Example 5.2.0.5 (The Monodromy Representation). Let $f: X \rightarrow S$ be a covering map of topological spaces. Applying Proposition 5.2.0.3 to the induced map $\operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$, we obtain a functor from the fundamental groupoid $\pi _{\leq 1}(S)$ to the category of sets, which we will denote by $\operatorname{hTr}_{X/S}: \pi _{\leq 1}(S) \rightarrow \operatorname{Set}$ and refer to as the *monodromy representation of $f$*. Concretely, it is given on objects by the formula $\operatorname{hTr}_{X/S}(s) = \{ s\} \times _{S} X$.

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