If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between their underlying effective orbifolds? At first, I thought it should always be true, but now that I think about it, you might need $f$ to be open (in this case, I can prove it). Is this known? (by this I mean, is it known to always hold, even without this open assumption?) Note, this really has nothing to do with the differentiable structure, so you may as well ask this for proper etale topological stacks, in fact, I doubt properness plays a role.



When do maps of ineffective orbifolds descend to their effective part?
Not true:  

Take *X* to be a non-trivial *Z*/2-gerbe on S^2, take *Y* to be a faithful vector bundle over *X*, and take *f* to be the inclusion of the zero section.


The effective quotient of *X* is S^2, and it has no map back to *X*.
In particular, it has no map to *Y* that's compatible with *f*.


