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.

3more comments