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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.
I think I convinced myself that an argument similar to my Lemma 9.13 as well as Proposition 4.23 in Christensen–Wu does show that any smooth manifold is cofibrant.
What’s more, I now think that the Christensen–Wu model structure does exist, is cartesian, and any smooth embedding is a cofibration.
Do you think this may be worthy of writing down as a separate paper?
Definitely!
Yes!
We’d have a neat application of this result to the problem of relating orbifold cohomology to equivariant cohomology:
There, abstract arguments in equivariant cohesion show that the equivariant homotopy type of a general cohesive orbifold looks just like that of a topological $G$-space, but with the system of topological spaces of $K$-fixed loci all replaced by the shape of the $K$-fixed loci of the underlying concrete cohesive space. If your claims are true, this would imply that, in the case of smooth cohesion, this latter system is again equivalent to that of an actual topological $G$-space, namely that which is the derived D-topology underlying the diffeological space which is the concrete cohesive covering space of the given orbifold.
All we’d need to complete this argument is to cite results as you just stated. :-)
Is the functor $D(-)$ (assigning underlying D-topological spaces) left Quillen, in that would-be model structure?
The Christensen-Wu model structure is transferred from the Quillen model structure on simplicial sets via the smooth singular simplicial set functor.
Its generating (acyclic) cofibrations are smooth geometric realizations of (acyclic) cofibrations of simplicial sets.
The functor D is cocontinuous, so it sends these generating (acyclic) realizations to the ordinary geometric realizations of (acyclic) cofibrations of simplicial sets. The latter are indeed (acyclic) cofibrations.
So the functor D is a left adjoint functor that preserves (acyclic) cofibrations, hence a left Quillen functor.
That would be a plausible strategy to check it, but don’t we need some Lemma that $D(\left\vert \Delta^n \right\vert)$ is what one would hope it is? A priori the topology could end up being funny.
don’t we need some Lemma that D(|Δ n|)D(\left\vert \Delta^n \right\vert) is what one would hope it is?
This follows from the definition of the D-topology. Recall (Definition 3.6) in Christensen–Wu that the D-topology on |Δ^n| is the final topology induced by its plots, where the domain of each plot is equipped with the standard topology on R^n.
But by Definition 4.3 in Christensen–Wu, the smooth geometric realization of Δ^n is precisely the extended smooth n-simplex with its standard diffeology.
And by Example 3.7 in Christensen–Wu, the D-topology on a smooth manifold with the standard diffeology coincides with the usual topology on the manifold.
Okay, great. Glad you have thought this through. :-)
I’ll go ahead then citing an upcoming theorem of yours in what I am writing up regardng orbifold cohomology. I’ll show you what we need once it is in readable form. Hopefully in a week or two.
One more question: Is there, in the would-be model structure under discussion, a functorial cofibrant replacement which is the identity on manifolds?
The standard way to produce functorial factorizations is the small object arguments of Quillen and Garner.
Both arguments produce huge cofibrant replacements, and I do not see how to reduce their size functorially.
Why do you need a functorial replacement of this type anyway?
What I strictly need in applications is just this:
Given a diffeological space equipped with the action of a finite group, I need that group action to extend to its cofibrant replacement. That’s why I am concerned with functorial replacement.
But in addition to that, I had the vague feeling that I’d rather keep a given difeological space intact (as arising from some differential geometric problem) as much as possible, instead of feeding it into a blind replacement machine such as the small object argument.
Can we maybe see concretely geometrically what Christensen-Wu cofibrancy is about? I am vaguely imagining one might identify “singular” subloci inside a diffeological space such that a kind of blowup of their vicinity restores cofibrancy. Maybe?
But this may be more my unenlightened prejudice than actual necessity.
Not to distract from this discussion, but just to log some edits to the entry:
Started a section Relation to topological spaces (already last week, but I had left the edit invisible for a while to showcase the redirects bug).
Also did a fair bit of editing on the section References – General:
Added missing publication data and DOI-s to a bunch of items, added missing references such as to Souriau’s second original articles, adjusted the order of the articles (now it goes Chen $\to$ Souriau $\to$ Iglesias-Zemmour).
In particular, the previous pointer to
I have expanded out to
and moved up to join the other articles by Chen.
By the way, it’s most curious: Even in this collection of texts in topos theory and sheaf theory, both Chen and his editors (!?) manage to still avoid recognizing that Chen is secretly talking about sheaves.
added a section Topological homotopy type and diffeological shape
Another dumb question:
Do we know whether for Fréchet manifolds
$X \in FrechetManifolds \hookrightarrow DiffeologicalSpaces \hookrightarrow SmoothGroupoids_\infty$the cohesive shape (i.e. $S^D(X)$ in Christensen-Wu notation) coincides with the underlying topological homotopy type?
