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\begin{document}

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\section*{diffeological space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{EmbeddingOfSmoothManifoldsIntoDiffeoloticalSpaces}{Embedding of smooth manifolds into diffeological spaces}\dotfill \pageref*{EmbeddingOfSmoothManifoldsIntoDiffeoloticalSpaces} \linebreak
\noindent\hyperlink{embedding_of_smooth_manifolds_with_boundary_into_diffeological_spaces}{Embedding of smooth manifolds with boundary into diffeological spaces}\dotfill \pageref*{embedding_of_smooth_manifolds_with_boundary_into_diffeological_spaces} \linebreak
\noindent\hyperlink{EmbeddingOfBanachManifoldsIntoDiffeologicalSpaces}{Embedding of Banach manifolds into diffeological spaces}\dotfill \pageref*{EmbeddingOfBanachManifoldsIntoDiffeologicalSpaces} \linebreak
\noindent\hyperlink{RelationBetweenDeffeologicalAndFrechetStructure}{Embedding of Fr\'e{}chet manifolds into diffeological spaces}\dotfill \pageref*{RelationBetweenDeffeologicalAndFrechetStructure} \linebreak
\noindent\hyperlink{EmbeddingOfDiffeologicalSpacesIntoTheSheafTopos}{Embedding of diffeological spaces into smooth sets}\dotfill \pageref*{EmbeddingOfDiffeologicalSpacesIntoTheSheafTopos} \linebreak
\noindent\hyperlink{EmbeddingOfDiffeologicalSpacesIntoHigherDifferentialGeometry}{Embedding of diffeological spaces into higher differential geometry}\dotfill \pageref*{EmbeddingOfDiffeologicalSpacesIntoHigherDifferentialGeometry} \linebreak
\noindent\hyperlink{distribution_theory}{Distribution theory}\dotfill \pageref*{distribution_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{full_subcategories}{Full subcategories}\dotfill \pageref*{full_subcategories} \linebreak
\noindent\hyperlink{ReferencesForOrbifolds}{For orbifolds}\dotfill \pageref*{ReferencesForOrbifolds} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{diffeological space} is a type of [[generalized smooth space]]. As with the other variants, it subsumes the notion of [[smooth manifold]] but also naturally captures other spaces that one would like to think of as smooth spaces but aren't manifolds; for example, the space of all smooth maps between two smooth manifolds can be made into a diffeological space. (These mapping spaces are rarely manifolds themselves, see [[manifolds of mapping spaces]].)

In a little more detail, a \textbf{diffeology}, $\mathcal{D}$ on a set $X$ is a [[presheaf]] on the category of open subsets of Euclidean spaces with smooth maps as morphisms. To each open set $U \subseteq \mathbb{R}^n$, it assigns a subset of $\Set(U,X)$. The functions in $\Set(U,X)$ are to be regarded as the ``smooth functions'' from $U$ to $X$. A \textbf{diffeological space} is then a set together with a diffeology on it.

Diffeological spaces were originally introduced in (\hyperlink{Souriau79}{Souriau 79}). They have subsequently been developed in the textbook (\hyperlink{PIZ}{Iglesias-Zemmour 13})

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{DiffSp}\hypertarget{DiffSp}{}
Let $\mathcal{Op}$ denote the [[site]] whose objects are the open subsets of the Euclidean spaces $\mathbb{R}^n$ and whose morphisms are [[smooth map]]s between these.

A \textbf{diffeological space} is a pair $(X,\mathcal{D})$ where

\begin{itemize}%
\item $X$ is a set


\item and $\mathcal{D} \in Sh(\mathcal{Op})$ is a \textbf{[[diffeology]]} on $X$:

\begin{itemize}%
\item a [[subobject|subsheaf]] of the sheaf $U \mapsto Hom_{Set}(U,X)$ with $\mathcal{D}(*) = X$


\item equivalently: a [[concrete sheaf]] on the [[site]] $\mathcal{Op}$ such that $\mathcal{D}(*) = X$ - a [[concrete object|concrete]] [[smooth space]] (see there for more details).



\end{itemize}


\end{itemize}
A [[morphism]] of diffeological spaces is a morphism of the corresponding [[sheaves]]: we take $DiffeologicalSp \hookrightarrow Sh(CartSp)$ to be the full [[subcategory]] on the diffeological spaces in the sheaf topos.

