1. Introduction
Diffeological spaces were introduced by Souriau [
5] as a generalization of differentiable manifolds. This setting includes not only finitedimensional manifolds, but also manifolds with boundary, infinitedimensional manifolds, leaf spaces of foliations or spaces of differentiable maps. The main reference is IglesiasZemmour’s book [
1]. Many fundamental results are exposed in nLab [
3] too.
A diffeological structure on a set
X is given by declaring which maps from open subsets of Euclidean spaces
into X are considered to be smooth. These maps are called the
plots of the diffeology (
Section 2). They can be seen as
ndimensional curves,
$n\ge 0$, on
X. This contravariant idea differs from the covariant classical one of declaring which real maps
from a manifold
X are smooth. On the other hand, the idea of an atlas on a manifold is generalized by the notion of a
generating family of plots (
Section 4).
Our main result (Theorem 6.1) will be to give a categorical interpretation of a generating family, namely by proving that the diffeological space is the join of the family, that is, the pushout of the pullback.
Most examples will be related to finitedimensional manifolds.
2. Diffeological Spaces
A diffeology on the set X is a family $\mathcal{D}$ of set maps $\alpha :U\to X$ called plots such that:
each plot $\alpha $ is defined on an open subset $U\subset {\mathbb{R}}^{n}$ of some Euclidean space ${\mathbb{R}}^{n}$, $n\ge 0$;
any constant map $\alpha :U\subset {\mathbb{R}}^{n}\to X$ belongs to $\mathcal{D}$;
if $\alpha :U\subset {\mathbb{R}}^{n}\to X$ belongs to $\mathcal{D}$ and if $h:V\subset {\mathbb{R}}^{m}\to U\subset {\mathbb{R}}^{n}$ is a ${C}^{\infty}$map, then the composition $\alpha \circ h$ belongs to $\mathcal{D}$;
the map $\alpha :U\to X$ belongs to $\mathcal{D}$ if and only if it locally belongs to $\mathcal{D}$, that is, for each $p\in U$ there exists some open subset $p\in V\subset U$ such that ${\alpha}_{\mid V}$ belongs to $\mathcal{D}$.
Note that the domain U and the Euclidean space ${\mathbb{R}}^{n}$, $n\ge 0$, depend on the plot $\alpha $.
A diffeological space $(X,\mathcal{D})$ is a set X endowed with a diffeology $\mathcal{D}$.
Remark 2.1.
Diffeological spaces are considered to be of class ${\mathcal{C}}^{\infty}$. By changing ${\mathcal{C}}^{\infty}$ by ${\mathcal{C}}^{r}$ in Axiom (3) for some $0\le r\le \omega $ we could obtain a theory for diffeological spaces of class ${\mathcal{C}}^{r}$.
Example 2.2. Let M be a finitedimensional manifold. The manifold diffeology ${\mathcal{D}}_{M}$ is the collection of all ${\mathcal{C}}^{\infty}$maps $U\subset {\mathbb{R}}^{m}\to M$ defined on open subsets of Euclidean spaces with values in M.
Example 2.3. Let R be any equivalence relation on the manifold M and let $\pi :M\to M/R$ be the quotient map. We endow $M/R$ with the quotient diffeology ${\mathcal{D}}_{M}/R$ where the map $\alpha :U\to M/R$ belongs to ${\mathcal{D}}_{M}/R$ if it locally factors through some plot of the manifold diffeology ${\mathcal{D}}_{M}$ on M.
Example 2.4. Let $N\subset M$ be any subset of the manifold M. We endow N with the subspace diffeology formed by the plots $\alpha :U\to M$ in the manifold diffeology ${\mathcal{D}}_{M}$ such that $\alpha \left(U\right)\subset N$.
The last two examples show that diffeology is a much more flexible setting than the classical one.
3. Smooth Maps
Definition 3.1. Let $(X,{\mathcal{D}}_{X})$, $(Y,{\mathcal{D}}_{Y})$ be two diffeological spaces. A set map $f:X\to Y$ is smooth when $\alpha \in {\mathcal{D}}_{X}$ implies $f\circ \alpha \in {\mathcal{D}}_{Y}$.
Example 3.2. If $M,N$ are ${\mathcal{C}}^{\infty}$manifolds endowed with the manifold diffeologies ${\mathcal{D}}_{M}$, ${\mathcal{D}}_{N}$, respectively, then the smooth maps between M and N as diffeological spaces are the ${\mathcal{C}}^{\infty}$maps as differentiable manifolds.
Proposition 3.3. The composition of smooth maps is a smooth map.
Proposition 3.4. The quotient map $\pi :M\to M/R$ of Example 2.3 is smooth. Moreover, a map $f:M/R\to N$ is smooth if and only if the composition $f\circ \pi :M\to N$ is smooth.
Corollary 3.5. A quotient map $\alpha :U\to X$ is a diffeomorphism if and only if it is bijective.
