# Stacks as local quotients or via atlases

If one looks up the definition of a Deligne--Mumford stack or an Artin stack, one usually finds something like:

A DM (resp. Artin) stack is a stack $$X$$ satisfying [insert condition in the diagonal here] and for which there exists a representable etale (resp. smooth) surjection $$U\to X$$ from a scheme $$U$$.

On the other hand, we are also told that we should think of a DM stack as something which is locally isomorphic to $$Y/G$$ where $$G$$ is a finite group acting on a scheme $$Y$$, and that we should think of an Artin stack as something which is locally isomorphic to $$Y/G$$ where $$G$$ is an algebraic group acting on a scheme $$Y$$. This discussion is not limited to algebraic stacks: one can have a similar discussion in the context of differentiable or topological stacks.

I have found relatively little material, however, on comparing and contrasting these two styles of definitions. The definition in terms of atlases seems by far the most common in the literature. I am thus lead to ask:

Is there a particular reason to prefer the definition in terms of atlases over the definition in terms of local quotients? At what point in the theory and/or applications of DM/Artin stacks is the distinction between the two styles of definitions relevant?

(Possibly relevant subquestion, but not the main question: what is involved in passing between the two styles of definitions?)

I am actually most interested in the answer to this question in the context of topological stacks, but I would also be happy to have an answer in the algebraic context.

• I don't have enough experience to say anything serious. In the case of differentiable stacks, not every stack can be representable locally as quotient. For any geometric/differentiable stack $\mathcal{D}$, one can associate a Lie groupoid $\mathcal{G}$, with $\mathcal{D}\cong B\mathcal{G}$. A theorem of Moerdijk and Pronk says that only proper and etale Lie groupoids are representable as quotients of Lie group actions on manifolds. – Praphulla Koushik Nov 16 '19 at 8:29
• So, atleast in differentiable stacks, there are interesting Lie groupoids which are not representable locally as Lie group actions on manifolds. May be even for proper etale Lie groupoids, there are some constructions which does not give proper etale Lie groupoids. So, restricting to proper etale Lie groupoids might not be sufficient. So, restricting to stacks that are represented locally as Lie group actions on manifolds (Algebriac group actions on schemes). – Praphulla Koushik Nov 16 '19 at 8:29
• One reason is Artin's axioms for a groupoid valued functor $X$ on schemes to be an algebraic stack. The proof involves a step-by-step construction of the local charts $U \to X$. To the best of my knowledge, this is the main technique used to prove moduli stacks are algebraic. In practice, these axioms can often be checked easily by constructing a suitable deformation theory for the objects being parametrized by $X$. It would be substantially harder (and seemingly irrelevant) to prove $X$ was locally $U/G$ in applications. – witt-voodoo Nov 16 '19 at 14:59

I'm not entirely sure if this qualifies as an answer, but it is certainly too long for a comment. I hope that someone else will give a better answer.

If you want to define an algebraic stack as something that is locally isomorphic to $$[Y/G]$$, then you have to say what you mean by this isomorphism. So you need an a priori definition of the objects you're working with, in the same way that you need to know what a topological space is before you can say that a manifold is something that is locally homeomorphic to an open in $$\mathbf R^n$$. So at the very least you cannot do away with the generalities about stacks.

(I guess you could define a manifold by only Čech data, but then you have an awkward time proving things do not depend on the chosen Čech cover, and so on so forth. Similarly, you could try to give a concrete model for what glueing data for stacks might look like, but again this leads to all sorts of problems. This is not a reasonable way to do geometry.)

However, there are actually some highly nontrivial results that many stacks are actually locally quotient stacks. In its most general form:

Theorem (Alper, Hall, and Rydh). Let $$S$$ be a quasi-separated algebraic space, let $$\mathscr X$$ be an algebraic stack that is locally of finite presentation and quasi-separated over $$S$$ whose stabilisers are linearly reductive. Then $$\mathscr X$$ is étale-locally a quotient stack, i.e. for every $$x \in X$$ (closed in its fibre of $$\mathscr X \to S$$) there exists an étale map $$[\operatorname{Spec} A/\operatorname{GL}_n] \to \mathscr X$$ whose image contains $$x$$.

The linear reductivity assumption is a bit strong (especially in characteristic $$p$$), but it seems likely to me that at least some assumption is needed. In their forthcoming Annals paper (which deals with the case where the base $$S$$ is an algebraically closed field) they give examples that a more precise version of the theorem is false if the stabiliser is not reductive at $$x$$, or reductive at $$x$$ but not affine in a neighbourhood of $$x$$.

I would actually be interested as well in a concrete example where the more general definition is useful for some geometric problem (stated outside the world of stacks). In practice many applications of algebraic stacks (notably moduli problems) rely on some concrete construction that locally boils down to some quotient by an action of a reasonable algebraic group.

One final comment as that you can already ask your question for algebraic spaces, which are defined by an étale equivalence relation. This is more general than a finite group action, so you could ask in what cases we're interested in something where there's no group.

• By "Čech data" data do you mean an equivalence relation (or groupoid) on an atlas? You can form the formal quotient by groupoids without having an existing category of stacks for it to live in. It's really not that bad. Some people do it. I forget where, I want to say "non-commutative geometry," that that has another option of functions on the groupoid. Given a groupoid, you can define a subobject as an equivariant subobject. And then you can define morphisms between groupoids as subobjects of the quotient that look like graphs of morphisms. – Ben Wieland Nov 18 '19 at 1:27