Talk:Diffeology

I am pretty sure, that the real vector spaces form a proper class. As every constant map has to be a plot there is at least one map for any real vector space. Plots therefore cannot form a set but a proper class.

According to Introduction to Diffeology (a working document) the domain of a plot has to be an open subset of Rⁿ, n in N₀. Now as N₀ is a set the Rⁿ-s form a set and the powerset of this set is also a set and the plots actually form a set. Markus Schmaus 13:29, 11 July 2005 (UTC)

Complex and analytic manifolds

Currently, the article lists complex manifolds and analytic manifolds as examples of spaces that "have natural diffeologies consisting of the maps preserving the extra structure". I don't think this is true; for example, on an analytic manifold $M$ the set of all analytic maps $U\subseteq \mathbb {R} ^{n}\to M$ does not form a diffeology, because it isn't closed under composition with arbitrary smooth maps $V\subseteq \mathbb {R} ^{m}\to U$, only analytic ones. If I am seeing things correctly one can't even usefully take the diffeology generated by all those maps, because it ends up containing all smooth maps $U\to M$, thus not capturing the extra structure at all. Am I missing something, or should this "example" be removed / turned into a counterexample? Peabrainiac (talk) 02:51, 3 November 2024 (UTC)

I agree that "the maps preserving the extra structure" does not define a diffeology in these cases. I will remove analytic and complex manifolds from the example section.

They are good examples to illustrate that plots are not "local models" of a diffeological space. Maybe a section with non-examples, aimed to dispel common (but understandable) misconceptions, could be helpful. But it should probably contain more than just these cases. SkiingArcher (talk) 12:49, 22 January 2025 (UTC)

Is this right?

In this passage:

"The differential calculus on $\mathbb {R} ^{n}$, or, more generally, on finite dimensional vector spaces, is one of the most impactful successes of modern mathematics."

I do not believe there is any meaningful differential calculus that can be defined on a vector space over a finite field. Maybe on the p-adic field $\mathbb {Q} _{p}$ or on any field that is metrically complete.

I hope that this issue can be addressed by someone knowledgeable about diffeologies. — Preceding unsigned comment added by ~2026-38980-22 (talk) 15:02, 14 July 2026 (UTC)

I suggest to resolve the issue by specifying "...[real] finite dimensional vector spaces..." SkiingArcher (talk) 16:49, 22 July 2026 (UTC)