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I have seen interesting geometric intuition answers about flatness, for example here and here.

I am not sure whether it still applies for morphisms of stack, and if yes how? In partitcular,

  • Can we say something about equidimensinal fibers of a flat morphism of stacks?

  • Can we say something about the "mass" of fibers in a flat morphism of stacks $f: M \to \Bbb A^1$? (the fibers seen as groupoids in the moduli space $M$ have a cardinality called mass)

Thank you!

Conjecture
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    A flat morphism is, pretty much by definition, a morphism whose fibers and derived fibers agree. That’s the significance of flatness in general; it’s that you don’t have to take derived tensor products to get the “morally correct” fibers. – Qiaochu Yuan Oct 27 '20 at 18:33
  • Many thanks @QiaochuYuan, could you please help me understand the definition of "derived" fibers? is it related to derived categories? I have some notions on stacks. Also, when the fibers and derived fibers agree, do you mean regarding cardinality or dimension? If I look at https://stacks.math.columbia.edu/tag/06FM, it looks like flatness of a morphism of stacks is defined as base change of a flat morphism of algebraic spaces so I am quite confused. – Conjecture Oct 27 '20 at 21:32
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    I mean they agree as in they’re canonically isomorphic. By derived fiber I mean taking derived fiber products; for affine things this means taking derived tensor products instead of ordinary ones, and flatness means exactly that the two are the same. – Qiaochu Yuan Oct 27 '20 at 21:46
  • I see now @QiaochuYuan, many thanks for your help! – Conjecture Oct 27 '20 at 22:49

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