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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.
I forget if the following is known, and where it is proven:
The homotopy type of a diffeological space (D-topology) is equivalently its cohesive shape (when regarded as a concrete 0-truncated objects in the cohesive $\infty$-topos over smooth manifolds).
(?)
Re #20: Yes.
By Proposition 3.1 in https://arxiv.org/abs/1010.3336 we have a left adjoint functor Diff→Top that sends a diffeological space to its underlying topological space equipped with the D-topology.
This left adjoint functor is a left Quillen functor because it sends generating (acyclic) cofibrations in Diff to (acyclic) cofibrations in Top.
Thus, the functor Diff→Top is homotopy cocontinuous.
The cohesive shape is also homotopy cocontinuous.
These two cocontinuous functors take contractible values on R^n.
Hence, they are weakly equivalent.
Thanks!
But help me, you seem to be using one more bit of information that I am lacking.
Explicitly, I am asking about the functor
$DiffeologicalSpaces \hookrightarrow Sh(CartSp) \hookrightarrow Sh_\infty(CartSp) \overset{Shape}{\longrightarrow} \infty Groupoids$whether it’s naturally equivalent to
$DiffeologicalSpaces \overset{D-topology}{\longrightarrow} TopologicalSpaces \overset{L_{whe}}{\longrightarrow} \infty Groupoids$You seem to be appealing to a homotopical structure on diffeological spaces being compatible with the first of these functors?
[later edit: ah, no, I misread Prop. 3.10 in Christensen-Wu, as per the warning on the next page – it does not hold generally for diffeological spaces – so the following does not work]
Let me see:
From your theorem about shape via cohesive path ∞-groupoid it follows that the first functor in #22 is equivalently the one called $S^D$ (Def. 4.3) in
The second functor in #22 would be called $S\circ D$ there.
So in the notation of that article I am asking for validity/proof of
$S^D \;\overset{?}{\simeq}\; S \circ D \,.$I don’t see exactly that statement in the article, but something close:
Theorem 4.11 together with Prop. 3.10 there says that the homotopy groups of the results of both functors agree assuming they are evaluated on a fibrant diffeological space $X_{fibr}$ (which is one whose smooth singular simpliciat set $S^D$ is Kan, Def. 4.8):
$\pi_n \circ S^D(X_{fibr}) \;\simeq\; \pi_n S \circ D(X_{fibr}) \,.$This is two steps away from the previous statement:
if this isomorphism of homotopy groups is/were induced by a morphism of simplicial sets, then it would constitute a weak homotopy equivalence. This is probably implicit in the proofs, I should chase through them.
if the assumption of fibrancy were unnecessary, we’d be done. Now, this would again follow from your theorem of shape via path $\infty$-groupoids, IF we knew there is fibrant replacement for diffeological spaces in the sense of Christensen – but that they explicitly do not prove.
[edit: ah, looks like both these steps are filled in in H. Kihara, Model category of diffeological spaces (arXiv:1605.06794), in Theorem 1.4 there, using the proof starting p. 33]
Re #23: I would argue as follows.
The Kihara model structure on diffeological spaces is transferred via the smooth singular simplicial set functor Diff→sSet.
The Quillen model structure on topological spaces is transferred via the singular simplicial set functor Top→sSet.
Furthermore, the composition of left adjoints sSet→Diff→Top equals the left adjoint sSet→Top.
The left Quillen functors sSet→Diff and sSet→Top are Quillen equivalences.
Therefore, the left Quillen functor Diff→Top is a Quillen equivalence by the 2-out-of-3 property, hence a homotopy cocontinuous functor.
The Kihara model structure on diffeological spaces is transferred via the smooth singular simplicial set functor Diff→sSet.
But Kihara defines a variant of smooth singular simplicial sets, by using a variant diffeology on standard simplices, in order to enforce existence of horn fillers.
The singular simplicial complex that corresponds to cohesive shape, the one also considered in your concordance article, that’s instead the one that Christensen-Wu use (their Def. 4.3). Isn’t it?
But with this definition, their Theorem 4.10 together with their (counter-)examles of smooth $\pi_n$ differing from D-topological $\pi_n$ proves that the desired equivalence fails.
It seems to me.
But Kihara defines a variant of smooth singular simplicial sets, by using a variant diffeology on standard simplices, in order to enforce existence of horn fillers.
Yes, it looks like my memory of Kihara’s paper was not entirely correct.
So really we need the Christensen-Wu construction, which gives the same weak equivalences, but different cofibrations. They do not prove it is a model structure, however, this is basically what we do in our paper. In fact, in our paper, Dan, Pedro, and I prove precisely the necessary lemmas that Christensen and Wu are missing, see Section 4.c, in particular, Lemma 4.13 is precisely the missing part necessary to complete the construction of a model structure, as Christensen and Wu point out themselves in Remark 4.9 in their paper.
Also, Proposition 4.10 shows that two different geometric realization functors by Kihara and Christensen-Wu are weakly equivalent by constructing an explicit homotopy equivalence between them.
Okay, I’ll have another look at your article.
But do you agree that Christensen-Wu’s results prove that the equivalence $S^D \overset{?}{\simeq} S \circ D$ fails?
They prove
$\pi_n^D(X) \simeq \pi_n S^D(X)$ for every diffeological space $X$ (Theorem 4.11),
$\pi_n^D(X) \neq \pi_n(S \circ D(X))$ for some diffeological spaces $X$ (Example 3.12, 3.20)
So it follows that
For my own future reference,
π_n^D is the nth homotopy group defined by mapping representable spheres into a diffeological space,
π_n S D is the nth continuous homotopy group of the D-topology,
π_n(S^D) is the nth simplicial homotopy group of the smooth singular simplicial set.
But do you agree that Christensen-Wu’s results prove that the equivalence S D≃?S∘DS^D \overset{?}{\simeq} S \circ D fails?
Yes, I obviously forgot to derive the D-topology functor, since not all diffeological spaces are cofibrant (in fact, in Example 4.29 they give the same example as in 3.20).
So I would say that the D-topology functor must be left derived in order for your statement to be true.
Note that Theorem 4.11 is stated for fibrant diffeological spaces.
However, my work with Dan and Pedro show that fibrancy is redundant, see 4.3 and 4.7.
Thanks for the comments!
Okay, you are pointing me to the conclusion in the last sentence of Remark 4.7 in arXiv:1912.10544… Ah, I see. That’s most useful.
Okay, I’ll try to get a feeling now for the cofibrant replacement of diffeological spaces, to see if this is of any use in my intended application (generalized orbifold cohomology).
If it is, I’ll want to state/quote as a proposition that $S\circ D((-)_{cof}) \simeq S^D(-)$. I’d be happy to cite you for this if you write it down somewhere.
Do you know if all smooth manifolds are Christensen-Wu cofibrant as diffeological spaces? (They leave this as a conjecture, p. 18.)
Re #30: It is easy to prove that any smooth manifold is concordance equivalent to to a cofibrant diffeological space, namely, the realization of the simplicial set K underlying some smooth triangulation of M.
This is precisely Lemma 9.13 in my draft.
I believe this will suffice for your purposes, since the D-topology functor sends concordance equivalences to homotopy equivalences of topological spaces.
Yes, I know that the cohesive shape of a smooth manifold is equivalent to its underlying (D-)topological homotopy type.
But it would be useful to know that smooth manifolds are actually Christensen-Wu cofibrant, so that a cofibrant replacement functor could be asked to preserve them. For if not, the homotopy types would be me made to work only at the expense of breaking the differential geometry of the core class of examples, and that would be besides the point.
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