Timeline for Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
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| Jul 11, 2022 at 16:50 | vote | accept | Chris Schommer-Pries | ||
| Jul 1, 2022 at 20:49 | answer | added | Dan Petersen | timeline score: 5 | |
| Mar 12, 2017 at 14:36 | history | edited | Willie Wong |
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| Mar 27, 2016 at 19:14 | comment | added | Jonathan Gleason | @IlanBarnea It's interesting that you phrase it like this. My perspective was what you might say is the dual: that pro-$C^*$-algebras are needed to extend Gelfand Duality to quasitopological spaces (or really, just beyond compact spaces). (Of course, there is a Gelfand Duality for locally compact spaces, but this requires what to my mind is an awkward choice of morphisms if you want the duality to be a categorical one.) | |
| Aug 17, 2015 at 21:56 | comment | added | Ilan Barnea | I was quite surprised to know that quasitopological spaces really needed to extend the Gelfand-duality to pro-$C^*$-algebras: mathoverflow.net/questions/214566/…. Maybe there is some connection? | |
| Mar 15, 2013 at 21:25 | comment | added | Chris Schommer-Pries | Hi Oscar! If you find the time, can you elaborate a bit more? Your comment raises two questions for me. (1) Why do we care that maps to the colimit are representable over a single neighborhood? My guess is that this could be problematic for the fibration property. Is that what you had in mind? (2) If so, is there an easy example of a family of germs of functions (a map to the colimit) which fails to lift? I can construct such families with open parameter spaces, but I haven't come up with one where that parameter space is a disc, which is all we care about for Serre fibrations. | |
| Mar 15, 2013 at 17:40 | comment | added | Oscar Randal-Williams | Briefly: one wants a compact family of sections over a closed set C to be represented by a family of sections over a single open neighbourhod U of C. But the maps in the direct system will typically not be injective, so maps from a compactum to the colimit will not typically be representable over a single neighbourhood U. | |
| Mar 15, 2013 at 16:55 | comment | added | Dan Ramras | David Ayala once explained to me why he needed quasi-topological spaces in order to deal with cobordism categories in which the manifolds were equipped with extra geometric data on a bundle over the manifold (I think he was particularly interested in equipping the manifolds with a principal bundle and a flat connection on that bundle). I do not recall his explanation, though, but at least I would suggest you ask him your question directly. | |
| Mar 15, 2013 at 16:15 | history | edited | Chris Schommer-Pries | CC BY-SA 3.0 |
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| Mar 15, 2013 at 13:47 | history | asked | Chris Schommer-Pries | CC BY-SA 3.0 |