(It feels like I knew this once, but I forget.)
Re #46: This basically amounts to saying that any continuous disk with a smooth boundary can be deformed relative boundary to a smooth disk.
This is probably established somewhere in the literature on Fréchet manifolds.
Hmm, yes. There was a recent paper by Glöckner on smoothing operators for functions valued in lctvs, but it’s not quite in the right setting (and doesn’t seem to do the relative case).
Glöckner’s result seems like a massive overkill anyway: we only need a single deformation, not a whole smoothing operator.
Yeah, but it indicates that current technology is much stronger than you’d need, evidence that smoothing for a single map to a Fréchet space should be known.
But contractibility of these disks is only the first step. Next we need to know that 2) there are good open covers or hypercovers by disjoint unions of such open disks and then 3) a suitable nerve theorem.
How much of a condition is paracompactness on an infinite-dimensional Fréchet manifold?
Re #51: Why do you want all these things?!
To show that the cohesive shape (i.e., S^D(X) in the Christensen-Wu notation) coincides with the underlying topological homotopy type, it suffices to show that the canonical map S^D(X) → Sing(D(X)) is a simplicial weak equivalence.
Both simplicial sets are Kan complexes, so by the simplicial Whitehead theorem, it suffices to show that for any map ∂Δ^n → S^D(X) together with a filling of its image in Sing(D(X)) by Δ^n, we can deform the filling relative boundary to another disk that lifts to S^D(X).
But this is exactly the disk deformation condition that I mentioned above.
Okay, I see that I wasn’t properly reading all the qualifications in #47.
So is this a consequence of Glöckner or not?
I am just trying to find out if you or somebody essentially knows the answer already, not just an idea for a strategy, or if I’d need to dive into it myself.
Re #53: I think an even easier argument is possible, one that does not require any smoothing arguments.
It suffices to observe that any Fréchet manifold has an atlas of Fréchet coordinate charts, in particular, is the homotopy colimit of the diagram consisting of its open subsets that are diffeomorphic to Fréchet vector spaces.
Thus, it suffices to show that S^D(X) → Sing(D(X)) is a simplicial weak equivalence whenever X is a Fréchet vector space. But this is trivial because both sides are contractible.
This now sounds like the beginning of the argument along the lines of #51 after all:
If we replace the manifold by a simplicial object of local charts and their intersections, or more generally by a hypercover by local charts, then we still need to argue that passing to the resulting simplicial set obtained by contracting each local chart to a point represents the homotopy type of the underlying topological space. This is intuituvely suggestive but needs a proof. If our space is paracompact and we can arrange for a good cover, then one such proof is Borsuk’s nerve theorem.
I trust there are other way’s to argue this, but some argument seems to be needed. But let me know if I am missing the obvious.
The Convenient Setting of Global Analysis has the result (Theorem 16.10) that nuclear Fréchet spaces are all smoothly paracompact, as well as “strict inductive limits of sequences of such spaces”. Lindelöf and smoothly regular is also sufficient. Countable products of smoothly paracompact Fréchet spaces (being metrizable) are smoothly paracompact (Corollary 16.17). It seems one could just assume separability on the Fréchet space, instead of nuclearity. So it seems spaces of smooth functions on compact manifolds to fin.dim. manifolds, as Fréchet manifolds/diffeological spaces, do indeed satisfy what you are looking for.
Theorem 16.15 looks potentially relevant, too.
Thanks. I’d like to check whether the proof of the full inclusion of Fréchet manifolds into diffeological spaces might not secretly assume paracompactness anyway(?).
The critical point seems, to me, to be the existence of good open covers. But I see that Fréchet manifolds are still metrizable if (and only if) they are paracompact. With a kind of infinite-dimensional Riemannian metric in hand, the usual proof of existence of good open covers might just go through.
Just to say that I see now that Kihara has an article whose abstract sounds like it has the proof:
Smooth Homotopy of Infinite-Dimensional $C^\infty$-Manifolds (arXiv:2002.03618)
But i haven’t dug into it yet.
[ edit: Ah, too bad: Theorem 1.1 in that article would be the desired statement… were it not for the fact that it’s using the non-standard diffeology on simplices, following arXiv:1605.06794.)
Re #55:
If we replace the manifold by a simplicial object of local charts and their intersections, or more generally by a hypercover by local charts, then we still need to argue that passing to the resulting simplicial set obtained by contracting each local chart to a point represents the homotopy type of the underlying topological space. This is intuituvely suggestive but needs a proof. If our space is paracompact and we can arrange for a good cover, then one such proof is Borsuk’s nerve theorem.
I think there is a very simple argument for this.
