Talk:WikiJournal of Science/Diffeology
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WikiJournal of Science
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It was adapted from the Wikipedia page Diffeology and contains some or all of that page's content licensed under a CC BY-SA license. Post-publication review comments or direct edits can be left at the version as it appears on Wikipedia.
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DOI: 10.15347/WJS/2026.002
QID: Q137667600
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Francesco Cattafi; David Miyamoto (18 April 2026). "Diffeology". WikiJournal of Science. doi:10.15347/WJS/2026.002. Wikidata Q137667600. ISSN 2470-6345.
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Wikipedia: This work is adapted from the Wikipedia article Diffeology (CC BY-SA). Content has also subsequently been used to update that same Wikipedia article Diffeology.
License: CC-BY-SA 4.0
Editors:Henry Hoff 
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John M. Lee
Jordan Watts
Article information
Submitting authors: Francesco Cattafi[a][i] , David Miyamoto[b][ii]
Additional contributors: Wikipedia community
Plagiarism check
Pass. Report from WMF copyvios tool identified no issues. OhanaUnitedTalk page 15:53, 2 January 2026 (UTC)
Peer Review 1
Review by anonymous peer reviewer , I have written a paper about diffeological spaces and used them in other papers
These assessment comments were submitted on , and refer to this previous version of the article
The article is generally fine as far as it goes. However:
1) one gap is that it doesn't say why diffeological spaces are an important alternative to the much more popular concept of smooth manifolds. Without some answer to this question, the concept may seem like abstraction for abstraction's sake.
The main reason diffeological spaces are useful is that the world of smooth manifolds lacks many of the constructions that are listed in this article for diffeological spaces. This means that starting with manifolds, many things we might want to do take us outside the world of smooth manifolds and into the larger world of diffeological spaces.
Anyone who knows a little category theory would like to see this sentence: the category of manifolds is not complete, is not cocomplete, and is not cartesian closed, but the category of diffeological spaces is. The ideas here should also be explained in humbler terms. For example, the set of solutions of equations between smooth maps out of a manifold is not generally a manifold, gluing together smooth manifolds with arbitrary smooth manifolds does not give a manifold, the set of smooth maps between smooth manifolds is not a manifold, etc. (These 3 example show the category of smooth manifolds lacks equalizers, lacks pushouts and is not cartesian closed.)
2) Saying there is not a "canonical definition" of tangent or cotangent space for a diffeological space is somewhat misleading. Yes, there are multiple definitions, but there are definitions with very good properties, which have been studied, and which are useful in applications.
OhanaUnitedTalk page 15:55, 2 January 2026 (UTC)
- We thank the referee for their comments and suggestions, which we have implemented.
- 1) We have expanded the previous "Motivating example" subsection into a longer "Motivation" subsection, which includes examples of subsets, quotients and mapping spaces of smooth manifolds which are not smooth manifolds, but they are natural non-trivial diffeological spaces. We have also commented on the underlying categorical aspects.
- 2) We have rewritten that paragraph in a more neutral way. Francesco Cattafi (discuss • contribs) 17:35, 16 February 2026 (UTC)
Peer Review 2
Review by John M Lee 
, University of Washington, Seattle
These assessment comments were submitted on , and refer to this previous version of the article
I do have a couple of small comments about the introductory section, which is about smooth manifolds, something I do know about. In the section "Calculus on smooth spaces," the authors use the term "chart" to refer to a map from an open subset of R^n into the manifold. But the near-universal convention in the current literature is to consider a chart to be a map from an open subset of the manifold to R^n. Maps in the opposite direction are called 'local parametrizations" (or, less frequently, "coordinate patches"). Also in that section, they refer to curves as maps from R to M, and homotopies as maps from R^2 to M. But we frequently need to consider curves defined only on a subinterval of R, and homotopies are always defined on [0,1]^2, not on all of R^2.
OhanaUnitedTalk page 01:11, 5 January 2026 (UTC)
- We thank the referee for his comments and suggestions, which we have implemented.
