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Unfortunately, I am lacking chocolat medals (as well as the authority to award them), but thanks to the author (presumably Todd) who graced Karoubi envelope with the proof that smooth manifolds result from open sets by idempotent splitting.
I have added a reference to Lawvere’s Perugia notes where this appeared as an exercise.
Entre parenthèses: it appears to me that it’d be better to have the proof at the page for smooth manifolds and to mention the result only at Karoubi envelope as I think this is kind of a butterfly at Karoubi though a beautiful one but an important result for manifolds.
It would be fine with me either to move it to or reproduce it at smooth manifold, but it would also be interesting to think of how this result relates to results already there at that article.
I’ve reproduced the proof at manifold as this seems to me strategically the best place in proximity of the general disucssion on piecing together there. Feel free to rectify my remarks or headings or placement there!
A notice of the result should be dropped at smooth manifold as well. As the proof requires a lemma on Karoubi envelopes I thought it best to leave a complete version of the proof there.
Yeah, I was hoping not to put it at manifold, since that article discusses a very general notion of manifold going well beyond smooth manifold, and the result doesn’t carry over to the general notion. For example, it doesn’t hold for topological manifolds.
It would be nice to determine how general the result can be made.
This was what I meant with ’strategically’: people who look up the discussion at manifold are more likely to wonder how this fits into the general philosophy of ’manifold’ and what is the generality of the result. Someone who just wants to look up the usual definition gets a pointer to idempotent splittings and the Lawvere remark sits nicely before the tangent bundle section which struggles with charts whereas smooth manifold is geared to $\infty$-theory. But as the content is mainly your work I don’t want to impose, so if you feel uncomfortable with this just give a sign and I will restore the entry.
I agree that smooth manifold is a better place for it.
I’ve moved the text from manfold to smooth manifold.
Thanks for the nice edits!
The text should link to Whitney embedding theorem. (Myself, I am quasi-offline for the time being.)
I am merely copy&paste here, the text is almost 100% Trimble not to forget this guy ’Zack’ from MO who provided the keys to the proof. a really nice piece!
I might eventually provide the link, though I would have to look into the entries to see what you have in mind exactly.
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