Kerodon

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

Example 5.6.1.3 (The Monodromy Representation). Let $f: X \rightarrow S$ be a covering map of topological spaces. Then $\operatorname{Sing}_{\bullet }(\pi ): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a covering map of simplicial sets (Proposition 3.1.4.9). Applying Construction 5.6.1.2 to $\operatorname{Sing}_{\bullet }(\pi )$, we obtain a functor from the fundamental groupoid $\pi _{\leq 1}(S)$ to the category of sets, which we will denote by $\operatorname{Tr}_{X/S}: \pi _{\leq 1}(S) \rightarrow \operatorname{Set}$ and refer to as the monodromy representation of $f$. Concretely, it is given on objects by the formula $\operatorname{Tr}_{X/S}(s) = \{ s\} \times _{S} X$.