
For some applications, it will be convenient to work with a reformulation of Definition 8.6.3.1.
Definition 8.6.3.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. We say that a morphism of simplicial sets $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ if the following conditions are satisfied:
- $(a)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the induced map $\mathscr {K}_{C}: \operatorname{\mathcal{E}}^{\vee }_{C} \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}$ is a balanced profunctor (see Definition 8.3.2.18).
- $(b)$
Let $f: X \rightarrow Y$ be a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ and let $f^{\vee }: X^{\vee } \rightarrow Y^{\vee }$ be a $U^{\vee }$-cocartesian edge of $\operatorname{\mathcal{E}}^{\vee }$ having the same image $e: C \rightarrow D$ in $\operatorname{\mathcal{C}}$. Then the map of Kan complexes
\[ \mathscr {K}( f^{\vee }, f): \mathscr {K}_{C}( X^{\vee }, X) \rightarrow \mathscr {K}_{D}( Y^{\vee }, Y ) \]
carries universal vertices of $\mathscr {K}_{C}(X^{\vee }, X))$ to universal vertices of $\mathscr {K}_{D}(Y^{\vee }, Y)$ (see Definition 8.3.2.7).
Variant 8.6.3.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. In the formulation of Definition 8.6.3.12, we have implicitly assumed that for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-categories $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{C}}^{\vee }_{C}$ are locally small (if this condition is not satisfied, then a balanced profunctor $\mathscr {K}_{C}: \operatorname{\mathcal{E}}^{\vee }_{C} \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}$ cannot exist). However, we will sometimes apply the theory of cocartesian duality in situations where this condition is not satisfied. If $\kappa $ is an uncountable cardinal (not necessarily small), we will say that a morphism $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ if it satisfies conditions $(a)$ and $(b)$ of Definition 8.6.3.12. In this case, we can take $\kappa $ to be any uncountable cardinal having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-categories $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{E}}^{\vee }_{C}$ are locally $\kappa $-small.
Combining Remark 8.6.3.14 with the classification of left fibrations (Corollary 5.6.0.6), we obtain the following:
Proposition 8.6.3.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of $\infty $-categories. Let $\kappa $ be an uncountable cardinal with the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-categories $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{E}}^{\vee }_{C}$ are locally $\kappa $-small. Then $U^{\vee }$ is a cocartesian dual of $U$ if and only if there exists a morphism $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ which exhibits $U^{\vee }$ as a cocartesian dual of $U$, in the sense of Definition 8.6.3.12.
We now give some examples of cocartesian duality.
Proposition 8.6.3.16. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of simplicial sets, and set $\widetilde{\operatorname{\mathcal{E}}} = \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}})$. Then the evaluation maps $\operatorname{ev}_0, \operatorname{ev}_1: \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}$ determine a left fibration $\lambda : \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}\times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ which exhibits $U$ as a cocartesian dual of itself.
Proof.
The morphism $\lambda $ is a pullback of the restriction map
\[ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}), \]
and is therefore a left fibration by virtue of Proposition 4.2.5.1. For every vertex $C \in \operatorname{\mathcal{C}}$, we can identify $\lambda _{C}$ with the coupling
\[ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}_{C} ) \rightarrow \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{E}}_{C} ) \times \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{E}}_{C} ). \]
It follows from Example 8.2.6.3 that each $\lambda _{C}$ is a balanced coupling, so that $\lambda $ satisfies condition $(a)$ of Definition 8.6.3.1. Moreover, every object of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}_{C} )$ is universal for the coupling $\lambda _{C}$, so that condition $(b)$ of Definition 8.6.3.1 is vacuous.
$\square$
Corollary 8.6.3.17. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of simplicial sets. Then $U$ is a cocartesian dual of itself.
Example 8.6.3.18. In the special case $\operatorname{\mathcal{C}}= \Delta ^0$, Corollary 8.6.3.17 asserts that every Kan complex $X = \operatorname{\mathcal{E}}$ is homotopy equivalent to the opposite Kan complex $X^{\operatorname{op}}$. This can also be deduced from Theorem 3.6.0.1, since the geometric realizations $|X|$ and $|X^{\operatorname{op}}|$ are homeomorphic.
Proposition 8.6.3.19. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories, and let $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be the functor given on objects by $C \mapsto \mathscr {F}(C)^{\operatorname{op}}$. Then (the nerves of) the fibrations
\[ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}\quad \quad \int _{\operatorname{\mathcal{C}}} \mathscr {F}' \rightarrow \operatorname{\mathcal{C}} \]
are cocartesian dual to one another.
