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I've heard people talk about the Euler characteristic of a stack, and I understand that it is sometimes fractional. This tells me that it is not defined in any usual sense like $\sum (-1)^i\dim H^i(X)$. What is the definition? What are good references for this? What is the inspiration for the definition?

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Stacks are closely related to orbifolds, and orbifolds have a definition of Euler characteristic. I think one of the key motivations for orbifold Euler characteristic is that it should be multiplicative under finite covering spaces. –  Ryan Budney Aug 31 '11 at 3:30

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