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Let $\mathcal{G}=(G_{0},G_{1})$ be a groupoid, where $G_{0}$ is the space of objects and $G_{1}$ is the space of morphisms. $\mathcal{G}$ is called etale if both the source and target maps $$ s,t:G_{1}\rightarrow G_{0} $$ are local diffeomorphisms. Being stale is $not$ invariant under Morita equivalence (equivalence of categories).

Could anyone give me a simple example of this fact?

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Take the action of $R$ on itself. The associated groupoid is the equivalence relation. This is Morita equivalent to a point (the action is free and proper). $R\times R$ is not etale.

A better example would be to look at the full holonomy groupoid vs the restriction to a complete transversal.

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