Étale maps of spaces and topoi

As we will see below in Example 6.22, every continuous map of topological spaces \(f\colon X \to Y\) gives rise to a morphism of topoi \(f_*\colon \Shv(X) \to \Shv(Y)\) between their sheaf categories. One can show that \(f\) is an étale morphism of topological spaces (that is, a local homeomorphism) if and only if \(f_*\) is an étale morphism of topoi, justifying the terminology.