Timeline for Categorical construction of the category of schemes?
Current License: CC BY-SA 3.0
16 events
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| Aug 22, 2017 at 19:42 | history | edited | Harry Gindi | CC BY-SA 3.0 |
Fixed broken link.
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| Aug 22, 2017 at 19:42 | comment | added | Harry Gindi | @JohnGowers It's the "Master Course on Stacks". It's been uploaded to the nlab. Going to fix the link now. | |
| Apr 9, 2015 at 17:17 | comment | added | John Gowers | @Harry - The link is down again. Was the set of notes the same as his PhD thesis (perso.math.univ-toulouse.fr/btoen/files/2015/02/These.pdf)? | |
| Mar 1, 2011 at 3:44 | comment | added | David Roberts♦ | @Harry - Toen's page has moved, would you mind updating the link to the notes. I presume this is the new url: ens.math.univ-montp2.fr/~toen/m2.html | |
| May 31, 2010 at 0:55 | comment | added | David Carchedi | @Harry: Ok, thanks. Nonetheless, it still makes sense to consider "Artin stacks" on the site of topological spaces, as simply being "topological stacks" (stacks of principal bundles for topological groupoids). This was my source of confusion. | |
| May 31, 2010 at 0:54 | comment | added | Harry Gindi | Topological manifolds and differential manifolds are the two he gives as examples. To my knowledge, there is no reason why we should expect it to work for arbitrary topological spaces. | |
| May 31, 2010 at 0:47 | comment | added | David Carchedi | @Harry: Since I'm not very familiar with Toen's work, I just wanted to make sure I didn't say anything wrong here, so, please correct me if I did. For example, are topological spaces and manifolds examples of geometric contexts? I believe that you mentioned they were when we discussed extending Artin stacks to different sites a few weeks ago. | |
| May 31, 2010 at 0:41 | comment | added | Harry Gindi | @BCnrd: In HAG II, T-V try out the theory with a notion of complicial algebraic geometry, where SSets are replaced with Complicial sets, and Brave New Algebraic geometry, where SSets are replaced with spectra (in the algebraic topology sense). These are (according to T-V) very interesting and very different from the ordinary and simplicial (derived) cases. | |
| May 31, 2010 at 0:31 | comment | added | BCnrd | @David: fair enough, I had in mind the setting of alg. spaces/stacks as in the question. For Toen et al., can one not copy the def'ns of Artin: replace "scheme" with "(nice) topological space", "smooth manifold", etc.? To use functors on affines it can be appealing to replace schemes with open affine covers so as to be "more categorical". But for the interesting cases of whatever Toen is doing, does one lose something by not going to that extreme and taking as known a more interesting class of building blocks (like manifolds, etc.) as functors on some basic class of test objects? | |
| May 31, 2010 at 0:17 | comment | added | David Carchedi | @BCnrd, I am certainly not an expert either, but, I don't think Toen's definition is meant to REPLACE that of a scheme, but rather, translate it into a different language- that of category theory? Why? Because you can then "categorify it"- so you can, in some sense, talk about "schemes up to homotopy". Also, as Harry points out with Geometric contexts, since category theory is a universal language, such a definition allows you to extend ideas of algebraic geometry to other fields (e.g. topology and differential geometry). | |
| May 31, 2010 at 0:15 | comment | added | Harry Gindi | Well, at least in the context of HAG (the tiny part of it that I've read), one doesn't have the option of using actual geometric things like locally ringed spaces, since one is not working with CRings. One instead works with an arbitrary symmetric monoidal model category, which forces one to define things like flatness, faithfulness, smoothness, unramifiedness, finite presentation, etc. directly by their functorial properties. I'm no expert, and I don't know about the value of it to algebraic geometers, so I'll defer to your expertise on that question. | |
| May 31, 2010 at 0:00 | comment | added | BCnrd | Harry, I won't pass judgement on HAG since I know nothing about it, but I am suspicious of value of def'ns which don't allow to do something hard to express without them. For alg. space/stacks, the "proof of concept" is seen geometrically by accepting schemes as known and considering functors or fibered categories which admit "covering" by scheme for suitable topology: similar to atlas def'n of manifold. I don't know any advantage for alg. spaces/stacks which is attained by less geometric def'ns; for many serious foundational proofs one reduces to thms for (non-affine) schemes anyway. | |
| May 30, 2010 at 23:07 | history | edited | Harry Gindi | CC BY-SA 2.5 |
added 461 characters in body
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| May 30, 2010 at 22:56 | comment | added | Harry Gindi | @BCnrd: Obviously you would know much better than I would, but it is my understanding that the point of developing it in this way is as something of a proof of concept of the vastly more general framework developed in HAG. | |
| May 30, 2010 at 22:53 | comment | added | BCnrd | But this is just dressing up the "old-fashioned" ringed space definition in fancy language, so is there any purpose to it beyond obsfucation of the geometric origins of the subject? | |
| May 30, 2010 at 22:49 | history | answered | Harry Gindi | CC BY-SA 2.5 |