Timeline for Stacks in the Zariski topology?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 16, 2010 at 16:02 | vote | accept | David Steinberg | ||
| Apr 16, 2010 at 11:59 | comment | added | Harry Gindi | Ah, I linked Toën's notes in my post. Cours 2 describes a general theory of "geometric contexts", which basically gives you the necessary structure to define "geometric varieties" which correspond to manifolds and schemes, "geometric spaces", which correspond to algebraic spaces and those space-like sheaves of Mumford from geometric invariant theory, and geometric stacks, which correspond to differentiable, topological, or algebraic (Artin) stacks. | |
| Apr 16, 2010 at 11:52 | comment | added | David Carchedi | @Harry, my other comment (about simplicial objects) is not quite what I meant to say. But, what I should have said is that for these stacks to behave nicely (other than what I said above), sometimes the structure maps of the groupoid are required to be some sort of "local fibration". To see this in the topological context, see: Noohi's Foundations of Topological Stacks I. | |
| Apr 16, 2010 at 11:45 | comment | added | David Carchedi | If you have groupoid objects (which means that certain pullbacks need to exist), you can look at those stacks arising as torsors of groupoid objects (these will be the stackification of pseudo-functors of the form Hom( ,G)). Whether or not these deserve to be called Artin stacks in general may indeed be contentious, but this is why I used quotation marks. I agree that you will need additional axioms for these "Artin stacks" to be equivalent to a bicategory of groupoids and torsors, so maybe this is what you mean. What is "Cours 2"? I would be interested in looking at what these conditions are. | |
| Apr 16, 2010 at 10:10 | comment | added | Harry Gindi | I don't know what you're talking about, but it sounds very cool. Where can I read about it? | |
| Apr 16, 2010 at 10:02 | comment | added | David Carchedi | Well, I suppose that yes, I do need a class of groupoid objects to exist such that when I take their enriched simplicial nerve, each structure map admits setions with respect to the Grothendieck topology. | |
| Apr 16, 2010 at 10:00 | history | edited | David Carchedi | CC BY-SA 2.5 |
added 61 characters in body
|
| Apr 16, 2010 at 2:27 | comment | added | Harry Gindi | What you said about Artin stacks being definable over any site is not true. You need a "geometric context" in the sense of Toën. That is, you need a distinguished class of morphisms that commutes with finite sums, is squarable, has finite pullbacks, is compatible with the topology etc. Then an Artin stack is a a stack representable by a "geometric space" in a suitable sense. Your examples are true, but the total statement is not. It's in Cours 2. | |
| Apr 15, 2010 at 20:06 | history | answered | David Carchedi | CC BY-SA 2.5 |