On the Wikipedia page Sheaf, under the section "The etale space of a sheaf," the author claims that the etale space of the sheaf of (continuous) sections of a continuous map $Y \to X$ is (homeomorphic to) $Y$ if and only if $Y \to X$ is a covering map. This doesn't seem quite right to me. Isn't the appropriate condition weaker? I think that it should be something like a local homeomorphism with discrete fibers, unless I misunderstood the Wikipedia article. Am I correct?
etale space v. covering space
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sheaf-theory
1 Answers
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You are right. The condition is that $Y\to X$ is an etale map, i.e. a local homeomorphism. (it is the standard equivalence between sheaves and etale maps)
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1I see. I was somewhat bothered by the fact that the etale space of a sheaf has discrete fibers, but I see now that the local homeomorphism condition actually implies this. – 2011-06-13