Recall that an orbifold is an etale and proper differentiable stack $X$. Etale means that it admits an etale atlas $M \to X$ from a manifold $M$ (which is to say it is represented by an etale Lie groupoid $G$ with $M$ as its space of objects, and here etale means that the source map is a local diffeomorphism). At least in the smooth setting, this is also equivalent to the more intrinsic characterization that the automorphism groups of each point of the stack $X$ are all discrete (and that X is a differentiable stack, i.e. has SOME atlas). Proper means that the the diagnoal map $X \to X \times X$ is proper, which is a nice intrinsic discription. In terms of Lie groupoids, this means that $G \to M \times M$ via $s \times t$ (source and target) is a proper map of manifolds. Now, an orbifold is called effective if it is the stack of torsors for a proper etale EFFECTIVE etale Lie groupoid (I will explain below what that means). My question: Is there a way to say this in terms of properties of the stack itself without mentioning any Lie groupoids (such as having each automorphism group act effectively somewhere)? I actually care about this intrinsic characterization when $X$ is not assumed to be proper, but merely etale. I also care about the topological case.

Explanation of what effective means for Lie groupoids:

Given any space $M$, we can construct a (highly non-Hausdorff) Lie groupoid $\Gamma(M)$ out of germs of local diffeomorphims on $M$, given the arrow space the etale space topology associated to the canonical sheaf of local diffeos. Given a Lie groupoid with objects $M$, an arrow $g:x \to y$ induces a germ of a local diffeomorphism from $x$ to $y$ as follows: Let $U$ be a neighborhood of $g$ so small that the source and target maps $s$ and $t$ are diffeomorphisms on it. Then take the germ of $t \circ \left(s|_{U}\right)^{-1}$. This produces a homomorphism $G \to \Gamma(M)$. $G$ is effective if this homomorphism is injective, i.e., if every arrow is determined uniquely by its germ.

UPDATE: Actually, it would be more helpful to know how to "naturally" extract the "effective PART" of an etale stack, without going back to the groupoid. The effective part is the image of $G$ under the canonical map to $\Gamma(G)$. This is indeed functorial at the level of stacks, but, I was hoping for a nice description of this functor in stacky language, instead of passing to a presenting groupoid. Once this is done, being effective is just the same as being equal to your effective PART. Maybe this can be done topos-theoretically, since etale stacks are equivalent to etendue.

  • $\begingroup$ I'm not sure if this is exactly right, but I'll take a swing here. The intrinsic characterisation should probably be phrased as some property of the morphism from the inertia stack $\Lambda X$ to $X$. I think effective is equivalent to this morphism being etale, but I haven't checked it so I could easily be wrong. $\endgroup$ – Jeffrey Giansiracusa Sep 21 '10 at 20:04
  • $\begingroup$ I don't think that can be correct, really, though it depends on which map from the inertia stack you mean. Consider for example $B\mathbb{Z}/2$. Then the inertia stack is just the disjoint union of two copies of this, and if the map is just the obvious projection, then this is a disconnected 2:1 cover, which is (unless I'm way off base) etale. $\endgroup$ – Simon Rose Sep 21 '10 at 20:24
  • $\begingroup$ Properness of the diagonal means $X$ is separated, not proper. $\endgroup$ – Laurent Moret-Bailly Sep 22 '10 at 9:40
  • $\begingroup$ @Laurent: For ALGEBRAIC stacks, I believe you are correct. But for topological/differentiable stacks, you want the diagonal to be a (representable) proper map, i.e., the pullback of the map along any map coming from a space becomes a proper map of spaces. (The diagonal is alway representable since it comes from a groupoid) $\endgroup$ – David Carchedi Sep 22 '10 at 12:39
  • $\begingroup$ @Laurent: Did I misunderstand? Are you just saying the terminology of having a proper diagonal is that $X$ is called separated? $\endgroup$ – David Carchedi Sep 22 '10 at 14:47

I answer this in http://arxiv.org/abs/1212.2282. An etale stack $\mathscr{X}$ is effective if and only if the substack assigning each manifold $M$ the groupoid of local diffeomorphisms $$M \to \mathscr{X}$$ is actually a sheaf, i.e. if and only if this groupoid is (equivalent to) a set.


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