When do maps of ineffective orbifolds descend to their effective part? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:50:32Z http://mathoverflow.net/feeds/question/45983 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45983/when-do-maps-of-ineffective-orbifolds-descend-to-their-effective-part When do maps of ineffective orbifolds descend to their effective part? David Carchedi 2010-11-13T22:23:31Z 2010-11-23T21:03:41Z <p>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.</p> http://mathoverflow.net/questions/45983/when-do-maps-of-ineffective-orbifolds-descend-to-their-effective-part/47147#47147 Answer by AndrĂ© Henriques for When do maps of ineffective orbifolds descend to their effective part? AndrĂ© Henriques 2010-11-23T21:03:41Z 2010-11-23T21:03:41Z <p>Not true:<br> Take <i>X</i> to be a non-trivial <i>Z</i>/2-gerbe on S^2, take <i>Y</i> to be a faithful vector bundle over <i>X</i>, and take <i>f</i> to be the inclusion of the zero section.</p> <p>The effective quotient of <i>X</i> is S^2, and it has no map back to <i>X</i>. In particular, it has no map to <i>Y</i> that's compatible with <i>f</i>.</p>