Theorem 8.6.4.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then $U$ admits a cocartesian dual $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$, which is uniquely determined up to equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
8.6.4 Existence of Dual Fibrations
The goal of this section is to prove the following:
We will give the proof of Theorem 8.6.4.1 at the end of this section.
Corollary 8.6.4.2. For every simplicial set $\operatorname{\mathcal{C}}$, the formation of cocartesian duals induces a bijection
\[ \xymatrix@C =50pt@R=50pt{ \{ \textnormal{Cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} / \textnormal{Equivalence} \ar [d]^{\theta } \\ \{ \textnormal{Cocartesian fibrations $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$} \} / \textnormal{Equivalence.} } \]
Proof. Theorem 8.6.4.1 implies that $\theta $ is well-defined, and Remark 8.6.3.3 implies that $\theta \circ \theta $ is the identity; in particular, $\theta $ is a bijection. $\square$
Variant 8.6.4.3 (Cartesian Duality). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be cartesian fibrations of simplicial sets. We say that $U'$ is a cartesian dual of $U$ if the cocartesian fibration $U'^{\operatorname{op}}: \operatorname{\mathcal{E}}'^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cocartesian dual of $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$. It follows from Theorem 8.6.4.1 that every cartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ admits a cartesian dual $U': \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ which is uniquely determined up to equivalence. Moreover, Corollary 8.6.3.10 implies that the (contravariant) homotopy transport representation of $U'$ is given by the composition In particular, for every vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}'_{C} \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ is equivalent to the opposite of the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
\[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \xrightarrow { \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} } \mathrm{h} \mathit{\operatorname{QCat}} \xrightarrow { \operatorname{\mathcal{A}}\mapsto \operatorname{\mathcal{A}}^{\operatorname{op}} } \mathrm{h} \mathit{\operatorname{QCat}}. \]
Warning 8.6.4.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets which is both a cartesian fibration and a cocartesian fibration. Then $U$ admits a both a cocartesian dual $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ and a cartesian dual $U'': \operatorname{\mathcal{E}}'' \rightarrow \operatorname{\mathcal{C}}$. For every vertex $C \in \operatorname{\mathcal{C}}$, there are equivalences of $\infty $-categories $\operatorname{\mathcal{E}}'_{C} \simeq \operatorname{\mathcal{E}}_{C}^{\operatorname{op}} \simeq \operatorname{\mathcal{E}}''_{C}$. Beware that the fibrations $U'$ and $U''$ are generally not equivalent to one another (see Example 8.6.4.5).
Example 8.6.4.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ be a cocartesian fibration of $\infty $-categories. By virtue of Remark 5.2.4.3, the cocartesian fibration $U$ can be recovered (up to equivalence) from its homotopy transport representation, which we can identify with the functor $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ given by covariant transport along the nondegenerate edge of $\Delta ^1$. The fibration $U$ then a cocartesian dual $U': \operatorname{\mathcal{E}}' \rightarrow \Delta ^1$, whose covariant transport functor can be identified with the composition (Corollary 8.6.3.10). Applying Proposition 6.2.3.5, we deduce the following:
The cocartesian fibration $U$ is a cartesian fibration if and only if the functor $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ admits a right adjoint.
The cocartesian fibration $U'$ is a cartesian fibration if and only if the functor $F^{\operatorname{op}}: \operatorname{\mathcal{E}}_0^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}_{1}^{\operatorname{op}}$ admits a right adjoint: that is, if and only if the functor $F$ admits a left adjoint.
\[ \operatorname{\mathcal{E}}'_{0} \simeq \operatorname{\mathcal{E}}_{0}^{\operatorname{op}} \xrightarrow { F^{\operatorname{op}} } \operatorname{\mathcal{E}}_{1}^{\operatorname{op}} \simeq \operatorname{\mathcal{E}}'_{1} \]
Note that conditions $(a)$ and $(b)$ are not equivalent. If $(a)$ is satisfied and $(b)$ is not, then $U$ admits a cartesian dual $U'': \operatorname{\mathcal{E}}'' \rightarrow \Delta ^1$ which cannot be equivalent to $U'$ (since $U''$ is a cartesian fibration and $U'$ is not).
