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 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>