diffeology

The differential calculus on $\backslash mathbb\{R\}^n$, or, more generally, on finite dimensional vector spaces, is one of the most impactful successes of modern mathematics. Fundamental to its basic definitions and theorems is the linear structure of the underlying space.

The field of differential geometry establishes and studies the extension of the classical differential calculus to non-linear spaces. This extension is made possible by the definition of a smooth manifold, which is also the starting point for diffeological spaces.

A smooth $n$-dimensional manifold is a set $M$ equipped with a maximal smooth atlas, which consists of injective functions, called charts, of the form $\backslash phi:U\; \backslash to\; M$, where $U$ is an open subset of $\backslash mathbb\{R\}^n$, satisfying some mutual-compatibility relations. The charts of a manifold perform two distinct functions, which are often syncretized:

• They dictate the local structure of the manifold. The chart $\backslash phi:U\; \backslash to\; M$ identifies its image in $M$ with its domain $U$. This is convenient because the latter is simply an open subset of a Euclidean space.
• They define the class of smooth maps between manifolds. These are the maps to which the differential calculus extends. In particular, the charts determine smooth functions (smooth maps $M\; \backslash to\; \backslash mathbb\{R\}$), smooth curves (smooth maps $\backslash mathbb\{R\}\; \backslash to\; M$), smooth homotopies (smooth maps $\backslash mathbb\{R\}^2\; \backslash to\; M$), etc.

A diffeology generalizes the structure of a smooth manifold by abandoning the first requirement for an atlas, namely that the charts give a local model of the space, while retaining the ability to discuss smooth maps into the space.

A diffeological space is a set $X$ equipped with a diffeology: a collection of maps$$\backslash \{p:U\; \backslash to\; X\backslash mid\; U\; \backslash text\{\; is\; an\; open\; subset\; of\; \}\backslash mathbb\{R\}^n,\; \backslash text\{\; and\; \}\; n\; \backslash geq\; 0\backslash \},$$whose members are called plots, that satisfies some axioms. The plots are not required to be injective, and can (indeed, must) have as domains the open subsets of arbitrary Euclidean spaces.

A smooth manifold can be viewed as a diffeological space which is locally diffeomorphic to $\backslash mathbb\{R\}^n$. In general, while not giving local models for the space, the axioms of a diffeology still ensure that the plots induce a coherent notion of smooth functions, smooth curves, smooth homotopies, etc. Diffeology is therefore suitable to treat objects more general than manifolds.

Let $M$ and $N$ be smooth manifolds. A smooth homotopy of maps $M\; \backslash to\; N$ is a smooth map $H:\backslash mathbb\{R\}\; \backslash times\; M\; \backslash to\; N$. For each $t\; \backslash in\; \backslash mathbb\{R\}$, the map $H\_t\; :=\; H(t,\; \backslash cdot):M\; \backslash to\; N$ is smooth, and the intuition behind a smooth homotopy is that it is a smooth curve into the space of smooth functions $\backslash mathcal\{C\}^\backslash infty(M,N)$ connecting, say, $H\_0$ and $H\_1$. But $\backslash mathcal\{C\}^\backslash infty(M,N)$ is not a finite-dimensional smooth manifold, so formally we cannot yet speak of smooth curves into it.

On the other hand, the collection of maps

$$\backslash \{p:U\; \backslash to\; \backslash mathcal\{C\}^\backslash infty(M,N)\; \backslash mid\; \backslash text\{\; the\; map\; \}U\; \backslash times\; M\; \backslash to\; N,\; \backslash \; (r,x)\; \backslash mapsto\; p(r)(x)\; \backslash text\{\; is\; smooth\}\backslash \}$$

is a diffeology on $\backslash mathcal\{C\}^\backslash infty(M,N)$. With this structure, the smooth curves (a notion which is now rigorously defined) correspond precisely to the smooth homotopies.

The concept of diffeology was first introduced by Jean-Marie Souriau in the 1980s under the name espace différentiel. Souriau's motivating application for diffeology was to uniformly handle the infinite-dimensional groups arising from his work in geometric quantization. Thus the notion of diffeological group preceded the more general concept of a diffeological space. Souriau's diffeological program was taken up by his students, particularly Paul Donato and Patrick Iglesias-Zemmour, who completed early pioneering work in the field.