We will deduce Proposition 8.6.3.19 from a more precise result. To formulate it, we need to introduce a bit of notation.
Construction 8.6.3.20. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories, and let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the category of elements of $\mathscr {F}$ (Definition 5.6.1.1); we identify objects of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ with pairs $(C,X)$, where $C$ is an object of $\operatorname{\mathcal{B}}$ and $X$ is an object of the category $\mathscr {F}(C)$. Let $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ denote the functor given on objects by $\mathscr {F}'(C) = \mathscr {F}(C)^{\operatorname{op}}$. We define a functor
\[ \mathscr {K}: \int _{\operatorname{\mathcal{C}}} \mathscr {F}' \times _{ \operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{Set} \]
as follows:
On objects, $\mathscr {K}$ is given by the formula $\mathscr {K}( (C,X'), (C,X) ) = \operatorname{Hom}_{ \mathscr {F}(C) }( X', X )$.
Let $f: (C,X) \rightarrow (D,Y)$ be a morphism in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ and let $f': (C,X') \rightarrow (D,Y')$ be a morphism in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}'$ having the same image $u: C \rightarrow D$ in $\operatorname{\mathcal{B}}$. Let us identify $f$ and $f'$ with morphisms $g: \mathscr {F}(u)(X) \rightarrow Y$ and $g': Y' \rightarrow \mathscr {F}(u)(Y)$ in the category $\mathscr {F}(D)$. Then the function $\mathscr {K}(f',f): \mathscr {K}( (C,X'), (C,X) ) \rightarrow \mathscr {K}( (D,Y'), (D,Y) )$ is given by the composition
\[ \operatorname{Hom}_{ \mathscr {F}(C) }( X', X) \xrightarrow { \mathscr {F}(u) } \operatorname{Hom}_{ \mathscr {F}(D) }( \mathscr {F}(u)(X'), \mathscr {F}(u)(X) ) \xrightarrow { g \circ \bullet \circ g' } \operatorname{Hom}_{ \mathscr {F}(D) }( Y', Y ). \]
Proposition 8.6.3.21. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories. Then the functor
\[ \operatorname{N}_{\bullet }( \mathscr {K} ): \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F}') \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{Set}) \subset \operatorname{\mathcal{S}} \]
of Construction 8.6.3.20 exhibits the projection map $U': \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F}' ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$ as a cocartesian dual of the projection map $U: \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$.
Proof.
For each object $C \in \operatorname{\mathcal{C}}$, the restriction of $\mathscr {K}$ to the fiber over $C$ is given concretely by the functor
\[ \mathscr {K}_{C}: \mathscr {F}(C)^{\operatorname{op}} \times \mathscr {F}(C) \rightarrow \operatorname{Set}\quad \quad (X',X) \mapsto \operatorname{Hom}_{ \mathscr {F}(C) }(X',X). \]
Example 8.3.3.4 implies that $\operatorname{N}_{\bullet }( \mathscr {K}_{C} )$ is a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{N}_{\bullet }( \mathscr {F}(C) )$ and is therefore a balanced profunctor (Proposition 8.3.3.8). Let $u: C \rightarrow D$ be a morphism in the category $\operatorname{\mathcal{C}}$, and let $f: (C,X) \rightarrow (D,Y)$ and $f': (C,X') \rightarrow (D,Y')$ be lifts of $u$ to the categories $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ and $\int _{\operatorname{\mathcal{C}}} \mathscr {F}'$, respectively. We wish to show that, if $f$ is $U$-cocartesian and $f'$ is $U'$-cocartesian, then the induced map
\[ \mathscr {K}(f,f'): \mathscr {K}_{C}( X',X ) \rightarrow \mathscr {K}_{D}( Y', Y ) \]
carries universal elements of $\mathscr {K}_{C}(X',X)$ to universal elements of $\mathscr {K}_{D}(Y',Y)$. Let us identify $f$ and $f'$ with morphisms $g: \mathscr {F}(u)(X) \rightarrow Y$ and $g': Y' \rightarrow \mathscr {F}(u)(Y)$ in the category $\mathscr {F}(D)$, so that $\mathscr {K}(f',f)$ is given by the composition
\[ \operatorname{Hom}_{ \mathscr {F}(C) }( X', X) \xrightarrow { \mathscr {F}(u) } \operatorname{Hom}_{ \mathscr {F}(D) }( \mathscr {F}(u)(X'), \mathscr {F}(u)(X) ) \xrightarrow { g \circ - \circ g' } \operatorname{Hom}_{ \mathscr {F}(D) }( Y', Y ). \]
Our assumption that $f$ is $U$-cocartesian guarantees that $g$ is an isomorphism in the category $\mathscr {F}(D)$, and our assumption that $f'$ is a $U'$-cocartesian guarantees that $g'$ is an isomorphism in the category $\mathscr {F}(D)$. The desired result now follows from the observation that if $e: X' \rightarrow X$ is an isomorphism in the category $\mathscr {F}(C)$, then the composition $g \circ \mathscr {F}(u)(e) \circ g'$ is an isomorphism in the category $\mathscr {F}(D)$.