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Theorem 8.6.4.1 implies that $U$ admits a cocartesian dual $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$. To prove this, it will be convenient to use the formulation of cocartesian duality supplied by Definition 8.6.3.12. In the special case where $\operatorname{\mathcal{C}}= \Delta ^0$ and $\operatorname{\mathcal{E}}$ is locally small, we wish to show that there exists an $\infty $-category $\operatorname{\mathcal{E}}^{\vee }$ and a balanced profunctor $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$. This is a special case of Corollary 8.3.2.21: in fact, we can take $\operatorname{\mathcal{E}}^{\vee }$ to be the $\infty $-category $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}})$ of corepresentable functors from $\operatorname{\mathcal{E}}$ to $\operatorname{\mathcal{S}}$, and $\mathscr {K}$ to be the evaluation functor
\[ \operatorname{ev}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}}) \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}\quad \quad (\mathscr {F},X) \mapsto \mathscr {F}(X). \]
To handle the general case we will use a variant of this construction, defined using the relative exponential introduced in ยง4.5.9.
Construction 8.6.4.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $\kappa $ be an uncountable cardinal, and let let $\operatorname{\mathcal{S}}^{< \kappa }$ denote the $\infty $-category of essentially $\kappa $-small spaces. We let $\operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$ denote the relative exponential of Construction 4.5.9.1. By construction, we can identify vertices of $\operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$ with pairs $(C, \mathscr {F}_{C} )$, where $C$ is a vertex of $\operatorname{\mathcal{C}}$ and $\mathscr {F}_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is a functor of $\infty $-categories. We let $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ denote the full simplicial subset of $\operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$ spanned by those vertices $(C, \mathscr {F}_{C} )$ where the functor $\mathscr {F}_{C}$ is corepresentable by an object of the $\infty $-category $\operatorname{\mathcal{E}}_{C}$. In what follows, we will generally write $\pi : \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{ < \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ for the projection map, and $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{C}}$ for the restriction of $\pi $ to the simplicial subset $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \subseteq \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{ < \kappa } )$.
Remark 8.6.4.7. Construction 8.6.4.6 is independent of the choice of the cardinal $\kappa $, provided that each of the $\infty $-categories $\operatorname{\mathcal{E}}_{C}$ is locally $\kappa $-small. If this condition is satisfied and $\lambda \geq \kappa $, then every corepresentable functor $\mathscr {F}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ factors through $\operatorname{\mathcal{S}}^{< \kappa }$. It follows that $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) = \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$.
The existence assertion of Theorem 8.6.4.1 is a consequence of the following more precise result:
Proposition 8.6.4.8. Let $\kappa $ be an uncountable cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets which is locally $\kappa $-small (Variant 4.7.9.2). Then the evaluation map exhibits the projection map $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{<\kappa }) \rightarrow \operatorname{\mathcal{C}}$ as a cocartesian dual of $U$ (in the sense of Variant 8.6.3.13).
\[ \operatorname{ev}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad ((C, \mathscr {F}_{C}), X) \mapsto \mathscr {F}_{C}(X) \]
Our first goal is to show that, in the situation of Proposition 8.6.4.8, the projection map $\pi ^{\operatorname{corep}}$ is a cocartesian fibration of simplicial sets. We begin with some more general remarks.
Proposition 8.6.4.9. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets which is essentially $\kappa $-small, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $\kappa $-cocomplete. Then the projection map $\pi : \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of simplicial sets.
Proof. It follows from Corollary 5.3.6.8 that $\pi $ is a cartesian fibration of simplicial sets. Let $e: C \rightarrow C'$ be an edge of the simplicial set $\operatorname{\mathcal{C}}$, and let $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ be the functor given by covariant transport along $e$ (for the cocartesian fibration $U$). Then precomposition with $e_{!}$ determines a functor
\[ e^{\ast }: \{ C'\} \times _{\operatorname{\mathcal{C}}} \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) = \operatorname{Fun}( \operatorname{\mathcal{E}}_{C'}, \operatorname{\mathcal{D}}) \xrightarrow {\circ e_{!} } \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{D}}) = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}). \]
Proposition 5.3.6.9 guarantees that the functor $e^{\ast }$ is given by contravariant transport along $e$ (for the cartesian fibration $\pi $). Using Proposition 6.2.3.5, we see that $\pi $ is a cocartesian fibration if and only if the functor $e^{\ast }$ has a left adjoint (for every edge $e$ of $\operatorname{\mathcal{C}}$). By virtue of Corollary 7.3.6.3, it will suffice to show that every functor $F: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{D}}$ admits a left Kan extension along the functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$. This is a special case of Proposition 7.6.7.13, by virtue of our assumptions on the cardinal $\kappa $. $\square$
Remark 8.6.4.10. In the situation of Proposition 8.6.4.9, let $e: C \rightarrow C'$ be an edge of $\operatorname{\mathcal{C}}$ and let be functors given by covariant transport along $e$ (for the cocartesian fibrations $U$ and $\pi $, respectively). Then the functor $e'_{!}$ is given by left Kan extension along $e_{!}$.