A structure similar to diffeology was introduced by Kuo-Tsaï Chen (陳國才, Chen Guocai) in the 1970s, in order to formalize certain computations with path integrals. Chen's definition used convex sets instead of open sets for the domains of the plots. The similarity between diffeological and "Chen" structures can be made precise by viewing both as concrete sheaves over the appropriate concrete site.

A diffeology on a set $X$ consists of a collection of maps, called plots or parametrizations, from open subsets of $\backslash mathbb\{R\}^n$ (for all $n\; \backslash geq\; 0$) to $X$ such that the following axioms hold:

• Covering axiom: every constant map is a plot.
• Locality axiom: for a given map $p:\; U\; \backslash to\; X$, if every point in $U$ has a neighborhood $V\; \backslash subset\; U$ such that $p|\_V$ is a plot, then $p$ itself is a plot.
• Smooth compatibility axiom: if $p$ is a plot, and $F$ is a smooth function from an open subset of some $\backslash mathbb\{R\}^m$ into the domain of $p$, then the composite $p\; \backslash circ\; F$ is a plot.

Note that the domains of different plots can be subsets of $\backslash mathbb\{R\}^n$ for different values of $n$; in particular, any diffeology contains the elements of its underlying set as the plots with $n\; =\; 0$. A set together with a diffeology is called a diffeological space.

More abstractly, a diffeological space is a concrete sheaf on the site of open subsets of $\backslash mathbb\{R\}^n$, for all $n\; \backslash geq\; 0$, and open covers.

A map between diffeological spaces is called smooth if and only if its composite with any plot of the first space is a plot of the second space. It is called a diffeomorphism if it is smooth, bijective, and its inverse is also smooth. Equipping the open subsets of Euclidean spaces with their standard diffeology (as defined in the next section), the plots into a diffeological space $X$ are precisely the smooth maps from $U$ to $X$.

Diffeological spaces constitute the objects of a category, denoted by $\backslash mathsf\{Dflg\}$, whose morphisms are smooth maps. The category $\backslash mathsf\{Dflg\}$ is closed under many categorical operations: for instance, it is Cartesian closed, complete and cocomplete, and more generally it is a quasitopos.

Any diffeological space is a topological space when equipped with the D-topology: the final topology such that all plots are continuous (with respect to the Euclidean topology on $\backslash mathbb\{R\}^n$).

In other words, a subset $U\; \backslash subset\; X$ is open if and only if $p^\{-1\}(U)$ is open for any plot $p$ on $X$. Actually, the D-topology is completely determined by smooth curves, i.e. a subset $U\; \backslash subset\; X$ is open if and only if $c^\{-1\}(U)$ is open for any smooth map $c:\; \backslash mathbb\{R\}\; \backslash to\; X$. The D-topology is automatically locally path-connected

A smooth map between diffeological spaces is automatically continuous between their D-topologies. Therefore we have the functor $D:\backslash mathsf\{Dflg\}\; \backslash to\; \backslash mathsf\{Top\}$, from the category of diffeological spaces to the category of topological spaces, which assigns to a diffeological space its D-topology. This functor realizes $\backslash mathsf\{Dflg\}$ as a concrete category over $\backslash mathsf\{Top\}$.

A Cartan-De Rham calculus can be developed in the framework of diffeologies, as well as a suitable adaptation of the notions of fiber bundles, homotopy, etc. However, there is not a canonical definition of tangent spaces and tangent bundles for diffeological spaces.

Any set carries at least two diffeologies:

• the coarse (or trivial, or indiscrete) diffeology, consisting of every map into the set. This is the largest possible diffeology. The corresponding D-topology is the trivial topology.
• the discrete (or fine) diffeology, consisting of the locally constant maps into the set. This is the smallest possible diffeology. The corresponding D-topology is the discrete topology.

Any topological space can be endowed with the continuous diffeology, whose plots are the continuous maps.

The Euclidean space $\backslash mathbb\{R\}^n$admits several diffeologies beyond those listed above.

