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I'm not an algebraic geometer, but I've been doing a little armchair reading on stacks just to see what all the fuss is about. It appears that stacks are used in algebra in a similar way that groupoids are used in analysis, e.g. to handle pathological group actions and moduli spaces geometrically. I'm wondering if the language of groupoids is also available in algebraic geometry (it seems like it should be) and, if so, what some of the technical advantages and disadvantages it has in comparison to stacks.

Thanks!

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Stacks in general are used to formalize gluing while preserving more structure, not to deal with pathological group actions. Algebraic (and DM) stacks are used for this purpose (they are stacks of groupoids!), but abstract stacks are much more general animals. –  Harry Gindi Apr 20 '10 at 16:26
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Algebraic stacks are traditionally usually expressed in the language of groupoids. However, more than one groupoid can present the same stack. All this works the same way in algebraic, differential and topological setup. See for example the article of Moerdijk on orbifolds as Lie groupoids (orbifolds being a very special class of differentiable stacks), and compare with Robbin-Salamon on Deligne-Mumford orbifold. –  Zoran Skoda Apr 20 '10 at 16:28
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Harry, of course, the original motivation of Grothendieck was that whenever he wanted to create some moduli space he had trouble with automorphism groups and not having nice quotients to describe this; so he decided that it is a universal phenomenon and he had to solve it by universal trick: putting the automorphism into the definition. So pathological group(oid) actions are one of the principal purposes of introducing stacks. –  Zoran Skoda Apr 20 '10 at 16:31
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(I'm by no means an expert! But) I agree with Skoda about stack describing quotients by bad group actions. - @Harry: what do you mean by "a way to get a better handle on glueing"? –  Qfwfq Apr 20 '10 at 17:08
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Paul, working with the groupoid is sort of like working with an atlas, so it's useful for some computations and constructions but the stack is the more fundamental object...because one can really do geometry with it (and take its cohomology, do intersection theory on it, etc.); that is really the point of making a theory about the stacks instead of working just with "groupoid presentations". Since Gindi's focus is more on categorical/algebraic formalism and less on geometry, his comments are not based on relevant experience for answering your geometrically-motivated question. –  BCnrd Apr 20 '10 at 17:42

1 Answer 1

I will try to answer this question in a way relevant to more than one field, however, to be honest, I'm rather unconventional in the sense that my experience in this area stems from topological and differentiable stacks rather than algebriaic ones. However, from a formal view point, everything is the same.

So let us work in a "background Grothendieck site", which can be topological spaces, differentiable manifolds, or schemes over a fixed base (the first two with the "open cover topology"). Let's call an object in category a "space".

If G is a group object, and X is a space with an action of G, we can take the corase qoutient. However, this is generally not a "nice space" in the senes that the quotient loses a lot of information about the action. In the context of topological spaces, a "nice quotient" would be one that makes the map $X \to X/G$ into a principal G-bundle. However, you need some really nice conditions on the action for this to work in general. E.g., the action needs to be free.

Note, if we consider the projection $X \to X/G$ "coming from the left and the top" and take the pullback, the action is free if and only if the pullback is $G \times X$.

Now, from G acting on X, we can construct the so-called "action groupoid", which has objects X, and arrows $G \times X$, where $(g,x):x \to gx$. This is a groupoid object in spaces, denote it by Act_G(X). Given a space T, we can pretend it's a groupoid object, with all idenity arrows. We can consider Hom(T,Act_G(X)), where the Hom is taken in the 2-category of groupoid objecs, hence, this Hom gives a groupoid, not just a set (the 2-cells are internal natural transormations). The assignment $T \mapsto Hom(T,Act_G(X))$ defines a presheaf of groupoid on spaces. Moreover, there is a canonical morphism $X \to Hom(Blank,Act_G(X))$ of presheaves of groupoids (where X is identified with its representable presheaf). If form a weak 2-pullback by having this morphism "coming from the left and the top", the pullback becomes $G \times X$, one projetion becomes the "source map" and another the "target map". If we say that $Hom(Blank,Act_G(X))$ is our new qoutient, then "the action becomes weakly free".

So far, everything I did was using groupoids. So where to stacks enter the game? Well, $Hom(Blank,Act_G(X))$ is not a very good quotient because if Y is another space, maps from Y to it don't see $Hom(Blank,Act_G(X))$ as "being like a space". E.g. if we are in topological spaces, we can't define maps from Y into it by defining them on the opens of Y in a way that agrees. (For more explanation see my answer to Stacks in the Zariski topology?). What we have to do is "stackify" the presheaf of groupoids $Hom(Blank,Act_G(X))$, (call its stackification X//G). This makes X//G behaves like a spacein the sense that, e.g. in topological spaces, we can defined maps into it by mapping out of opens in a way that agrees. Since stackification preserves finite weak 2-limits, if we form the same pullback diagram but insetad with respect to $X \to X//G$, we still recover the action grouoid and the action is still "weakly free". Morevoer, the projection $X \to X//G$ becomes a G-torosor (principal G-bundle).

So, just using the groupoid, allowed us to keep track of the isotropy data, but not in a way that we get something like a space. For that we need stacks.

If instead of using the groupoid $Act_G(X)$, we used any groupoid object, we can still stackify its associated presheaf of groupoids. The stacks we get in this way are "geometrical", and give rise to topological, differentiable, and Artin stacks respectively.

A final remark. In the comments, it was said that in some sense groupoids are "atlases for stacks". To see this, let's go to manifolds. Given a manifold M described in terms of an atlas, we can construct a Lie groupoid whose objects are the disjoint union of the elements of the atlas, and where we have an arrows from (x,U_a) to (x,U_b) whenever x is in the intersection of these two. This Lie groupoid's associated stack is the same as the manifold M. More generally, given an orbifold described in terms of charts, we can also construct a Lie groupoid with respect to these charts, and its associated stack "represents the orbifold". In general, you can think of Lie groupoids as "generalized atlases" which describe the geometric object which is their associated stack. Of course, just as a manifold can be described by more than one atlas, a differenitbale stack can be described by more than one Lie groupoid.

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