- We replaced the word "chart" with the more appropriate "local parametrization" (and mentioned that they are the inverse of ordinary charts). Moreover, we also defined the domains of curves and homotopies as [0,1] and [0,1]^2, respectively. Francesco Cattafi (discuss • contribs) 17:38, 16 February 2026 (UTC)
Peer Review 3
Review by Jordan Watts , Central Michigan University, Mt. Pleasant, Michigan, USA
These assessment comments were submitted on , and refer to this previous version of the article
Overall, this is an excellent article on diffeology, covering aspects of its history and motivation, basic constructions, and simple yet important examples. My complaints below are mostly nit-picky, but I think addressing them would lead to a clearer presentation.
1) 2nd bullet of paragraph 3: Smooth homotopies can have more general domains that [0,1]^2, especially if one considers a smooth homotopy between two smooth maps between two diffeological spaces X and Y. Perhaps just say "smooth homotopies between smooth curves".
2) Motivation, paragraph 3: Perhaps also mention how the irrational torus shows up as the structure group of principal bundles arising in theory of geometric quantization, in which the symplectic form is not assumed to be integral (the textbook of Iglesias-Zemmour would suffice as a reference). In other words, the irrational torus is not a pathological example; it shows up naturally.
3) Last sentence of Motivation: Perhaps, "...can be shown to correspond precisely to the smooth homotopies between smooth functions $M\to N$."
4) Suggestion: for "smooth functions $M\to N$", I recommend "smooth maps" instead. Often times, a "smooth function" makes one think of a scalar-valued function. This is only a personal preference, however; what the authors wrote is correct.
5) Formal definition, first sentence: remove "or parametrizations". A parametrization is more general than a plot: it is any set-theoretic function from a Euclidean open set into the set $X$. While every plot is a parametrization, not every parametrization is a plot. Thus its inclusion here is confusing.
6) Second-last sentence: "...in particular, any diffeology contains the inclusions of the elements of the underlying set as the plots with n=0." (The points themselves are not plots.)
7) Suggestion: Last sentence of Additional Structures: Other potential references for tangent spaces/bundles to consider adding: - Masaki Taho, "Tangent spaces of diffeological spaces and their variants", https://doi.org/10.1016/j.topol.2026.109741 - Christian Blohmann, "Elastic diffeological spaces", - https://doi.org/10.1090/conm/794/15925
8) First Examples: The empty set and a singleton set only carry one diffeology (the coarse and trivial are the same). Any set with at least two points will have at least two (different) diffeologies.
9) First Examples, wire diffeology, last sentence: "...for instance, the identity $\mathbb{R}^n\to X=\mathbb{R}^n$ where the domain is equipped with the standard smooth structure is not a plot for the wire diffeology." (This just helps the reader parse what is written a little faster.)
10) Relation to other smooth spaces, first paragraph, second sentence: remove "infinite-dimensional". While Frechet manifolds generalize finite-dimensional manifolds, infinite-dimensional Frechet manifolds do not.
11) Relation to other smooth spaces, the part on orbifolds, last sentence. The references should also include: https://doi.org/10.1216/RMJ-2017-47-1-289 (Theorem B is essentially this sentence, and Section 7 discusses the lack of fullness and faithfulness. This is a self-plug, but I believe a justified one, as it's from 2017.)
--Jordan A. Watts (discuss • contribs) 21:18, 25 February 2026 (UTC)
- We thank the referee for his comments and suggestions, which we have implemented entirely or with slight changes (see below):
- 2) we added a sentence explaining that irrational tori appear naturally in geometric quantization
- 4) we fully agree with the subtle linguistic distinction (since we also used ourselves in a few instances the expression "smooth function" meaning "smooth scalar function"), and we uniformized the terminology in the rest of the paper
- 6) we made the sentence even more explicit, writing the constant maps from R^0 to X
- 8) we added "with at least two elements" in the first sentence, and mentioned the uniqueness of the diffeology on empty set and singletons in a further sentence below
- 9) we clarified the paragraph: we now state that the identity is not a plot for the wire diffeology, and also that it is not smooth (as map between the standard plane and the wire plane) Francesco Cattafi (discuss • contribs) 22:54, 3 March 2026 (UTC)