• The standard diffeology on $\backslash mathbb\{R\}^n$ consists of those maps $p:U\; \backslash to\; \backslash mathbb\{R\}^n$ which are smooth in the usual sense of multivariable calculus.
• The wire (or spaghetti) diffeology on $\backslash mathbb\{R\}^n$ is the diffeology whose plots factor locally through $\backslash mathbb\{R\}$. More precisely, a map $p:\; U\; \backslash to\; \backslash mathbb\{R\}^n$ is a plot if and only if for every $u\; \backslash in\; U$ there is an open neighbourhood $V\; \backslash subseteq\; U$ of $u$ such that $p|\_V\; =\; q\; \backslash circ\; F$ for two smooth functions $F:\; V\; \backslash to\; \backslash mathbb\{R\}$ and $q:\; \backslash mathbb\{R\}\; \backslash to\; \backslash mathbb\{R\}^n$. This diffeology does not coincide with the standard diffeology on $\backslash mathbb\{R\}^n$ when $n\backslash geq\; 2$: for instance, the identity $\backslash mathbb\{R\}^n\; \backslash to\; X=\; \backslash mathbb\{R\}^n$ is not a plot for the wire diffeology.
• The previous example can be enlarged to diffeologies whose plots factor locally through $\backslash mathbb\{R\}^r$, yielding the rank-$r$-restricted diffeology on a smooth manifold $M$: a map $U\; \backslash to\; M$ is a plot if and only if it is smooth and the rank of its differential is less than or equal than $r$. For $r=1$ one recovers the wire diffeology.

Diffeological spaces generalize manifolds, but they are far from the only mathematical objects to do so. For instance manifolds with corners, orbifolds, and infinite-dimensional Fréchet manifolds are all well-established alternatives. This subsection makes precise the extent to which these spaces are diffeological.

We view $\backslash mathsf\{Dflg\}$ as a concrete category over the category of topological spaces $\backslash mathsf\{Top\}$ via the D-topology functor $D:\backslash mathsf\{Dflg\}\; \backslash to\; \backslash mathsf\{Top\}$. If $U:\backslash mathsf\{C\}\; \backslash to\; \backslash mathsf\{Top\}$ is another concrete category over $\backslash mathsf\{Top\}$, we say that a functor $E:\backslash mathsf\{C\}\; \backslash to\; \backslash mathsf\{Dflg\}$ is an embedding (of concrete categories) if it is injective on objects and faithful, and $D\; \backslash circ\; E\; =\; U$. To specify an embedding, we need only describe it on objects; it is necessarily the identity map on arrows.

We will say that a diffeological space $X$ is locally modeled by a collection of diffeological spaces $\backslash mathcal\{E\}$ if around every point $x\; \backslash in\; X$, there is a D-open neighbourhood $U$, a D-open subset $V$ of some $E\; \backslash in\; \backslash mathcal\{E\}$, and a diffeological diffeomorphism $U\; \backslash to\; V$.

The category of finite-dimensional smooth manifolds (allowing those with connected components of different dimensions) fully embeds into $\backslash mathsf\{Dflg\}$. The embedding $y$ assigns to a smooth manifold $M$ the canonical diffeology$$\backslash \{p:U\; \backslash to\; M\; \backslash mid\; p\; \backslash text\{\; is\; smooth\; in\; the\; usual\; sense\}\backslash \}.$$In particular, a diffeologically smooth map between manifolds is smooth in the usual sense, and the D-topology of $y(M)$ is the original topology of $M$. The essential image of this embedding consists of those diffeological spaces that are locally modeled by the collection $\backslash \{y(\backslash mathbb\{R\}^n)\backslash \}$, and whose D-topology is Hausdorff and second-countable.

The category of finite-dimensional smooth manifolds with boundary (allowing those with connected components of different dimensions) similarly fully embeds into $\backslash mathsf\{Dflg\}$. The embedding is defined identically to the smooth case, except "smooth in the usual sense" refers to the standard definition of smooth maps between manifolds with boundary. The essential image of this embedding consists of those diffeological spaces that are locally modeled by the collection $\backslash \{y(O)\; \backslash mid\; O\; \backslash text\{\; is\; a\; half-space\}\backslash \}$, and whose D-topology is Hausdorff and second-countable. The same can be done in more generality for manifolds with corners, using the collection $\backslash \{y(O)\; \backslash mid\; O\; \backslash text\{\; is\; an\; orthant\}\backslash \}$.

The category of Fréchet manifolds similarly fully embeds into $\backslash mathsf\{Dflg\}$. Once again, the embedding is defined identically to the smooth case, except "smooth in the usual sense" refers to the standard definition of smooth maps between Fréchet spaces. The essential image of this embedding consists of those diffeological spaces that are locally modeled by the collection $\backslash \{y(E)\; \backslash mid\; E\; \backslash text\{\; is\; a\; Fréchet\; space\}\backslash \}$, and whose D-topology is Hausdorff.