\[ e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'} \quad \quad e'_{!}: \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C'}, \operatorname{\mathcal{D}}) \]
Variant 8.6.4.11. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an exponentiable inner fibration which is essentially $\kappa $-small, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $\kappa $-cocomplete. Then the projection map $\pi : \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration. Moreover, an edge $\widetilde{e}$ of $\operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})$ is $\pi $-cocartesian if and only if satisfies the following condition:
Write $\widetilde{e} = (e, F_ e)$, where $e$ is an edge of $\operatorname{\mathcal{C}}$ and $F_{e}: \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories. Then $F_{e}$ is left Kan extended from the full subcategory $\{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
Proof. Corollary 4.5.9.18 guarantees that $\pi $ is an isofibration, and Corollary 7.3.7.9 guarantees that every edge of $\operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})$ which satisfies condition $(\ast )$ is $\pi $-cocartesian. Suppose we are given a vertex $\widetilde{C} = (C, F_ C)$ of $\operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, where $C$ is a vertex of $\operatorname{\mathcal{C}}$ and $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories. If $e: C \rightarrow C'$ is an edge of $\operatorname{\mathcal{C}}$, then Proposition 7.6.7.13 guarantees that $F_{C}$ admits a left Kan extension $F_{e}: \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$, which we can identify with an edge $\widetilde{e}$ of $\operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ satisfying $\pi ( \widetilde{e} ) = e$. By construction, the morphism $\widetilde{e}$ satisfies condition $(\ast )$, and is therefore $\pi $-cocartesian by virtue of Corollary 7.3.7.9. Allowing $\widetilde{C}$ and $e$ to vary, we conclude that $\pi $ is a cocartesian fibration. To complete the proof, it will suffice to show that every $\pi $-cocartesian edge $\widetilde{e}'$ of $\operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ satisfies condition $(\ast )$. Let us identify $\widetilde{e}'$ with a pair $(e, F'_{e})$, where $e$ is an edge of $\operatorname{\mathcal{B}}$ and $F'_{e}: \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ is a functor. Using the preceding argument, we see that the restriction $F'_{e}|_{ \{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}}$ admits a left Kan extension $F_{e}: \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$, corresponding to another edge $\widetilde{e}$ of $\operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})$. By construction, $\widetilde{e}$ satisfies condition $(\ast )$ and is therefore $\pi $-cocartesian. Invoking the uniqueness of cocartesian lifts (Remark 5.1.3.8), we deduce that the functors $F_{e}$ and $F'_{e}$ are isomorphic. It follows that $F'_{e}$ is also left Kan extended from $\{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (Remark 7.3.3.17), so that $\widetilde{e}'$ satisfies condition $(\ast )$ as desired. $\square$
Proposition 8.6.4.12. Let $\kappa $ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets which is essentially $\kappa $-small. Then:
The projection map $\pi : \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ is both a cartesian fibration and a cocartesian fibration.
Let $\widetilde{e}$ be a $\pi $-cocartesian edge of the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{<\kappa } )$. If the source of $\widetilde{e}$ belongs to the simplicial subset $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$, then the target of $\widetilde{e}$ also belongs to the simplicial subset $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$.
The morphism $\pi $ restricts to a cocartesian fibration $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{C}}$. Moreover, an edge of $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is $\pi ^{\operatorname{corep}}$-cocartesian if and only if it is $\pi $-cocartesian.
