Automorphism groups and etale topological stacks - MathOverflow

Automorphism groups and etale topological stacks

2011-05-08T01:01:49Z

Recall that an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a topological space. Equivalently, it is a topological stack arising from an etale topological groupoid. It is well known that a differentiable stack is etale if and only if all of its automorphism groups are discrete, but the proof involves foliation theory. It seems this proof cannot be extended to the topological setting. However, clearly every etale topological stack has discrete isotropy groups. This begs the question:

If a topological stack has all of its isotropy groups discrete, is it necessarily etale?

EDIT: By a topological stack, I mean a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable epimorphism $X \to \mathscr{X}$ (not necessarily a local homeomorphism). This is equivalent to saying $\mathscr{X}$ is the stack of torsors for a topological groupoid.

Remark: This question is equivalent to asking if a topological groupoid all of whose isotropy groups are discrete must be Morita equivalent to an etale topological groupoid.

Answer by Angelo for Automorphism groups and etale topological stacks

2011-05-08T11:56:25Z

I don't think this is true. Let $X$ be the quotient of the action of $\mathbb Q$ on $\mathbb R$ by translation. This is a sheaf, and its automorphism groups are trivial. Suppose that there exist a local homeomorphism $U \to X$, where $U$ is non-empty a topological space. Let $V \to U$ be the pullback to $U$ of the $\mathbb Q$-torsor $\mathbb R \to X$; then $V\to \mathbb R$ is a local homeomorphism. By restricting $U$ we may assume that $V = U \times \mathbb Q$; but then $V$ can't be locally connected, and this is a contradiction.

Answer by André Henriques for Automorphism groups and etale topological stacks

2011-05-08T12:27:05Z

Here's a counterexample: the stack associated to the relative pair groupoid of the map $$ ([0,1]\times\{0\}) \cup (\{1\}\times[0,1]) \cup ([1,2]\times\{1\})\;\;\to\; [0,2] \qquad\qquad\qquad\qquad\qquad $$

$$\qquad\qquad\qquad\qquad(x,y)\qquad\qquad\mapsto\;\;\; x$$

Equivalently, this stack can be described as the pushout in the 2-category of stacks of the diagram $[0,1]\leftarrow \{1\} \rightarrow [1,2]$ (where we identify a space with the stack it represents).