The embedding restricts to one of the category of Banach manifolds. Historically, the case of Banach manifolds was proved first, by Hain, and the case of Fréchet manifolds was treated later, by Losik. The category of manifolds modeled on convenient vector spaces also similarly embeds into $\backslash mathsf\{Dflg\}$.

A (classical) orbifold $X$ is a space that is locally modeled by quotients of the form $\backslash mathbb\{R\}^n/\backslash Gamma$, where $\backslash Gamma$ is a finite subgroup of linear transformations. On the other hand, each model $\backslash mathbb\{R\}^n/\backslash Gamma$ is naturally a diffeological space (with the quotient diffeology discussed below), and therefore the orbifold charts generate a diffeology on $X$. This diffeology is uniquely determined by the orbifold structure of $X$.

Conversely, a diffeological space that is locally modeled by the collection $\backslash \{\backslash mathbb\{R\}^n/\backslash Gamma\backslash \}$ (and with Hausdorff D-topology) carries a classical orbifold structure that induces the original diffeology, wherein the local diffeomorphisms are the orbifold charts. Such a space is called a diffeological orbifold.

Whereas diffeological orbifolds automatically have a notion of smooth map between them (namely diffeologically smooth maps in $\backslash mathsf\{Dflg\}$), the notion of a smooth map between classical orbifolds is not standardized.

If orbifolds are viewed as differentiable stacks presented by étale proper Lie groupoids, then there is a functor from the underlying 1-category of orbifolds, and equivalent maps-of-stacks between them, to $\backslash mathsf\{Dflg\}$. Its essential image consists of diffeological orbifolds, but the functor is neither faithful nor full.

If a set $X$ is given two different diffeologies, their intersection is a diffeology on $X$, called the intersection diffeology, which is finer than both starting diffeologies. The D-topology of the intersection diffeology is finer than the intersection of the D-topologies of the original diffeologies.

If $X$ and $Y$ are diffeological spaces, then the product diffeology on the Cartesian product $X\; \backslash times\; Y$ is the diffeology generated by all products of plots of $X$ and of $Y$. Precisely, a map $p:U\; \backslash to\; X\; \backslash times\; Y$ necessarily has the form $p(u)\; =\; (x(u),y(u))$ for maps $x:U\; \backslash to\; X$ and $y:U\; \backslash to\; Y$. The map $p$ is a plot in the product diffeology if and only if $x$ and $y$ are plots of $X$ and $Y$, respectively. This generalizes to products of arbitrary collections of spaces.

The D-topology of $X\; \backslash times\; Y$ is the coarsest delta-generated topology containing the product topology of the D-topologies of $X$ and $Y$; it is equal to the product topology when $X$ or $Y$ is locally compact, but may be finer in general.

Given a map $f:\; X\; \backslash to\; Y$ from a set $X$ to a diffeological space $Y$, the pullback diffeology on $X$ consists of those maps $p:U\; \backslash to\; X$ such that the composition $f\; \backslash circ\; p$ is a plot of $Y$. In other words, the pullback diffeology is the smallest diffeology on $X$ making $f$ smooth.

If $X$ is a subset of the diffeological space $Y$, then the subspace diffeology on $X$ is the pullback diffeology induced by the inclusion $X\; \backslash hookrightarrow\; Y$. In this case, the D-topology of $X$ is equal to the subspace topology of the D-topology of $Y$ if $Y$ is open, but may be finer in general.

Given a map $f:\; X\; \backslash to\; Y$ from diffeological space $X$ to a set $Y$, the pushforward diffeology on $Y$ is the diffeology generated by the compositions $f\; \backslash circ\; p$, for plots $p:U\; \backslash to\; X$ of $X$. In other words, the pushforward diffeology is the smallest diffeology on $Y$ making $f$ smooth.

If $X$ is a diffeological space and $\backslash sim$ is an equivalence relation on $X$, then the quotient diffeology on the quotient set $X/\{\backslash sim\}$ is the pushforward diffeology induced by the quotient map $X\; \backslash to\; X/\{\backslash sim\}$. The D-topology on $X/\{\backslash sim\}$ is the quotient topology of the D-topology of $X$. Note that this topology may be trivial without the diffeology being trivial.