Proof. Assertion $(1)$ follows from Corollary 5.3.6.8 and Proposition 8.6.4.9 (since the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$ admits $\kappa $-small colimits; see Remark 7.4.5.7). We will prove $(2)$. Let $\widetilde{e}: (C,\mathscr {F}_ C) \rightarrow (C', \mathscr {F}_{C'})$ be an edge of the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$ having image $e: C \rightarrow C'$ in $\operatorname{\mathcal{C}}$. Let $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ be given by covariant transport along $e$ for the cocartesian fibration $U$. If $\widetilde{e}$ is $\pi $-cocartesian, then we can identify $\mathscr {F}_{C'}$ with a left Kan extension of $\mathscr {F}_{C}$ along the functor $e_{!}$ (Remark 8.6.4.10). In particular, if the functor $\mathscr {F}_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}$ is corepresentable by an object $X \in \operatorname{\mathcal{E}}_{C}$, then $\mathscr {F}_{C'}$ is corepresentable by the image $e_{!}(X) \in \operatorname{\mathcal{C}}_{C'}$ (Corollary 8.3.1.9). This proves assertion $(2)$, and assertion $(3)$ is a formal consequence (see Proposition 5.1.4.16). $\square$
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. To show that the projection map $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration (at least for $\kappa \gg 0$), we used the fact that the collection of corepresentable functors is closed under the formation of left Kan extensions. To prove Proposition 8.6.4.8, we will need to characterize the collection of $\pi ^{\operatorname{corep}}$-cocartesian edges of the simplicial set $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ more explicitly.
Lemma 8.6.4.13. Let $\kappa $ be an uncountable cardinal, let $\operatorname{\mathcal{E}}$ be an $\infty $-category which is locally $\kappa $-small, and let $\mathscr {F}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor. Suppose we are given a full subcategory $\operatorname{\mathcal{E}}_{0} \subseteq \operatorname{\mathcal{E}}$, an object $X \in \operatorname{\mathcal{E}}_0$, and a vertex $\eta \in \mathscr {F}(X)$. Then $\eta $ exhibits the $\mathscr {F}$ as corepresented by the object $X$ if and only if the following conditions are satisfied:
The vertex $\eta $ exhibits $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{E}}_0}$ as corepresented by the object $X$.
The functor $\mathscr {F}$ is left Kan extended from $\operatorname{\mathcal{E}}_0$.
Proof. It follows immediately from the definition that if $\eta $ exhibits $\mathscr {F}$ as corepresented by $X$, then it also exhibits $\mathscr {F}_0$ as corepresented by $X$. We may therefore assume without loss of generality that condition $(a)$ is satisfied. In this case, the desired equivalence follows immediately by combining the criterion of Lemma 8.3.1.7 with the transitivity property of Kan extensions (Proposition 7.3.8.18). $\square$
Lemma 8.6.4.14. Let $\kappa $ be an uncountable cardinal, let $\operatorname{\mathcal{E}}$ be an $\infty $-category which is locally $\kappa $-small, and let $\mathscr {F}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor. Suppose we are given a functor $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ and a $U$-cocartesian morphism $e: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ satisfying $U(X) = 0$ and $U(Y) = 1$. Write $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{ \Delta ^1} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_1 = \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$, and let $\eta \in \mathscr {F}(X)$ be a vertex which exhibits the functor $\mathscr {F}_0 = \mathscr {F}|_{ \operatorname{\mathcal{E}}_0 }$ as corepresented by the object $X$. The following conditions are equivalent:
The functor $\mathscr {F}$ is left Kan extended from $\operatorname{\mathcal{E}}_0$.
The vertex $\eta $ exhibits the functor $\mathscr {F}$ as corepresented by $X$.
The vertex $\mathscr {F}(e)(\eta ) \in \mathscr {F}(Y)$ exhibits the functor $\mathscr {F}_1 = \mathscr {F}|_{ \operatorname{\mathcal{E}}_1 }$ as corepresented by the object $Y \in \operatorname{\mathcal{E}}_1$.
Proof. The equivalence of $(1)$ and $(2)$ follows from Lemma 8.6.4.13. We will show that $(2)$ and $(3)$ are equivalent. Fix an object $Z \in \operatorname{\mathcal{E}}_1$. Then the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}_1}(Y,Z) \ar [rr]^{ \circ [e] } \ar [dr] & & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z) \ar [dl] \\ & \mathscr {F}(Z) & } \]
commutes up to homotopy, where the right vertical map is determined by $\eta \mathscr {F}(X)$ and the left vertical map is determined by $\mathscr {F}(e)(\eta ) \in \mathscr {F}(Y)$. Our assumption that $e$ is $U$-cocartesian guarantees that the horizontal map is a homotopy equivalence (Corollary 5.1.2.3). It follows that the left vertical map is a homotopy equivalence if and only if the right vertical map is a homotopy equivalence. The desired result now follows by allowing the object $Z$ to vary. $\square$
Lemma 8.6.4.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories, let $\kappa $ be an uncountable cardinal such that each fiber of $U$ is locally $\kappa $-small. Let $\widetilde{e}$ be an edge of the simplicial set $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ corresponding to a pair $(e, \mathscr {F} )$, where $e: C \rightarrow D$ is an edge of $\operatorname{\mathcal{C}}$ and $\mathscr {F}: \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is a functor. The following conditions are equivalent:
The edge $\widetilde{e}$ is $\pi ^{\operatorname{corep}}$-cocartesian (where $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{C}}$ denotes the projection map).
