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This question is very naive, but it will help me a lot in getting in to the vast literature about stacks.

The question is this: there are many kinds of stacks (algebraic spaces, DM, algebraic stacks, Artin stacks, geometric stacks...). What would be a catch-phrase you would use to describe them, and more importantly to differentiate their uses one from the other?

I'll give you an example (which may be off because of my lack of familiarity with subject): it seems to me that the catchphrase for algebraic spaces is that they are the result of looking at the "orbit space" of the action of a finite group on a scheme.

What catchphrases would you give for the rest of them (and how would you change the catchphrase I gave for algebraic spaces, if at all) to best describe their utility?

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algebraic stack = categorified algebraic space; algebraic space : scheme = étale topology : Zariski topology –  Martin Brandenburg Jun 29 '11 at 20:13
    
I would prefer if you had this as an answer so people can comment on it. How would you describe the difference in utility between algebraic stacks and algebraic spaces? –  James D. Taylor Jun 29 '11 at 20:37
    
And don't forget non-geometric stacks, say stacks fibred in categories, like that of quasicoherent sheaves. –  David Roberts Jul 1 '11 at 7:22
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2 Answers 2

I just try to give my loose answer waiting for other, more knowledgeable, users to answer the question.

1) "Algebraic spaces are what you obtain by glueing affine schemes in an étale way"

2) "Algebraic stacks are stacks that have an atlas made of (finite dimensional) schemes"

3) "Smooth DM stacks are the algebro geometric equivalent of orbifolds in differential geometry; in general they look locally like the quotient of affine schemes by the action of finite groups"

4) "Artin stacks look locally like quotients of schemes by (possibly positive dimensional) algebraic groups"

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There are also ind-stacks, which are locally quotients by ind-groups. –  Thomas K Jun 29 '11 at 21:52
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Usually, Artin stacks denote the most general form of algebraic stacks. Therefore schemes are algebraic spaces, algebraic spaces are DM stacks, and DM stacks are Artin stacks, which is the most general category of algebraic stacks considered. –  Leo Alonso Jun 30 '11 at 8:19
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Algebraic spaces are non-stacky algebraic (Artin) stacks, and DM-stacks, although stacky, have only finite stabilizer groups (or étale stab. groups; I'm sloppy here) for all points on the "underlying space".

To get to a scheme from an algebraic space, one can either pass to an étale cover or just restrict oneself to an open dense subspace. To get to a scheme from a DM-stack, one can still pass to an étale cover, or, if one prefers finite group actions, pass to a Zariski open, which supports a $G$-bundle with total space an affine scheme, and $G$ is a finite group. The DM-stack is covered by such Zariski opens (though with different groups $G$). But for a general Artin stack there is really a long way to get to a scheme (or alg. space): if schemes are floating on the surface of the ocean, then Artin stacks rest deep in the ocean --- they are "covered" by schemes but the relative dimension are usually positive.

This somehow explains (at least to me) why for algebraic spaces (or more generally, DM-stacks) it suffices to use the étale topology to define and compute cohomology of sheaves, whereas for general Artin stacks, lisse-étale topology is necessary.

Edit: This is actually a comment for 4) in unknowngoogle's answer. It is possible to have a continuously varying family of algebraic groups parameterized by a space (i.e. not iso-trivial, even if one passes to an algebraic stratification of the base space). Therefore, I don't think all algebraic stacks (say over $k$) are locally quotients of $k$-schemes by $k$-algebraic groups: one needs group schemes over a larger base. There exists a stratification of any algebraic stack such that the inertia is flat over each stratum, but one cannot make this flat family a constant family.

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