Quotients often give rise to non-manifold diffeologies. For example, the set of real numbers $\backslash mathbb\{R\}$ is a smooth manifold. The quotient $\backslash mathbb\{R\}/(\backslash mathbb\{Z\}\; +\; \backslash alpha\; \backslash mathbb\{Z\})$, for some irrational $\backslash alpha$, called the irrational torus, is a diffeological space diffeomorphic to the quotient of the regular 2-torus $\backslash mathbb\{R\}^2/\backslash mathbb\{Z\}^2$ by a line of slope $\backslash alpha$. It has a non-trivial diffeology, although its D-topology is the trivial topology.

The functional diffeology on the set $\backslash mathcal\{C\}^\{\backslash infty\}(X,Y)$ of smooth maps between two diffeological spaces $X$ and $Y$ is the diffeology whose plots are the maps $\backslash phi:\; U\; \backslash to\; \backslash mathcal\{C\}^\{\backslash infty\}(X,Y)$ such that$$U\; \backslash times\; X\; \backslash to\; Y,\; \backslash quad\; (u,x)\; \backslash mapsto\; \backslash phi(u)(x)$$is smooth with respect to the product diffeology of $U\; \backslash times\; X$. When $X$ and $Y$ are manifolds, the D-topology of $\backslash mathcal\{C\}^\{\backslash infty\}(X,Y)$ is the smallest locally path-connected topology containing the Whitney $C^\backslash infty$ topology.

Taking the subspace diffeology of a functional diffeology, one can define diffeologies on the space of sections of a fibre bundle, or the space of bisections of a Lie groupoid, etc.

If $M$ is a compact smooth manifold, and $F\; \backslash to\; M$ is a smooth fiber bundle over $M$, then the space of smooth sections $\backslash Gamma(F)$ of the bundle is frequently equipped with the structure of a Fréchet manifold. Upon embedding this Fréchet manifold into the category of diffeological spaces, the resulting diffeology coincides with the subspace diffeology that $\backslash Gamma(F)$ inherits from the functional diffeology on $\backslash mathcal\{C\}^\backslash infty(M,F)$.

Analogous to the notions of submersions and immersions between manifolds, there are two special classes of morphisms between diffeological spaces. A subduction is a surjective function $f:\; X\; \backslash to\; Y$ between diffeological spaces such that the diffeology of $Y$ is the pushforward of the diffeology of $X$. Similarly, an induction is an injective function $f:\; X\; \backslash to\; Y$ between diffeological spaces such that the diffeology of $X$ is the pullback of the diffeology of $Y$. Subductions and inductions are automatically smooth.

It is instructive to consider the case where $X$ and $Y$ are smooth manifolds.

• Every surjective submersion $f:X\; \backslash to\; Y$ is a subduction.
• A subduction need not be a surjective submersion. One example is $$f:\backslash mathbb\{R\}^2\; \backslash to\; \backslash mathbb\{R\},\; \backslash quad\; f(x,y)\; :=\; xy.$$
• An injective immersion need not be an induction. One example is the parametrization of the "figure-eight,"

$$f:\backslash left(-\backslash frac\{\backslash pi\}\{2\},\; \backslash frac\{3\backslash pi\}\{2\}\backslash right)\; \backslash to\; \backslash mathbb\{R^2\},\; \backslash quad\; f(t)\; :=\; (2\backslash cos(t),\; \backslash sin(2t)).$$

• An induction need not be an injective immersion. One example is the "semi-cubic,"

$$f:\backslash mathbb\{R\}\; \backslash to\; \backslash mathbb\{R\}^2,\; \backslash quad\; f(t)\; :=\; (t^2,\; t^3).$$

In the category of diffeological spaces, subductions are precisely the strong epimorphisms, and inductions are precisely the strong monomorphisms. A map that is both a subduction and induction is a diffeomorphism.

