Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 8.6.0.1. Ultimately, each of the constructions described above gives rise to essentially the same object. However, it will be useful to consider all three, since each reveals different facets of the overall picture. Fix a cocartesian fibration of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.

  • The construction of §8.6.2 can be used to show that $U$ admits a cartesian conjugate $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ (Corollary 8.6.2.4), which is unique up to equivalence if $\operatorname{\mathcal{C}}$ is an $\infty $-category (Corollary 8.6.2.9). However, it is not obvious that the opposite fibration $U^{\dagger , \operatorname{op}}: \operatorname{\mathcal{E}}^{\dagger , \operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian dual of $U$ (in the sense of Definition 8.6.3.1).

  • The construction of §8.6.4 can be used to show that $U$ admits a cocartesian dual $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$, which is uniquely determined up to equivalence (Theorem 8.6.4.1). However, it is not obvious that the opposite fibration $U^{\vee ,\operatorname{op}}: \operatorname{\mathcal{E}}^{\vee ,\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian conjugate of $U$ (in the sense of Definition 8.6.1.1).

  • The construction of §8.6.5 produces a specific example of a fibration $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ which is a cocartesian dual of $U$. From this perspective, it is not obvious that the cocartesian dual is unique. However, it enjoys another form of uniqueness: in §8.6.6, we show that every cartesian conjugate of $U$ is equivalent to the fibration $U^{\vee , \operatorname{op}}$ (Proposition 8.6.6.6). Combining this with the existence of cartesian conjugates (obtained from §8.6.2) and the uniqueness of cocartesian duals (obtained from §8.6.4), we deduce that the diagram (8.77) is commutative: that is, a fibration $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian conjugate of $U$ if and only if $U^{\dagger , \operatorname{op}}: \operatorname{\mathcal{E}}^{\dagger , \operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian dual of $U$ (Proposition 8.6.6.1).