Added (and split off foundations in a separate section):

Cartan calculus for diffeological spaces is developed in

- Christian Blohmann,
*Elastic diffeological spaces*, arXiv:2301.02583.

And now the paper is on arXiv: https://arxiv.org/abs/2210.12845.

]]>Re #80: I added a remark about Cisinski’s result in the draft.

Urs, if you can think of any additional results/statements that would be of interest to you, let me know, I will be happy to add them to the paper; Proposition 10.3 already resolves one of your previous questions.

]]>Yes, I think this works fine for any first-countable topological vector space as long as smoothness holds, since such first-countable TVS admit countable fundamental systems of neighborhoods U_ε and we can take u_ε∈U_ε.

And for smoothness of curves, only smoothness at 0 is nontrivial, since at all other points we can use as coefficients smooth real-valued bump functions with disjoint supports instead of the piecewise linear construction, and these automatically yield a smooth curve away from 0.

In the Frechet case, we can choose seminorms to be exponentially decreasing as the parameter approaches 0, which guarantees smoothness.

]]>This appears to work also for Frechet spaces, since the topology of a Frechet space is induced by a countable system of seminorms.

Possibly even just using the fact the topology comes from a translation-invariant metric is enough: use $d(0,u_\varepsilon)$ instead of ‖u_ε‖<ε. I think the smooth curve construction should work basically the same (I imagine doing something like a piecewise linear continuous path joining the set of points $u_\varepsilon$, then smoothing it by a reparametrisation introducing flat points at the joins. Since the chain rule works for the usual calculus in Fréchet spaces this will still be smooth). Do you agree?

]]>An argument of an apparently different nature was supplied on Twitter.

]]>Do we know if Banach spaces are Δ\Delta-generated?

I think so. We need to show that given a Banach space B and a subset U⊂F, if the preimage of U under any smooth map R^n→B is open, then U is open.

Assume the converse: there is a point u∈U (wlog u=0) such that for any ε>0 there is a point u_ε∈B∖U such that ‖u_ε‖<ε.

Now use smooth bump functions to construct a smooth curve f:R→B such that f(ε)=u_ε for some set of ε that have 0 as an accumulation point.

We have a contradiction: 0∈f^{-1}U, but arbitrary small neighborhoods of 0 have points outside of f^{-1}U.

Thus, U is open in the norm topology.

This appears to work also for Frechet spaces, since the topology of a Frechet space is induced by a countable system of seminorms.

]]>It seems to be true at least in special cases: in conversation with Enxin Wu we agreed that the Fréchet space topology and the D-topology on $\prod_{\mathbb{N}} \mathbb{R}$ agree.

Do we know if Banach spaces are $\Delta$-generated?

]]>This amounts to saying that Frechet spaces are Δ-generated topological spaces. Is this known?

]]>A naive question that I’ve not seen addressed, and someone who’s published on diffeology seems to not know: is the D-topology on a the diffeological space associated to a Fréchet space the same as the original topology? I would be surprised if not. We seem to have danced around the issue earlier in the thread, but skimming through I only saw discussion of the shape working out correctly.

]]>How is the model structure on smooth sets which you consider related to that considered by Cisinski, as highlighted in Adrian CLough’s thesis?

Given the way you phrased this, may I point out that the nLab has a detailed article about Cisinski’s model structures on toposes: test topos, which you once created.

The weak equivalences are the same for the transferred model structure and Cisinski’s model structure.

Cofibrations in Cisinski’s model structure are precisely monomorphisms, whereas cofibrations in the transferred model structure are precisely retracts of transfinite compositions of cobase changes of smooth horn inclusions.

So fibrancy in the transferred model structure is something you can establish in practice, which is not really the case for Cisinski’s model structure.

]]>Hi Dmitri, re #74:

thanks for posting this! Looks really interesting.

I am on a brief family vacation and didn’t find time yet to really look at your pdf, nor may I find much time in the next week.

Just one quick question from the list of highlights:

How is the model structure on smooth sets which you consider related to that considered by Cisinski, as highlighted in Adrian CLough’s thesis?