{{Reflist|refs=

{{Cite book |last=Spivak |first=Michael |author-link=Michael Spivak |title=Calculus on manifolds: a modern approach to classical theorems of advanced calculus |title-link=Calculus on Manifolds (book) |date=1965 |publisher=Benjamin Cummings |isbn=978-0-8053-9021-6 |series=Mathematics monograph series |location=Redwood city (Calif.)}}

{{Cite book |last=Munkres |first=James R. |author-link=James Munkres |url=https://www.taylorfrancis.com/books/9780429962691 |title=Analysis on Manifolds |date=2018-02-19 |publisher=CRC Press |isbn=978-0-429-49414-7 |edition=1 |language=en |doi=10.1201/9780429494147}}

{{Cite book |last1=Kobayashi |first1=Shōshichi |author-link1=Shoshichi Kobayashi |title=Foundations of differential geometry. 1 |title-link=Foundations of Differential Geometry |last2=Nomizu |first2=Katsumi |author-link2=Katsumi Nomizu |date=1996 |publisher=Wiley |isbn=978-0-471-15733-5 |place=New York}}

{{Cite book |last=Tu |first=Loring W. |author-link=Loring W. Tu |url=https://link.springer.com/book/10.1007/978-1-4419-7400-6 |title=An Introduction to Manifolds |series=Universitext |date=2011 | publisher=Springer |language=en |doi=10.1007/978-1-4419-7400-6 |isbn=978-1-4419-7399-3 |issn=0172-5939}}

{{Cite book |last=Lee |first=John M. |author-link=John M. Lee |url=https://link.springer.com/book/10.1007/978-1-4419-9982-5 |title=Introduction to Smooth Manifolds |series=Graduate Texts in Mathematics |date=2012 |volume=218 |publisher=Springer |language=en |doi=10.1007/978-1-4419-9982-5 |isbn=978-1-4419-9981-8 |issn=0072-5285}}

{{Cite book |last=Iglesias-Zemmour |first=Patrick |url=https://www.ams.org/surv/185 |title=Diffeology |date=2013-04-09 |publisher=American Mathematical Society |isbn=978-0-8218-9131-5 |series=Mathematical Surveys and Monographs |volume=185 |language=en |doi=10.1090/surv/185}}

{{Citation |last=Iglesias-Zemmour |first=Patrick |title=An Introduction to Diffeology |date=2021 |work=New Spaces in Mathematics: Formal and Conceptual Reflections |volume=1 |pages=31–82 |editor-last=Catren |editor-first=Gabriel |url=https://math.huji.ac.il/~piz/documents/AITD.pdf |access-date=2025-03-17 |place=Cambridge |publisher=Cambridge University Press |doi=10.1017/9781108854429.003 |isbn=978-1-108-49063-4 |editor2-last=Anel |editor2-first=Mathieu}}

{{Cite book |last=Iglesias-Zemmour |first=Patrick |url=https://mp.weixin.qq.com/s?__biz=MjM5NzE3Nzc4MQ==&mid=2664511235&idx=2&sn=a2b55b7fc0e88a3ae4142a8f53077055 |title=Diffeology |date=2022 |publisher=Beijing World Publishing Corporation |isbn=9787519296087 |language=en}}

{{Citation |last=Souriau |first=J. M. |author-link=Jean-Marie Souriau |title=Groupes differentiels |date=1980 |work=Differential Geometrical Methods in Mathematical Physics |volume=836 |pages=91–128 |editor-last=García |editor-first=P. L. |url=http://link.springer.com/10.1007/BFb0089728 |access-date=2022-01-16 |series=Lecture Notes in Mathematics |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/bfb0089728 |isbn=978-3-540-10275-5 |editor2-last=Pérez-Rendón |editor2-first=A. |editor3-last=Souriau |editor3-first=J. M.|url-access=subscription }}

{{Citation |last=Souriau |first=Jean-Marie |author-link=Jean-Marie Souriau |title=Groupes différentiels et physique mathématique |date=1984 |work=Group Theoretical Methods in Physics |volume=201 |pages=511–513 |editor-last=Denardo |editor-first=G. |url=http://link.springer.com/10.1007/BFb0016198 |access-date=2022-01-16 |series=Lecture Notes in Physics |place=Berlin/Heidelberg |publisher=Springer-Verlag |language=en |doi=10.1007/bfb0016198 |isbn=978-3-540-13335-3 |editor2-last=Ghirardi |editor2-first=G. |editor3-last=Weber |editor3-first=T.|url-access=subscription }}

{{Cite book |last=Donato |first=Paul |title=Revêtement et groupe fondamental des espaces différentiels homogènes |publisher=ScD thesis, Université de Provence |year=1984 |location=Marseille |language=fr |trans-title=Coverings and fundamental groups of homogeneous differential spaces}}

