Kerodon

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

Exercise 5.1.5.3. Let $Q$ be a partially ordered set, let $\operatorname{Chain}[Q]$ denote the collection of all finite nonempty linearly ordered subsets of $Q$ (Notation 3.3.2.1), and let $\mathrm{Max}: \operatorname{Chain}[Q] \rightarrow Q$ be the map which carries each element $S \in \operatorname{Chain}[Q]$ to the largest element of $S$.

  • Show that the induced map of nerves $\operatorname{N}_{\bullet }(\mathrm{Max}): \operatorname{N}_{\bullet }(\operatorname{Chain}[Q]) \rightarrow \operatorname{N}_{\bullet }(Q)$ is a locally cocartesian fibration.

  • Show that, if $Q = [n]$ for $n \geq 2$, the functor $\operatorname{N}_{\bullet }(\mathrm{Max}): \operatorname{N}_{\bullet }(\operatorname{Chain}[Q]) \rightarrow \operatorname{N}_{\bullet }(Q)$ is not a cocartesian fibration.