$\square$
Proof of Proposition 8.6.3.19.
Combine Propositions 8.6.3.15 and 8.6.3.21.
$\square$
Let $\operatorname{QCat}$ be the (ordinary) category of $\infty $-categories, which we regard as a full subcategory of $\operatorname{Set_{\Delta }}$. If $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is a functor of ordinary categories, we let $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the weighted nerve of Definition 5.3.3.1. According to Corollary 5.3.3.16, the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$ is a cocartesian fibration, whose fiber over an object $C \in \operatorname{\mathcal{C}}$ can be identified with the $\infty $-category $\mathscr {F}(C)$. In this situation, it is easy to construct a cocartesian dual of $U$:
Proposition 8.6.3.22. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a functor of ordinary categories, and let $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ denote the functor given on objects by $C \mapsto \mathscr {F}(C)^{\operatorname{op}}$. Then the fibrations
\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) \leftarrow \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}) \]
are cocartesian dual to one another.
Proposition 8.6.3.22 is an immediate consequence of the following more precise result:
Proposition 8.6.3.23. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a functor of ordinary categories and let $\mathscr {F}', \operatorname{Tw}(\mathscr {F}): \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be the functors given on objects by the formulae $\mathscr {F}'(C) = \mathscr {F}(C)^{\operatorname{op}}$ and $\operatorname{Tw}(\mathscr {F})(C) = \operatorname{Tw}( \mathscr {F}(C) )$. Then the tautological map
\[ \lambda = (\lambda _{-},\lambda _{+} ): \operatorname{N}_{\bullet }^{\operatorname{Tw}(\mathscr {F})}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}^{\operatorname{op}}}(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \]
exhibits the fibration $U^{\vee }: \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ as a cocartesian dual of the fibration $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
Proof.
For each object $C \in \operatorname{\mathcal{C}}$, Proposition 8.1.1.11 guarantees that the morphism
\[ \lambda _{C}: \operatorname{Tw}( \mathscr {F}(C) ) \rightarrow \mathscr {F}(C)^{\operatorname{op}} \times \mathscr {F}(C) \]
is a left fibration of $\infty $-categories, which is a balanced coupling by virtue of Example 8.2.6.2. Applying Corollary 5.3.3.18, we deduce that $\lambda $ is a left fibration of $\infty $-categories. Let $U: \operatorname{N}_{\bullet }^{ \operatorname{Tw}(\mathscr {F}) }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ denote the projection map, and let $f: X \rightarrow Y$ be a morphism in the $\infty $-category $\operatorname{N}_{\bullet }^{ \operatorname{Tw}(\mathscr {F}) }(\operatorname{\mathcal{C}})$ having image $u: C \rightarrow D$ in $\operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that if $X$ is universal for the coupling $\lambda _{C}$ and $f$ is $U$-cocartesian, then $Y$ is universal for the coupling $\lambda _{D}$. Our assumption that $f$ is $U$-cocartesian guarantees that $Y$ is isomorphic to the image of $X$ under the functor $\operatorname{Tw}( \mathscr {F}(u) ): \operatorname{Tw}( \mathscr {F}(C) ) \rightarrow \operatorname{Tw}( \mathscr {F}(D) )$. The desired result now follows from Example 8.2.1.5, since the functor $\mathscr {F}(u)$ carries isomorphisms in the $\infty $-category $\mathscr {F}(C)$ to isomorphisms in the $\infty $-category $\mathscr {F}(D)$.
$\square$