First, given an open hypercover H of X, the canonical map
$hocolim H \to X$computed in the model category of topological spaces is a weak equivalence of topological spaces.
This is Lurie’s abstract Seifert–van Kampen theorem, see Theorem A.3.1 in HA.
The same statement is also true for the model category of diffeological spaces that I hope to finish writing down soon (if all arguments work out).
But then it remains to observe that any Fréchet manifold admits a good hypercover (all elements are diffeomorphic to Fréchet spaces). Indeed, start with some atlas, then choose an atlas for each intersection, etc.
All right.
By the way, did you see that there is also this article:
This seems to define smooth homotopy groups using maps out of $n$-cubes equipped with their standard diffeology. So that might already be the model structure in question. But I don’t know, have only glanced over the article so far. Also, this appears to remain unpublished (?)
Ah, right, Kihara 16 claims (p. 2) that
there exists a gap in the proof of [ Haraguchi-Shimakawa 13, Theorem 5.6]
But then later Haraguchi 18 seems to mean to address this, as he writes (p. 1):
We present the Quillen model structure on the category $Diff$ of diffeological spaces $[...]$ (cf. [ Haraguchi-Shimakawa 13, Theorem 5.6 and Theorem 6.2])
On the other hand, Haraguchi 18 also seems not to be published yet.
I am aware of this paper. Their argument is very technical, and they claim that the model structure is not cofibrantly generated, apparently.
What is (or would be) nice about this model structure is that it is compatible with that neat idempotent adjunction between topological spaces and diffeological spaces, in that it makes the factorization
$TopologicalSpaces \underoverset { \underset{ Cdfflg }{\longrightarrow} } { \overset{ }{\hookleftarrow} } {\phantom{AA}\bot\phantom{AA}} DTopologicalSpaces \underoverset { \underset{ }{\hookrightarrow} } { \overset{ Dtplg }{\longleftarrow} } {\phantom{AA}\bot\phantom{AA}} DiffeologicalSpaces$into a sequence of Quillen equivalences.
That should be rather useful, if true.
It is probably not of any immediate use to you, Urs, but by my thesis I think it is more or less immediate that one can put a Hurewicz model structure on D-topological spaces and diffeological spaces, both of which are Quillen equivalent to the Hurewicz model structure on topological spaces. All of this would be compatible with #63.
To get what you need from this, it might suffice to have some kind of Whitehead theorem for diffeological spaces. I.e. if two diffeological spaces are weakly equivalent in the sense you are looking at, then if one could show they are then actually homotopy equivalent in the sense of the Hurewicz model structure on diffeological spaces, one can use the Hurewicz Quillen equivalences to get what you need I think (if I am not overlooking something; the notation is a bit heavy, and I don’t really know anything about diffeological spaces, so I am somewhat guessing what you are looking to prove; Dmitri’s #28 was very helpful).
Edit: There is some kind of Whitehead theorem in Haraguchi’s article from 2018, maybe it is sufficient.
Hi Richard,
I have only glanced over your thesis (arXiv:1304.0867). Would have to dig deeper to see which statemen(s) one would need to quote to get the desired model structure. There seem to be a lot of technical conditions to be checked(?).
I never thought much about the Hurewicz model structure at all. But if you could deduce with ease a theorem for that case, I expect it would be of interest.
To get what you need from this, it might suffice…
Yeah, this is what the Haraguchi-Shimakawa-structure would (or will) give: Here diffeological homotopy type is detected on smooth homotopy groups, while the functor to underlying D-topological spaces is the left adjoint of a Quillen equivalence. Therefore the existence of this model structure would (or will) imply that the cohesive homotopy types of cofibrant diffeological spaces is in bijection to their underlying D-topological homotopy type.
Ok, so Haraguchi and Shimakawa have a new preprint out, claiming to fix the issues with the old, incorrect result on the model structure on diffeological spaces: https://arxiv.org/abs/2011.12842
Thanks for the alert. But maybe best to discuss in the thread for model structure on diffeological spaces, here.
Ok, thanks for the pointer.
Diffeologies coming out from singular statistical models were discussed this Wednesday, 10.3.2021. in the Prague-Hradec Králové seminar (Cohomology in algebra, geometry, physics and statistics) talk by Hông Vân Lê (Institute of Mathematics of the Czech Academy of Sciences), now on youtube
The slides are available from https://users.math.cas.cz/~hvle/PHK/Lediffeological10032021.pdf and there are two arXiv preprints,
I copy this information at Fisher metric.
I have added statement and proof (here) that the internal hom as diffeological spaces of any pair of D-topological spaces has the correct diffeological homotopy type.
This follows, I think, by combining a couple of statements from Shimakawa & Haraguchi with that proposition from Christensen & Wu (observing that the latter gives a natural weak equivalence).
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