\end{defn}
For $(X,\mathcal{D})$ a diffeological space, and for any $U \in \mathcal{Op}$, the set $\mathcal{D}(U)$ is also called the set of \textbf{plots} in $X$ on $U$. This is to be thought of as the set of ways of mapping $U$ smoothly into the would-be space $X$. This assignment \emph{defined} what it means for a map $U \to X$ of sets to be smooth.

For some comments on the reasoning behind this kind of definition of generalized [[space]]s see [[motivation for sheaves, cohomology and higher stacks]].

A sheaf on the site $\mathcal{Op}$ of open subsets of Euclidean spaces is completely specified by its restriction to [[CartSp]], the full subcategory of [[Cartesian space]]: The [[fully faithful functor]] $CartSp \hookrightarrow \mathcal{Op}$ is a [[dense subsite]]-inclusion. Therefore in the sequel we shall often restrict our attention to [[CartSp]].

One may define a \emph{[[smooth sets]]} to be \emph{any} sheaf of [[CartSp]]. A diffeological space is equivalently a [[concrete sheaf]] on the [[concrete site]] [[CartSp]]. (For details see \href{geometry+of+physics+--+smooth+sets#DiffeologicalSpacesAreTheConcreteSmoothSets}{this Prop.} at \emph{[[geometry of physics -- smooth sets]]}.)

The [[full subcategory]]

\begin{displaymath}
DiffeologicalSpace \hookrightarrow Sh(CartSp)
\end{displaymath}
on all concrete sheaves is not a [[topos]], but is a [[quasitopos]].

This is Prop. \ref{DiffeologicalSpacesAreTheConcreteSmoothSets} below.

The concreteness condition on the sheaf is a reiteration of the fact that a diffeological space is a subsheaf of the sheaf $U \mapsto X^{|U|}$. In this way, one does not have to explicitly mention the underlying set $X$ as it is determined by the sheaf on the one-point open subset of $\mathbb{R}^0$.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Every [[smooth manifold]] $X$, i.e. every object of [[Diff]], becomes a diffeological space by defining the plots on $U \in CartSp$ to be the ordinary [[smooth function]]s from $U$ to $X$, i.e. the morphisms in [[Diff]]:

\begin{displaymath}
X : U \mapsto Hom_{Diff}(U,X)
  \,.
\end{displaymath}

\item For $X$ and $Y$ two diffeological spaces, their [[product]] as sets $X \times Y$ becomes a diffeological space whose plots are pairs consisting of a plot into $X$ and one into $Y$

\begin{displaymath}
X \times Y : U \mapsto Hom_{DiffSp}(U,X) \times Hom_{DiffSp}(U,Y)
  \,.
\end{displaymath}

\item Given any two diffeological spaces $X$ and $Y$, the set of morphisms $Hom_{DiffSp}(X,Y)$ becomes a smooth space by taking the plots on some $U$ to be the smooth morphisms $X \times U \to Y$, i.e. the smooth $U$-parameterized families of smooth maps from $X$ to $Y$:

\begin{displaymath}
[X,Y] : U \mapsto Hom_{DiffSp}(X \times U, Y)
  \,.
\end{displaymath}
In this formula we regard $U \in CartSp \hookrightarrow Diff$ as a diffeological space according to the above example. In fact, we apply secretly here the [[Yoneda embedding]] and use the general formula for the cartesian [[closed monoidal structure on presheaves]].



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{EmbeddingOfSmoothManifoldsIntoDiffeoloticalSpaces}{}\subsubsection*{{Embedding of smooth manifolds into diffeological spaces}}\label{EmbeddingOfSmoothManifoldsIntoDiffeoloticalSpaces}

\begin{prop}
\label{}\hypertarget{}{}
The obvious functor from the category [[Diff]] of [[smooth manifold]]s to the category of diffeological spaces is a [[full and faithful functor]]

\begin{displaymath}
Diff \to DiffeologicalSpace
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This is a direct consequence of the fact that [[CartSp]]$_{smooth}$ is a [[dense sub-site]] of [[Diff]] and the [[Yoneda lemma]].

It may nevertheless be useful to spell out a pedestrian proof.

To see that the functor is faithful, notice that if $f,g : X \to Y$ are two [[smooth function]]s that differ at some point, then they must differ in some [[open neighbourhood]] of that point. This [[open ball]] is a plot, hence the corresponding diffeological spaces differ on that plot.

