Kerodon

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Theorem 5.2.7.1 (The Fundamental Theorem of Covering Space Theory). Let $S$ be a topological space which is semilocally simply connected. Then the construction $X \mapsto \operatorname{hTr}_{X/S}$ determines an equivalence of categories

$\{ \textnormal{Covering maps f: X \rightarrow S} \} \rightarrow \operatorname{Fun}( \pi _{\leq 1}(S), \operatorname{Set}).$