There exists an object $X \in \operatorname{\mathcal{E}}_{C}$, a vertex $\eta \in \mathscr {F}(X)$ which exhibits $\mathscr {F}|_{ \operatorname{\mathcal{E}}_{C} }$ as corepresented by the object $X$, and a $U$-cocartesian morphism $\overline{e}: X \rightarrow Y$ such that $U( \overline{e} ) = e$ and the vertex $\mathscr {F}( \overline{e} )( \eta ) \in \mathscr {F}(Y)$ exhibits $\mathscr {F}|_{ \operatorname{\mathcal{E}}_{D} }$ as corepresented by the object $Y$.
For every object $X \in \operatorname{\mathcal{E}}_{C}$, every vertex $\eta \in \mathscr {F}(X)$ which exhibits $\mathscr {F}|_{\operatorname{\mathcal{E}}_{C}}$ as corepresented by the object $X$, and every $U$-cocartesian morphism $\overline{e}: X \rightarrow Y$ satisfying $U( \overline{e} ) = e$, the vertex $\mathscr {F}( \overline{e} )( \eta ) \in \mathscr {F}(Y)$ exhibits $\mathscr {F}|_{ \operatorname{\mathcal{E}}_{D} }$ as corepresented by the object $Y$.
Proof. By virtue of Remark 8.6.4.7, we are free to enlarge the cardinal $\kappa $; we may therefore assume without loss of generality that $\kappa $ is regular and that the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small. In this case, Variant 8.6.4.11 shows that $(1)$ is equivalent to the following:
- $(1')$
The functor $\mathscr {F}$ is left Kan extended from the full subcategory $\operatorname{\mathcal{E}}_{C} \subseteq \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
The equivalences $(1') \Leftrightarrow (2) \Leftrightarrow (3)$ now follow from Lemma 8.6.4.14. $\square$
Proof of Proposition 8.6.4.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $\kappa $ be an uncountable cardinal such that each fiber of $U$ is locally $\kappa $-small. It follows from Proposition 8.6.4.12 that the projection map $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of simplicial sets. We wish to show that the evaluation map
\[ \operatorname{ev}: \operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad ( (C, \mathscr {F}_{C}), X) \mapsto \mathscr {F}_{C}(X) \]
satisfies conditions $(a)$ and $(b)$ of Definition 8.6.3.12. Condition $(a)$ asserts that, for each vertex $C \in \operatorname{\mathcal{C}}$, the evaluation map
\[ \operatorname{ev}_{C}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{S}}^{< \kappa } ) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad (\mathscr {F},X) \mapsto \mathscr {F}(X) \]
is a balanced profunctor; this follows from Corollary 8.3.2.21. Assertion $(b)$ is a restatement of the implication $(1) \Rightarrow (3)$ of Lemma 8.6.4.15. $\square$
Proposition 8.6.4.8 immediately implies the existence assertion of Theorem 8.6.4.1. To establish uniqueness, it will be convenient to introduce some terminology.
Definition 8.6.4.16. 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 $\kappa $ be an uncountable cardinal. We will say that a morphism $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is a weak $\operatorname{\mathcal{C}}$-family of corepresentable profunctors if, for every vertex $C \in \operatorname{\mathcal{C}}$, the induced map is a corepresentable profunctor (Definition 8.3.2.9). We say that $\mathscr {K}$ is a $\operatorname{\mathcal{C}}$-family of corepresentable profunctors if it is a weak $\operatorname{\mathcal{C}}$-family of corepresentable profunctors and satisfies the following additional condition:
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 $u: C \rightarrow D$ in $\operatorname{\mathcal{C}}$. Then the map of Kan complexes
carries couniversal vertices of $\mathscr {K}_{C}( X^{\vee }, X )$ to couniversal vertices of $\mathscr {K}_{D}( Y^{\vee }, Y)$.
