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Apr 13, 2017 at 12:58 history edited CommunityBot
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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
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
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
Aug 6, 2014 at 10:20 history edited David Carchedi CC BY-SA 3.0
added 93 characters in body
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
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