]]>So something like a simplicial Lie group, I guess? That’s useful to know.

]]>Re #77: As long as the latching maps (inclusions of degenerate simplices) of your simplicial manifold are cofibrations of diffeological spaces, the answer is affirmative: consider the skeletal filtration of the smooth realization; every step in the filtration is a cobase change of the pushout product of a smooth boundary inclusion and the corresponding latching map. Since the model structure is cartesian, the pushout product is a cofibration, and so is its cobase change. Transfinite compositions of cofibrations are cofibrations.

The latching maps are cofibration in many cases of interest. A trivial case is when the degenerate simplices are split. A less trivial case is when the latching map is a closed embedding of manifolds, since such maps are cofibrations by a relative version of Proposition 9.2.

]]>Hmm, interesting. Now I’m wondering about geometric realization of simplicial fin dim manifolds. If they were cofibrant that would be excellent.

]]>Do you know an explicit example of a cofibrant diffeological space that’s not a manifold?

Yes, of course: smooth realizations of simplicial sets are not manifolds. So as a completely explicit example, take the smooth realization of a simplicial 2-horn.

In general, cofibrant diffeological spaces will be smooth analogues of CW-complexes (or, more generally, retracts of transfinite compositions of cobase changes of smooth horn inclusions).

So it is not unreasonable to expect that we have smooth analogues of various results about certain spaces being CW-complexes.

]]>Do you know an explicit example of a cofibrant diffeological space that’s not a manifold?

]]>Re #35:

I am finalizing a paper for the arXiv: https://dmitripavlov.org/diffeo.pdf, which answers the questions about model structures on diffeological spaces posed above.

Some highlights:

Theorem 6.3: The category of diffeological spaces does

**not**admit a model structure transferred from simplicial sets via the smooth singular complex functor. This is caused by the highly pathological behavior of the concretization functor, which is used to compute colimtis of diffeological spaces. However, the smooth singular complex functor is a Dwyer–Kan equivalence of relative categories (Corollary 7.7).Theorem 7.4: The category of smooth sets

**does**admit a model structure transferred from simplicial sets via the smooth singular complex functor.All smooth manifolds are cofibrant.

This model structure is cartesian.

It is left proper, combinatorial, h-monoidal, flat, symmetric h-monoidal, all operads are admissible, etc.

The internal hom Hom(X,-) from any smooth manifold X preserves weak equivalences. This is just a reformulation of the smooth Oka principle.

Proposition 10.3 resolves the question in Remark 2.2.9 of the paper “Equivariant principal infinity-bundles”.

Finally, all of the above continues to hold if we replace (pre)sheaves of sets by presheaves valued in a left proper combinatorial model category V.

As an application, in Section 14 I prove classification results for principal G-bundles and bundle gerbes over arbitrary cofibrant diffeological spaces.

Losik’s paper bibliographic data updated:

- {#Losik92} Mark Losik,
*Fréchet manifolds as diffeologic spaces*, Russian Mathematics 36:5 (1992), 36–42. English translation: PDF. Russian original: (mathnet:ivm4812)

Added:

The Grothendieck topology on $\mathcal{Op}$ is generated by the coverage of open covers, i.e., a family of maps $\{U_i\to X\}_{i\in I}$ is a covering family if every map $U_i\to X$ is an open embedding and the union of the images of $U_i$ in $X$ equals $X$.

]]>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).

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

- Hông Vân Lê,
*Diffeological statistical models and diffeological Hausdorff measures*, yt

The slides are available from https://users.math.cas.cz/~hvle/PHK/Lediffeological10032021.pdf and there are two arXiv preprints,

- Hông Vân Lê, Alexey A. Tuzhilin,
*Nonparametric estimations and the diffeological Fisher metric*, arXiv:2011.13418 - Hông Vân Lê,
*Diffeological statistical models,the Fisher metric and probabilistic mappings*, Mathematics 2020, 8(2),167, arXiv:1912.02090

I copy this information at Fisher metric.

]]>added pointer to the new website:

]]>Ok, thanks for the pointer.

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