What is the local structure of a Lie groupoid?

Question

A manifold is locally $\mathbb R^n$. An orbifold is locally $\mathbb R^n/\{\text{finite group}\}$. Is there a similar way to think about the local structure of a Lie groupoid $G_1 \rightrightarrows G_0$?

For example, the Lie algebroid determines a distribution on $G_0$, and I think that it is locally integrable? What extra structure "local" structure of the groupoid is there (e.g. this distribution loses the data about the automorphisms of a point).

Finally, is the right notion of "local structure" well-behaved under equivalences of groupoids? If it is, then I really should change the title of the question to "What is the local structure of a smooth stack?".

Answer 1

This is roughly what I know about how to do devissage on algebraic stacks. It may or may not apply to differentiable stacks.

1. Given an Artin stack X (say of finite type over the base, a field if you like), there is an nonempty open substack U in X over which the inertia is flat. This means the automorphism groups (a family of Lie groups) is varying "smoothly" over points in U.

2. When the inertia is flat over X, one can take the "rigidification" of X with respect to the full inertia. This means we modulo the automorphisms, and get an algebraic space Y (non-stacky stack, which might be singular, so I don't want to call it a manifold).

3. The morphism X --> Y is called a gerbe, namely fibers are classifying spaces of the automorphism groups. Then one can use many techniques (like Leray spectral sequence etc.) to deduce/compute something (like cohomology) on X from Y and the fibers.