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I have been reading about stacks lately and in the definition of Artin stacks, one requires the diagonal to be representable and quasi compact with a smooth surjective morphism from a scheme to the stack. In Sorger's notes on principal bundles, the definition of algebraic stacks requires the additional condition of the diagonal being separated. I'm looking for interesting examples which are Artin stacks but not algebraic in the sense above or an explanation as to what difference does separatedness make.

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