Proof. Let ${\alpha}^{1}:X\to U$ be the inverse map. Since ${\alpha}^{1}\circ \alpha ={\mathrm{id}}_{X}$ is smooth, ${\alpha}^{1}$ is smooth. □
Example 3.6. If $N\subset M$ is a subset endowed with the subspace diffeology of Example 2.4 then the inclusion map ${i}_{N}:N\subset M$ is smooth. Moreover, a map $f:P\to N$ is smooth if and only if ${i}_{N}\circ f:P\to M$ is smooth.
Example 3.7. By Axiom 3 of diffeology, the plots $\alpha :U\to X$ of a diffeology are smooth maps.
Definition 3.8. A diffeomorphism is a smooth map with a smooth inverse.
4. Generating Families
Definition 4.1. It is easy to check that the intersection of diffeologies on a set X is a diffeology on X. Then if $\mathcal{F}$ is any family of set maps ${\alpha}_{i}:{U}_{i}\subset {\mathbb{R}}^{{n}_{i}}\to X$ we can consider the smallest diffeology $\mathcal{D}=\langle \mathcal{F}\rangle $ containing $\mathcal{F}$. We will say that $\mathcal{F}$ is a generating family for the diffeology $\mathcal{D}=\langle \mathcal{F}\rangle $.
We will always assume that the family $\mathcal{F}$ contains all the constant plots on X.
Example 4.2. Any atlas on a manifold M is a generating family of the manifold diffeology.
Example 4.3. Let $(X,{\mathcal{D}}_{X})$ be a diffeological space and let $f:X\to Y$ be a set map. The diffeology on Y generated by the maps $f\circ \alpha $, with $\alpha \in {\mathcal{D}}_{X}$, is called the final diffeology. It is formed by the maps that locally are of the form $f\circ \alpha $ for some $\alpha \in {\mathcal{D}}_{X}$.
Example 4.4. Let $(Y,{\mathcal{D}}_{Y})$ be a diffeological space and let $f:X\to Y$ be a set map. The initial diffeology on X is the diffeology generated by the set maps $\alpha :U\subset {\mathbb{R}}^{n}\to X$ such that $f\circ \alpha \in {\mathcal{D}}_{Y}$.
Example 4.5. Let $(X,{\mathcal{D}}_{X})$, $(Y,{\mathcal{D}}_{Y})$ be two diffeological spaces. The product diffeology on $X\times Y$ is the intersection of the initial diffeologies for the projections ${p}_{X}:X\times Y\to X$ and ${p}_{Y}:X\times Y\to Y$.
Example 4.6. The coproduct diffeology on the disjoint union $X\bigsqcup Y$ is the intersection of the final diffeologies for the inclusions ${i}_{X}:X\hookrightarrow X\bigsqcup Y$ and ${i}_{Y}:Y\hookrightarrow X\bigsqcup Y$.
It is easy to check that the product and coproduct diffeologies verify the usual universal properties.
The following criterion of generation is very useful.
Theorem 4.7 ([
1] Art. 1.68).
Let the diffeology $\mathcal{D}=\langle \mathcal{F}\rangle $ be generated by the family $\mathcal{F}$. A set map $\alpha :U\subset {\mathbb{R}}^{n}\to X$ belongs to $\mathcal{D}$ if and only if it locally factors through some element of $\mathcal{F}$ (we assume that constant plots are all contained in $\mathcal{F}$).
Moreover, a map $f:(X,{\mathcal{D}}_{X})\to (Y,{\mathcal{D}}_{Y})$ is smooth if and only if $f\circ \alpha \in {\mathcal{D}}_{Y}$ for any $\alpha \in \mathcal{F}$ in a generating family $\mathcal{F}$ of ${\mathcal{D}}_{X}$.
5. Categorical Constructions
The category of diffeological spaces and smooth maps has limits and colimits. Here, we will show how to construct the pullback, the pushout and the join of two smooth maps. Most times these maps will be plots of some diffeology on X.
5.1. PullBack
Let
$\alpha :U\to X$,
$\beta :V\to X$ be two smooth maps with the same codomain. The pullback of
$\alpha $ and
$\beta $ is the set
endowed with the subspace diffeology of the product diffeology on
$U\times V$.
The projections
${p}_{1}:U\times V\to U$ and
${p}_{2}:U\times V\to V$ induce smooth maps
which verify
$\alpha \circ {p}_{U}=\beta \circ {p}_{V}$ and the universal property of a pullback:
5.2. PushOut
Let
${p}_{U}:P\to U$,
${p}_{V}:P\to V$ be two smooth maps with the same domain. The pushout of
${p}_{U}$ and
${p}_{V}$ is the quotient
$J=(U\bigsqcup V)/R$ of the coproduct
$U\bigsqcup V$ by the equivalence relation
R generated by
The maps
${j}_{U}=\pi \circ {i}_{U}$ and
${j}_{V}=\pi \circ {i}_{V}$ verify
${j}_{U}\circ {p}_{U}={j}_{V}\circ {p}_{V}$ and the universal property of a pushout:
5.3. Join
Given two smooth maps
$\alpha :U\to X$ and
$\beta :V\to X$, it is an exercise to check that the pushout of the pullback of
$\alpha $ and
$\beta $ is diffeomorphic to the join
$U\phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}V$ of
$\alpha $ and
$\beta $ ([
4]), defined as follows: there is a welldefined smooth map
$\alpha \phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}\beta :U\phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}V\to X$ and maps
${j}_{U}:U\to U\phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}V$ and
${j}_{V}:V\to U\phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}V$ such that

$(\alpha \phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}\beta )\circ {j}_{U}=\alpha $, $(\alpha \phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}\beta )\circ {j}_{V}=\beta $,

they verify the universal property
Analogously, given k maps ${\alpha}_{i}:{U}_{i}\to X$, the join $U={U}_{1}\phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}{U}_{k}$ and the universal map $\alpha ={\alpha}_{1}\phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}{\alpha}_{k}:U={U}_{1}\phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}{U}_{k}\to X$ are defined by induction.