{{Cite book |last=Iglesias |first=Patrick |url=https://math.huji.ac.il/~piz/documents/TheseEtatPI.pdf |title=Fibrés difféologiques et homotopie |publisher=ScD thesis, Université de Provence |year=1985 |location=Marseille |language=fr |trans-title=Diffeological fiber bundles and homotopy}}

{{Cite journal |last=Chen |first=Kuo-Tsai |date=1977 |title=Iterated path integrals |url=https://www.ams.org/bull/1977-83-05/S0002-9904-1977-14320-6/ |journal=Bulletin of the American Mathematical Society |language=en |volume=83 |issue=5 |pages=831–879 |doi=10.1090/S0002-9904-1977-14320-6 |issn=0002-9904 |doi-access=free}}

{{Cite journal |last1=Baez |first1=John |last2=Hoffnung |first2=Alexander |date=2011 |title=Convenient categories of smooth spaces |url=https://www.ams.org/tran/2011-363-11/S0002-9947-2011-05107-X/ |journal=Transactions of the American Mathematical Society |language=en |volume=363 |issue=11 |pages=5789–5825 |arxiv=0807.1704 |doi=10.1090/S0002-9947-2011-05107-X |issn=0002-9947 |doi-access=free}}

{{Cite journal |last1=Christensen |first1=John Daniel |last2=Sinnamon |first2=Gordon |last3=Wu |first3=Enxin |date=2014-10-09 |title=The D -topology for diffeological spaces |url=http://www.msp.org/pjm/2014/272-1/p04.xhtml |journal=Pacific Journal of Mathematics |language=en |volume=272 |issue=1 |pages=87–110 |arxiv=1302.2935 |doi=10.2140/pjm.2014.272.87 |issn=0030-8730 |doi-access=free}}

{{Cite journal |last=Laubinger |first=Martin |date=2006 |title=Diffeological spaces |url=https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/1542 |journal=Proyecciones |language= |volume=25 |issue=2 |pages=151–178 |doi=10.4067/S0716-09172006000200003 |issn=0717-6279 |doi-access=free}}

{{Cite journal |last1=Christensen |first1=Daniel |last2=Wu |first2=Enxin |date=2016 |title=Tangent spaces and tangent bundles for diffeological spaces |url=https://cahierstgdc.com/wp-content/uploads/2017/11/ChristensenWu.pdf |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |volume=57 |issue=1 |pages=3–50 |arxiv=1411.5425}}

{{Citation |last=Blohmann |first=Christian |title=Elastic diffeological spaces |date=2024 |work=Recent advances in diffeologies and their applications |volume=794 |issue= |pages=49–86 |editor-last=Magnot |editor-first=Jean-Pierre |url=https://www.ams.org/books/conm/794/ |access-date=2025-03-17 |series=Contemporary Mathematics |place= |publisher=American Mathematical Society |arxiv=2301.02583 |doi=10.1090/conm/794 |isbn=978-1-4704-7254-2}}

{{Cite journal |last=van der Schaaf |first=Nesta |date=2021 |title=Diffeological Morita Equivalence |url=https://cahierstgdc.com/wp-content/uploads/2021/04/Van-der-Schaaf-_LXII-2.pdf |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |volume=LXII |issue=2 |pages=177–238 |arxiv=2007.09901}}

{{Cite journal |last1=Gürer |first1=Serap |last2=Iglesias-Zemmour |first2=Patrick |date=2019 |title=Differential forms on manifolds with boundary and corners |journal=Indagationes Mathematicae |language=en |volume=30 |issue=5 |pages=920–929 |doi=10.1016/j.indag.2019.07.004 |doi-access=free}}

{{Cite journal |last=Hain |first=Richard M. |date=1979 |title=A characterization of smooth functions defined on a Banach space |url=https://www.ams.org/proc/1979-077-01/S0002-9939-1979-0539632-8/ |journal=Proceedings of the American Mathematical Society |language=en |volume=77 |issue=1 |pages=63–67 |doi=10.1090/S0002-9939-1979-0539632-8 |issn=0002-9939 |doi-access=free|url-access=subscription }}

