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seeing Eric create diffeology I became annoyed by the poor state that the entry diffeological space was in. So I spent some minutes expanding and editing it. Still far from perfect, but a step in the right direction, I think.
(One day I should add details on how the various sites in use are equivalent to using CartSp)
I have expanded the Properties-section at diffeological space:
added the statement and proof of the full and faithful embedding of smooth manifolds into diffeological spaces;
split off a section of the properties of the ambient sheaf topos and how diffeological spaces sit inside there.
I created Boman’s theorem and added the link to the embedding proof on diffeological space (also corrected a couple of minor typos in the vicinity).
I created Boman’s theorem
Thanks! I was scanning your articles for it, but didn’t see it. Then I thought about it and figured that it is easy to prove (isn’t it? one needs to show that for each higher partial derivatives of a function one can find a curve such that the composite’s $n$-fold total derivative involves as a summand the partial derivatives in question. But that’s obvious.)
I have added that to the list of theorems in the floating differential geometry TOC.
also corrected a couple of minor typos in the vicinity
Thanks! I found some more ;-)
it is easy to prove
Not sure. I’ve not worked through the details myself. The proof in Kriegl and Michor is about a page long.
At diffeological space I have added the remark that the statement proven there, that smooth manifolds embed fully faithfully in diffeological spaces, is a direct consequence of the fact that $CartSp$ is a dense sub-site of $Diff$ and then of the Yoneda lemma.
One can see that this is effectively what the previous proof checks in a pedestrian fashion, but it is maybe useful to have the general abstract version, too.
I have added more of the original references to the References-section at diffeological space.
Andrew, when you have a second, maybe have a look to see if my attributions are precise.
For the purpose of pointers at MO, I have expanded slightly at diffeological space to make it have this series of sub-sections on embeddings of categories:
added also
(with just a pointer to a reference for the moment)
added also the embedding of locally convex vector spaces by cor 3.14 in Kriegl-Michor
Patrick Iglesias-Zemmour kindly pointed out to me by email that the latest version of this book Diffeology now contains, around exercise 72, a discussion of how Banach manifolds faithfully embed into diffeological spaces. So I have now added brief pointers to Banach manifold and to the relevant section of diffeological space. (This really deserves to be expanded on, but I don’t have the time.)
The entry diffeology didn't seem to serve any purpose, so now it redirects to diffeological space. (If somebody wants to revive it, its edit history is at diffeology > history.)
I’ve added a comment that Frölicher proved the full and faithful embedding of (paracompact) Fréchet spaces into diffeological spaces in 1981, and in fact I think he proved paracompact Fréchet manifolds also embed fully faithfully, but he has a funny extra condition to link with some functional/sequential notion of smoothness (see théorème 2 on this page)
On a different note, I’m not sure that convenient spaces do embed into diffeological spaces. My reading of corollary 3.14 at mentioned at #10 above is that it is just Boman’s theorem, and that the $c^\infty$ notion of smoothness agrees with the usual notion on cartesian spaces.
Thanks for further looking into this! This is useful.
Finally cleared this up. There is a faithful but non-full functor from lctvs into diffeological spaces, if we take MB-smooth maps as morphisms between the former, since there are non-continuous conveniently smooth maps. I still don’t know if diffeological isomorphisms are MB-smooth, though. I added to the page a reference to Gloecker’s counterexamples, and clarification about what is meant by smooth maps between lcvts.
added pointer to Patrick Iglesias-Zemmour’s lecture notes Iglesias-Zemmour 18
I have considerably trimmed down the section Embedding of diffeological spaces into smooth sets. It used to contain a proof that $Sh(CartSp)$ is cohesive, and had the result announced in its title only hidden somewhere in that discussion. But the cohesion of smooth sets should instead be discussed there, and so I removed it here and instead included (a complete rewrite of) the proof there.
Here I only kept the actual statement that diffeological spaces are the concrete smooth sets, with the minimum indication of the proof that used to be here. Below that I added pointer to a completely (maybe pedantically) detailed proof, which is now at this Prop. in geometry of physics – smooth sets.
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