Remark 5.1.
It is possible to define the join of an infinite number of plots, but we will not develop this idea, for the sake of simplicity.
Lemma 5.2. Let $\mathcal{F}=\{{\alpha}_{i}:{U}_{i}\to X\}$ be a generating family. Then the universal map $\alpha ={\ast}_{i}{\alpha}_{i}:U={\ast}_{i}{U}_{i}\to X$ of the join is a quotient map.
Proof. Let $\gamma :W\to X$ be a plot on X. By Criterion 4.7, for each $p\in W$ there exists a neighborhood ${W}_{p}\subset W$ such that $\gamma $ factors through some ${\alpha}_{i}$. Hence it factors through the disjoint union and consequently through the join. □
6. Main Result
Next theorem is our main result.
Theorem 6.1. Let $(X,\mathcal{D})$ be a diffeological space. A family $\mathcal{F}=\{{\alpha}_{i}:{U}_{i}\to X\}$ of plots is a generating family for $\mathcal{D}$ if and only if there is a diffeomorhism $\alpha ={\alpha}_{1}\phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}{\alpha}_{n}:U={U}_{1}\phantom{\rule{0.166667em}{0ex}}\ast \phantom{\rule{0.166667em}{0ex}}\cdots \ast {U}_{n}\to X$ commuting with the maps ${\alpha}_{i}$ and the natural maps ${j}_{i}:{U}_{i}\to U$, that is $\alpha \circ {j}_{i}={\alpha}_{i}$ for all i.
Proof. Let $\mathcal{F}$ be a generating family. The map $\alpha $ is surjective because generating families include all constant plots.
It is injective because $\alpha \left(\left[{u}_{i}\right]\right)=\alpha \left(\left[{u}_{j}\right]\right)$ means that either ${u}_{i}={u}_{j}$ or ${u}_{i}\in {U}_{i}$, ${u}_{j}\in {U}_{j}$ and ${\alpha}_{i}\left({u}_{i}\right)={\alpha}_{j}\left({u}_{j}\right)$, that is $\left[{u}_{i}\right]=\left[{u}_{j}\right]$. Then $\alpha $ is bijective and Corollary 3.5 applies. Hence $\alpha $ is a diffeomorphism.
The converse statement follows from Lemma 5.2. □
Example 6.2. Let X be the disjoint union of a line ${X}_{1}=\mathbb{R}$ and a point ${X}_{0}=\left\{0\right\}$, endowed with the diffeology generated by the constant plot ${\alpha}_{0}:{\mathbb{R}}^{0}\to X$, ${\alpha}_{0}\left(0\right)=0$, and the identity plot ${\alpha}_{1}:{\mathbb{R}}^{1}\to {X}_{1}\subset X$, ${\alpha}_{1}\left(t\right)=t$, respectively. Clearly, $X\cong {U}_{0}\ast {U}_{1}={U}_{0}\bigsqcup {U}_{1}$.
Example 6.3. Let
X be the set
We consider the diffeology on
X generated by the two plots
Then
$X\cong {U}_{1}\ast {U}_{2}$.
As we pointed out in [
2], this is not the
cross diffeology on
X induced by the manifold diffeology of
${\mathbb{R}}^{2}$.
Example 6.4. The manifold diffeology on a manifold
M is the join of any atlas. More precisely, let
$\{{\alpha}_{i}\subset {\mathbb{R}}^{m}:{U}_{i}\to M\}$ be an atlas on the manifold
${M}^{m}$. Then
where the open subsets
${U}_{i}\cong \alpha \left({U}_{i}\right)\subset M$ are glued by inclusions.
References
 IglesiasZemmour, Patrick. Diffeology. (Mathematical Surveys and Monographs 185. Providence, RI: American Mathematical Society (AMS) xxiii, 439 p. (2013).
 MacíasVirgós, Enrique; Mehrabi, Reihaneh. MayerVietoris sequence for generating families in diffeological spaces. Indag. Math., New Ser. 34, No. 4, 661672 (2023).
 nLab. Diffeological space. https://ncatlab.org/nlab/show/diffeological+space (viewed on August 28, 2023).
 Rijke, Egbert. The join construction, arXiv:1701.07538 (2017).
 Souriau, JeanMarie. Groupes différentiels, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., AixenProvence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128.

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