{{Cite journal |last=Losik |first=Mark |date=1992 |title=О многообразиях Фреше как диффеологических пространствах |trans-title=Fréchet manifolds as diffeological spaces |url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ivm&paperid=4812&option_lang=eng |journal=Izv. Vyssh. Uchebn. Zaved. Mat. |language=ru |volume=5 |pages=36–42 |via=All-Russian Mathematical Portal}}

{{Cite journal |last=Losik |first=Mark |date=1994 |title=Categorical differential geometry |url=http://www.numdam.org/item/CTGDC_1994__35_4_274_0/ |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |volume=35 |issue=4 |pages=274–290}}

{{Cite book |last1=Frölicher |first1=Alfred |author-link1=Alfred Frölicher |title=Linear spaces and differentiation theory |last2=Kriegl |first2=Andreas |date=1988 |publisher=J. Wiley & sons |isbn=978-0-471-91786-1 |series=Pure and applied mathematics |location=Chichester |language=en}}

{{Citation |last=Miyamoto |first=David |title=Lie algebras of quotient groups |date=2025-02-14 |url=https://arxiv.org/abs/2502.10260 |access-date=2025-03-21 |publisher= |arxiv=2502.10260 |id=}}

{{Cite journal |last1=Iglesias-Zemmour |first1=Patrick |last2=Karshon |first2=Yael |last3=Zadka |first3=Moshe |date=2010 |title=Orbifolds as diffeologies |url=https://www.ams.org/journals/tran/2010-362-06/S0002-9947-10-05006-3/S0002-9947-10-05006-3.pdf |journal=Transactions of the American Mathematical Society |volume=362 |issue=6 |pages=2811–2831 |doi=10.1090/S0002-9947-10-05006-3 |jstor=25677806 |s2cid=15210173}}

{{Citation |last=Miyamoto |first=David |title=Lie groupoids determined by their orbit spaces |date=2024-03-22 |url=https://arxiv.org/abs/2310.11968 |access-date=2025-03-21 |publisher= |arxiv=2310.11968 |id=}}

{{Cite journal |last1=Donato |first1=Paul |last2=Iglesias |first2=Patrick |date=1985 |title=Exemples de groupes difféologiques: flots irrationnels sur le tore |trans-title=Examples of diffeological groups: irrational flows on the torus |journal=C. R. Acad. Sci. Paris Sér. I |language=fr |volume=301 |issue=4 |pages=127–130 |mr=799609}}

{{Cite journal |last1=Karshon |first1=Yael |last2=Miyamoto |first2=David |last3=Watts |first3=Jordan |date=2024-10-01 |title=Diffeological submanifolds and their friends |url=https://linkinghub.elsevier.com/retrieve/pii/S0926224524000639 |journal=Differential Geometry and Its Applications |volume=96 |pages=102170 |arxiv=2204.10381 |doi=10.1016/j.difgeo.2024.102170 |issn=0926-2245}}

{{Cite journal |last=Joris |first=Henri |date=1982-09-01 |title=Une C∞-application non-immersive qui possède la propriété universelle des immersions |trans-title=A non-immersive C∞-map which possesses the universal property of immersions |url=https://doi.org/10.1007/BF01899535 |journal=Archiv der Mathematik |language=fr |volume=39 |issue=3 |pages=269–277 |doi=10.1007/BF01899535 |issn=1420-8938|url-access=subscription }}

{{cite journal |last1=Hamilton |first1=Richard S. |author-link=Richard S. Hamilton |title=The inverse function theorem of Nash and Moser |url = https://www.ams.org/journals/bull/1982-07-01/S0273-0979-1982-15004-2/S0273-0979-1982-15004-2.pdf |journal=Bulletin of the American Mathematical Society |date=1982 |volume=7 |issue=1 |pages=65–222 |doi=10.1090/S0273-0979-1982-15004-2}}

{{cite journal |last1=Waldorf |first1=Konrad |title=Transgression to loop spaces and its inverse, I: Diffeological bundles and fusion maps |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |date=2012 |volume=53 |issue=3 |pages=162–210 |issn=2681-2363}}

}}

• Patrick Iglesias-Zemmour: [http://math.huji.ac.il/~piz/ Diffeology (many documents)]
• [http://diffeology.net/ diffeology.net] Global hub on diffeology and related topics

{{Manifolds}}

Category:Differential geometry

Category:Functions and mappings

Chen, Guocai

Category:Smooth manifolds