When is a stack (NOT) geometric? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T20:47:45Z http://mathoverflow.net/feeds/question/27399 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27399/when-is-a-stack-not-geometric When is a stack (NOT) geometric? David Carchedi 2010-06-07T22:22:34Z 2010-06-08T16:53:00Z <p>Following the terminology of $n$-Lab, a geometric stack $\mathcal{X}$ on a site $\mathcal{(C,J)}$ is a stack for which there exists a representable epimorphism $X \to \mathcal{X}$ from an object $X$ of $\mathcal{C}$ (viewed as a representable presheaf). Equivalently, $\mathcal{X}$ is geometric if and only if there exists a (nice enough) groupoid object $\mathcal{G}$ in $\mathcal{C}$ such that $\mathcal{X}$ is (2-iso to) the stackification of the strict presheaf of groupoids $Hom(blank,\mathcal{G})$ (where nice enough essentially means that you can take enough iterated pullbacks in $\mathcal{C}$ to form a $\mathcal{C}$-enriched nerve).</p> <p>My question is, is there a more intrinsic definition of geometric stack? By "more-intrinsic" I mean a definition that does not use the existential quantifier. For example, if our site is topological spaces, we know a presheaf is representable if and only if it sends colimits in $Top$ to limits in $Set$. Since geometric stacks are in some sense a natural generalization of representable presheaves, it would seem natural to expect a similar characterization of geometric stacks (at least in the case when our site is nice enough, like $Top$).</p> <p>I ask this mostly because, although in some circumstances there is a natural atlas or a natural choice of representing groupoid object around to try to prove that something is a geometric stack, proving that a stack is NOT geometric becomes very difficult when the definition involves the EXISTENCE of a nice atlas.</p> <p>If someone only knows the answer for certain sites, this is still interesting to me.</p> http://mathoverflow.net/questions/27399/when-is-a-stack-not-geometric/27487#27487 Answer by Urs Schreiber for When is a stack (NOT) geometric? Urs Schreiber 2010-06-08T16:53:00Z 2010-06-08T16:53:00Z <p>One problem -- or at least one characteristic aspect -- of the notion of <a href="http://ncatlab.org/nlab/show/geometric%20stack" rel="nofollow">geometric stack</a> is of course that it makes explcit reference to a fixed chosen site. Different sites may give rise to equivalent toposes and still to different notions of geometric stacks.</p> <p>One approach is to make that extra information an explicit extra piece of data in a controlled way. This is effectively what is achieved by the notion of <a href="http://ncatlab.org/nlab/show/geometry+(for+structured+(infinity%2C1)-toposes)" rel="nofollow">geometry for a structured topos</a>. In terms of this one can then characterize geometric stacks fairly intrinsically. For instance in <a href="http://ncatlab.org/nlab/show/Structured+Spaces" rel="nofollow">Structured Spaces</a> it is shown that with a standard choice for "geometry" a <a href="http://ncatlab.org/nlab/show/Deligne-Mumford%20stack" rel="nofollow">Deligne-Mumford stack</a> is precisely a "2-scheme" in a suitable sense.</p> <p>Going beyond that, one could ask which "geometries" in this sense are naturally associated to a given topos, without choosing them by hand, such that the corresponding 2-schemes are the natural notion of geometric stack.</p> <p>I think a big step in that direction is achieved in Bertrand Toen's work <a href="http://arxiv.org/PS_cache/math/pdf/0012/0012219v6.pdf" rel="nofollow">Champs affine</a>. As reviewed at <a href="http://ncatlab.org/nlab/show/rational+homotopy+theory+in+an+(infinity%2C1)-topos" rel="nofollow">rational homotopy theory in an (oo,1)-topos</a>, Toen there shows that for stacks or higher stacks on the algebraic site, one can characterize "affine stacks" intrinsically, as the objects of the reflective sub-(oo,1)-category on objects that are local with respect to morphisms that induce isomorphisms in "rational cohomology", where "rational" is as seen by the ground field. </p> <p>Using that intrinsic notion of "affine stack", Toen then gives <a href="http://arxiv.org/PS_cache/math/pdf/0012/0012219v6.pdf#page=97" rel="nofollow">in section 4</a> a definition of geometric oo-stacks.</p> <p>This may or may not be exactly what you are asking for, but I think it does provide some noteworthy indications of the kind of approach that one should think about.</p>