What is the signficance of the existence of a moduli stack to a moduli problem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:40:12Z http://mathoverflow.net/feeds/question/47349 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47349/what-is-the-signficance-of-the-existence-of-a-moduli-stack-to-a-moduli-problem What is the signficance of the existence of a moduli stack to a moduli problem? James D. Taylor 2010-11-25T17:55:59Z 2010-11-25T17:55:59Z <p>This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me.</p> <p>Let's say we begin with a naive moduli problem: we want a moduli space (whatever space would mean) that would "classify" all gadgets.</p> <p>The first, obvious attempt, is to set \$F\$ as a functor from \$Schemes\$ to \$Sets\$ like so: \$F(S):=\$ all the families of gadgets parametrized by \$S\$. Sometimes \$F\$ will not be representable by a scheme.</p> <p>Now we may ask more cleverly, is there a stack fibered in groupoids given by \$F(S):=\$ all families of gadgets parametrized by \$S\$, with isomorphisms corresponding the various \$S\$-automorphisms.</p> <p>It seems that people think of stacks as the "natural" object for moduli problems, but I do occasionally see papers proving the existence of a moduli stack. So let me phrase it like this:</p> <p>The failure for the moduli space to be a scheme has to do with non-trivial automorphisms. What does the failure of the moduli space to be a stack indicative of? What does the existence of a moduli stack imply?</p>