\[ \mathscr {K}_{C}: \operatorname{\mathcal{E}}^{\vee }_{C} \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}^{< \kappa } \]
\[ \mathscr {K}( f^{\vee }, f ): \mathscr {K}_{C}( X^{\vee }, X) \rightarrow \mathscr {K}_{D}( Y^{\vee }, Y ) \]
Example 8.6.4.17. In the situation of Definition 8.6.4.16, the morphism $\mathscr {K}$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ (in the sense of Variant 8.6.3.13) if and only if it is a $\operatorname{\mathcal{C}}$-family of corepresentable profunctors having the further property that each of the profunctors $\mathscr {K}_{C}$ is balanced: that is, it is corepresentable by an equivalence of $\infty $-categories $(\operatorname{\mathcal{E}}^{\vee }_{C})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}_{C}$ (see Corollary 8.3.2.20).
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, let $\kappa $ be an uncountable cardinal, and let $\pi : \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ denote the projection map. For any morphism of simplicial sets $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$, we can identify morphisms $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ with morphisms $F: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$ satisfying $\pi \circ F = U$.
Proposition 8.6.4.18. 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. Let $\kappa $ be an uncountable cardinal such that $U$ is locally $\kappa $-small. Fix a morphism $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, which we identify with a morphism $F: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Then:
The morphism $\mathscr {K}$ is a weak $\operatorname{\mathcal{C}}$-family of corepresentable profunctors if and only if $F$ factors through the simplicial subset $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{<\kappa }) \subseteq \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$.
The morphism $\mathscr {K}$ is a $\operatorname{\mathcal{C}}$-family of corepresentable profunctors if and only if $F$ factors through $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ and carries $U^{\vee }$-cocartesian edges of $\operatorname{\mathcal{E}}^{\vee }$ to $\pi ^{\operatorname{corep}}$-cocartesian edges of $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$. Here $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{C}}$ denotes the cocartesian fibration of Proposition 8.6.4.12.
The morphism $\mathscr {K}$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ if and only if $F: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$.
Proof. Assertion $(1)$ is immediate from the definitions and assertion $(2)$ follows from Lemma 8.6.4.15. Assertion $(3)$ follows by combining $(2)$ with Example 8.6.4.17 (see Proposition 5.1.7.14). $\square$
Proof of Theorem 8.6.4.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Fix an uncountable cardinal $\kappa $ such that $U$ is locally $\kappa $-small. Proposition 8.6.4.8 implies that the projection map $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian dual of $U$, and Proposition 8.6.4.18 implies that any other cocartesian dual $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ is equivalent to $\pi ^{\operatorname{corep}}$. $\square$
Using Proposition 8.6.4.18, we can characterize the dual of a fibration by a universal mapping property.
Corollary 8.6.4.19. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$, and $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets, let $\kappa $ be an uncountable cardinal, and let $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ exhibit $U^{\vee }$ as a cocartesian dual of $U$. Then:
Composition with $\mathscr {K}$ induces a fully faithful functor
The essential image is spanned by the weak $\operatorname{\mathcal{C}}$-families of corepresentable profunctors.
A morphism $F \in \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}^{\vee } )$ carries $V$-cocartesian edges of $\operatorname{\mathcal{D}}$ to $U^{\vee }$-cocartesian edges of $\operatorname{\mathcal{E}}^{\vee }$ if and only if the composite map
is a $\operatorname{\mathcal{C}}$-family of corepresentable profunctors.
A morphism $F \in \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}^{\vee } )$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$ if and only if the composite map
exhibits $V$ as a cocartesian dual of $U$
\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}^{\vee } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}^{\vee }, \operatorname{\mathcal{S}}^{ < \kappa } ). \]
\[ \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\xrightarrow {F \times \operatorname{id}} \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\xrightarrow { \mathscr {K} } \operatorname{\mathcal{S}}^{<\kappa } \]
\[ \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\xrightarrow {F \times \operatorname{id}} \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\xrightarrow { \mathscr {K} } \operatorname{\mathcal{S}}^{<\kappa } \]
Proof. We can identify $\mathscr {K}$ with a morphism $G \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}^{\vee }, \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) )$. It follows from Proposition 8.6.4.18 that $G$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$. We can therefore replace $\operatorname{\mathcal{E}}^{\vee }$ by $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ and $\mathscr {K}$ by the evaluation map $\operatorname{ev}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$. In this case, assertions $(1)$, $(2)$, and $(3)$ follow immediately from the corresponding assertions of Proposition 8.6.4.18. $\square$