Kerodon

Remark 8.6.8.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets. Unwinding the definitions, we see that a diagram $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$ is a contravariant transport representation for $U$ if and only if there exists a commutative diagram of simplicial sets

which exhibits $U$ as a cartesian conjugate of $V$, in the sense of Definition 8.6.1.1. If this condition is satisfied, we say that $T$ exhibits $\mathscr {F}$ as a contravariant transport representation for $U$. In this case, for every vertex $C \in \operatorname{\mathcal{C}}$, we have equivalences of $\infty $-categories $\operatorname{\mathcal{E}}_{C} \xrightarrow {T_ C} \{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \int _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F} \leftarrow \mathscr {F}(C)$ (see Example 5.6.2.19).

Remark 8.6.8.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small cartesian fibration of simplicial sets. and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$ be its contravariant transport representation. Then the opposite functor $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is an essentially small cocartesian fibration which admits a covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}^{\operatorname{op}} / \operatorname{\mathcal{C}}^{\operatorname{op}} }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$. In this case, we can identify $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ with the composition $\sigma \circ \operatorname{Tr}_{\operatorname{\mathcal{E}}^{\operatorname{op}}/ \operatorname{\mathcal{C}}^{\operatorname{op}} }$, where $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ is the opposition functor introduced in Construction 8.6.7.6 (given on objects by the construction $\operatorname{\mathcal{D}}\mapsto \operatorname{\mathcal{D}}^{\operatorname{op}}$). This follows by combining Propositions 8.6.6.1 and 8.6.7.12.

Remark 8.6.8.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration which is conjugate to $U$. By definition, a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is a contravariant transport representation for $U^{\dagger }$ if and only if it is a covariant transport representation for $U$. In this case:

• Proposition 7.4.4.1 asserts that the limit $\varprojlim (\mathscr {F})$ can be computed as the $\infty $-category $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian sections of $U$.

• Proposition 8.6.8.11 asserts that the limit $\varprojlim (\mathscr {F})$ can be computed as the $\infty $-category $\operatorname{Fun}^{\operatorname{Cart}}_{/ \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}^{\dagger })$ of cartesian sections of $\operatorname{\mathcal{E}}^{\dagger }$.

To identify these conditions, it suffices to observe that the $\infty $-categories $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ and $\operatorname{Fun}^{\operatorname{Cart}}_{/ \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}^{\dagger })$ are equivalent. In fact, we can be more precise. Assume for simplicity that $\operatorname{\mathcal{C}}$ is an $\infty $-category, and fix a functor $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ which exhibits $U^{\dagger }$ as a cartesian conjugate of $U$ (Definition 8.6.1.1). Then $T$ determines a canonical equivalence $\Theta : \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}^{\operatorname{Cart}}_{/ \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}^{\dagger })$, which is characterized (up to isomorphism) by the requirement that the diagram of $\infty $-categories

commutes up to isomorphism. Here the left vertical map is given by precomposition with the projection $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$, and the right vertical map is given by pullback along the projection $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ followed by postcomposition with $T$. See Corollary 8.6.2.12 for a more general assertion.

Remark 8.6.8.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small cocartesian fibration of simplicial sets. Then $U$ is a cartesian fibration if and only if the covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ factors through the full subcategory $\operatorname{\mathcal{QC}}_{\operatorname{LAdj}} \subset \operatorname{\mathcal{QC}}$. Similarly, if $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ is an essentially small cartesian fibration, then $U'$ is a cocartesian fibration if and only if its contravariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$ factors through the subcategory $\operatorname{\mathcal{QC}}_{\operatorname{RAd}} \subset \operatorname{\mathcal{QC}}$. See Proposition 6.2.3.6.

Remark 8.6.8.17. The formation-of-adjoints functor $\Psi : \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}_{\operatorname{RAd}}$ induces a functor of homotopy categories $\mathrm{h} \mathit{\Psi }: \mathrm{h} \mathit{ \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}^{\operatorname{op}} } \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{QC}}_{\operatorname{RAd}}} \subset \mathrm{h} \mathit{\operatorname{\mathcal{QC}}}$, which is a homotopy transport representation for the cartesian fibration $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}^{0} \rightarrow \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}$ (Corollary 8.6.8.10). By virtue of Proposition 6.2.3.6, this functor can be described more concretely as follows:

• The functor $\mathrm{h} \mathit{\Psi }$ carries every small $\infty $-category $\operatorname{\mathcal{C}}$ (viewed as an object of $\operatorname{\mathcal{QC}}_{\operatorname{LAdj}}$) to itself (viewed as an object of $\operatorname{\mathcal{QC}}_{\operatorname{RAd}}$).

• If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor between small $\infty $-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, then the functor $\mathrm{h} \mathit{\Psi }$ carries the isomorphism class $[F]$ (viewed as a morphism in the homotopy cateogry $\mathrm{h} \mathit{\operatorname{\mathcal{QC}}_{\operatorname{LAdj}}}$) to the isomorphism class $[G]$ (viewed as a morphism in the homotopy category $\mathrm{h} \mathit{ \operatorname{\mathcal{QC}}_{\operatorname{RAd}} }$.

Remark 8.6.8.18. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small cocartesian fibration of simplicial sets, and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ denote the covariant transport representation for $U$. Assume that $U$ is also a cartesian fibration, so that $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ factors through the subcategory $\operatorname{\mathcal{QC}}_{\operatorname{LAdj}} \subset \operatorname{\mathcal{QC}}$. Then the contravariant transport representation of $U$ is given by the composition

where $\Psi $ is the formation-of-adjoints functor (Construction 8.6.8.16). The proof reduces to the universal case $\operatorname{\mathcal{C}}= \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}$, in which case it is immediate from the definition of $\Psi $.