Kerodon

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

Warning 7.4.2.3. The compatibility condition of Definition 7.4.2.2 depends not only on the morphisms $\gamma $ and $\Gamma $, but also on the choice of natural transformations

\[ \alpha : \underline{\Delta ^0}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}|_{\operatorname{\mathcal{E}}} \quad \quad \alpha ': \underline{\Delta ^0}_{\operatorname{\mathcal{E}}' } \rightarrow \mathscr {F}'|_{\operatorname{\mathcal{E}}'} \]

exhibiting $\mathscr {F}$ and $\mathscr {F}'$ as covariant transport representations for $U$ and $U'$, respectively. We will sometimes omit mention of these choices, particularly in situations where there are natural candidates for $\alpha $ and $\alpha '$. For example:

  • Suppose that $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ and $\operatorname{\mathcal{E}}' = \int _{\operatorname{\mathcal{C}}} \mathscr {F}'$ are the $\infty $-categories of elements of $\mathscr {F}$ and $\mathscr {F}'$, respectively. Then we can take $\alpha $ and $\alpha '$ to be the natural transformations of Example 7.4.1.10.

  • Suppose that $\mathscr {F}$ and $\mathscr {F}'$ are obtained from strictly commutative diagrams of Kan complexes $\mathscr {F}_0, \mathscr {F}'_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{Kan}$, and that $\operatorname{\mathcal{E}}= \operatorname{N}_{\bullet }^{\mathscr {F_0} }(\operatorname{\mathcal{C}}_0)$ and $\operatorname{\mathcal{E}}' = \operatorname{N}_{\bullet }^{\mathscr {F}'_0}(\operatorname{\mathcal{C}}_0)$ are the weighted nerves introduced in Definition 5.3.3.1. Then we can take $\alpha $ and $\alpha '$ to be the natural transformations given in Example 7.4.1.11.

  • Suppose $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ and that the diagrams $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ are the homotopy coherent nerve of the strict transport representations $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}_0}, \operatorname{sTr}_{\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}_0}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{Kan}$. In this case, we can take $\alpha $ and $\alpha '$ to be the natural transformations described in Example 7.4.1.12 (which are well-defined up to homotopy).