Timeline for Diffeology as a sheaf on the site of smooth manifolds
Current License: CC BY-SA 3.0
20 events
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Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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Aug 8, 2014 at 14:20 | comment | added | Todd Trimble | Yeah, I think I had already admitted that a bit more is needed along those lines. Anyway, FWIW, I wrote out something short for the 1-categorical case here: ncatlab.org/toddtrimble/published/Karoubi+envelope. It is basically a repetition of an argument already in the nLab here: ncatlab.org/nlab/show/Karoubi+envelope#InComponents I haven't gone through the details of making this higher-categorical. | |
Aug 7, 2014 at 13:00 | comment | added | David Carchedi | @Todd: I think a bit more is needed, because we are talking about the $2$-category of psuedo-functors into the $2$-category of groupoids, not the $1$-category of strict functors- but the "more that we need", I explain above, which works not just for groupoids, but for $n$-groupoids for any $n$, even $n=\infty$. But thanks for pointing out that the $1$-categorical result generalizes for arbitrary Cauchy-complete categories! I'll wait for your proof, since perhaps it directly generalizes to higher categories. | |
Aug 7, 2014 at 11:43 | comment | added | Todd Trimble | Okay, thanks for clarifying! But I think it ought to be true that for any Cauchy-complete $D$, the restriction functor $[\bar{C}, D] \to [C, D]$ is an equivalence. Then apply that to $D = Gpd$. (Well, that's a coarse 1-categorical statement, so something more is needed, but maybe not too much more.) | |
Aug 7, 2014 at 11:01 | comment | added | David Carchedi | @Todd: Great, thanks! What I did above, is just show that the set-valued presheaf result implies the infinity-presheaf result, which in particular, implied the groupoid-valued presheaf result that Eugene wanted. | |
Aug 7, 2014 at 10:26 | comment | added | Todd Trimble | Well, the set-valued presheaf result doesn't require a lengthy argument. If the argument isn't already in the nLab, I can add it later today. | |
Aug 7, 2014 at 8:35 | comment | added | David Carchedi | @Todd: The result about presheaves is a stronger result. Since the Grothendieck topology on $\mathbf{Open}$ is the same as the one induced by restriction from $\mathbf{Man}$, the result about presheaves implies the one about sheaves. | |
Aug 7, 2014 at 0:29 | comment | added | Todd Trimble | I didn't try to follow all the details of this post, which look complicated. But I didn't see where you got to the point of discussing sheaves as opposed to just presheaves. Did I miss something? | |
Aug 7, 2014 at 0:07 | history | edited | David Carchedi | CC BY-SA 3.0 |
re-ordered
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Aug 7, 2014 at 0:02 | comment | added | David Carchedi | OK, I found (and fixed) a small error in the proof. What would be helpful in the future, is constructive feedback however. | |
Aug 7, 2014 at 0:01 | history | edited | David Carchedi | CC BY-SA 3.0 |
added 458 characters in body
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Aug 6, 2014 at 22:28 | comment | added | David Carchedi | Could whoever downvoted, please explain to me why? Because if there is math error, I would like to know. | |
Aug 6, 2014 at 18:33 | history | edited | David Carchedi | CC BY-SA 3.0 |
found a hole in my argument, so gave a new one
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Aug 6, 2014 at 10:20 | history | edited | David Carchedi | CC BY-SA 3.0 |
added 93 characters in body
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Aug 6, 2014 at 7:33 | comment | added | David Carchedi | @AndréHenriques: I have edited my answer to address this. | |
Aug 6, 2014 at 7:32 | history | edited | David Carchedi | CC BY-SA 3.0 |
added more detail
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Aug 5, 2014 at 23:05 | comment | added | David Carchedi | The wiki-article claims that this is true for presheaves of sets (though I'd like a better reference). To go from here to presheaves of $n$-groupoids, one can use the model structure on simplicial presheaves, and Bousfield localize to get a presentation for presheaves of $n$-groupoids. | |
Aug 5, 2014 at 21:12 | comment | added | David Carchedi | Hi Andre, I was just signing on because of that reason. I thought this was true in general, but perhaps that's not right. I'll leave this up for now, so that someone can verify or deny this claim. | |
Aug 5, 2014 at 20:11 | comment | added | André Henriques | David: could you please elaborate on the "which implies that" part of your answer. | |
Aug 5, 2014 at 19:23 | history | answered | David Carchedi | CC BY-SA 3.0 |