To see that the functor is full, we need to show that a map of sets $f : X \to Y$ that sends plots to plots is necessarily a [[smooth function]], hence that all its [[derivative]]s exist. This can be tested already on all smooth curves $\gamma : (0,1) \to X$ in $X$. By [[Boman's theorem]], a function that takes all smooth curves to smooth curves is necessarily a smooth function. But curves are in particular plots, so a function that takes all plots of $X$ to plots of $Y$ must be smooth.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The proof shows that we could restrict attention to the full sub-site $CartSp_{dim \leq 1} \subset CartSp$ on the objects $\mathbb{R}^0$ and $\mathbb{R}^1$ and still have a full and faithful embedding

\begin{displaymath}
Diff \hookrightarrow Sh(CartSp_{dim \leq 1})
  \,.
\end{displaymath}
This fact plays a role in the definition of [[Frölicher space]]s, which are [[generalized smooth space]]s defined by plots by curves into and out of them.

While the site $CartSp_{dim \leq 1}$ is more convenient for some purposes, it is not so useful for other purposes, mostly when diffeological spaces are regarded from the point of view of the full sheaf topos: the sheaf topos $Sh(CartSp_{dim \leq 1})$ lacks some non-[[concrete sheaf|concrete]] sheaves of interest, such as the sheaves of differential forms of degree $\geq 2$.

\end{remark}
\hypertarget{embedding_of_smooth_manifolds_with_boundary_into_diffeological_spaces}{}\subsubsection*{{Embedding of smooth manifolds with boundary into diffeological spaces}}\label{embedding_of_smooth_manifolds_with_boundary_into_diffeological_spaces}

\begin{prop}
\label{SmoothManifoldsWithBoundaryEmbedIntoDiffeologicalSpaces}\hypertarget{SmoothManifoldsWithBoundaryEmbedIntoDiffeologicalSpaces}{}
\textbf{([[manifolds with boundaries and corners]] form [[full subcategory]] of [[diffeological spaces]])}

The evident [[functor]]

\begin{displaymath}
SmthMfdWBdrCrn \overset{\phantom{AAAA}}{\hookrightarrow} DiffeologicalSpaces
\end{displaymath}
from the [[category]] of [[smooth manifold|smooth]] [[manifolds with boundaries and corners]] to that of [[diffeological spaces]] is [[fully faithful functor|fully faithful]], hence is a [[full subcategory]]-embedding.

\end{prop}
(\hyperlink{PIZ}{Igresias-Zemmour 13, 4.16}, \hyperlink{GurerIZ19}{Gürer \& Iglesias-Zemmour 19})

\hypertarget{EmbeddingOfBanachManifoldsIntoDiffeologicalSpaces}{}\subsubsection*{{Embedding of Banach manifolds into diffeological spaces}}\label{EmbeddingOfBanachManifoldsIntoDiffeologicalSpaces}

Also [[Banach manifolds]] embed [[full and faithful functor|fully faithfully]] into the category of diffeological spaces. In (\hyperlink{Hain}{Hain}) this is discussed in terms of Chen smooth spaces.

\hypertarget{RelationBetweenDeffeologicalAndFrechetStructure}{}\subsubsection*{{Embedding of Fr\'e{}chet manifolds into diffeological spaces}}\label{RelationBetweenDeffeologicalAndFrechetStructure}

We discuss a natural embedding of [[Fréchet manifolds]] into the category of diffeological spaces.

\begin{defn}
\label{}\hypertarget{}{}
Define a [[functor]]

\begin{displaymath}
\iota \colon FrechetManifolds \to DiffeologicalSpaces
\end{displaymath}
in the evident way by taking for $X$ a [[Fréchet manifold]] for any $U \in$ [[CartSp]] the set of $U$-plots of $\iota(X)$ to be the set of smooth functions $U \to X$.

\end{defn}
\begin{prop}
\label{FrechetEmbedding}\hypertarget{FrechetEmbedding}{}
The functor $\iota \colon FrechetManifolds \hookrightarrow DiffeologicalSpaces$ is a [[full and faithful functor]].

\end{prop}
This appears as (\hyperlink{Losik}{Losik 94, theorem 3.1.1}), as variant of the analogous statement for [[Banach manifolds]] in (\hyperlink{Hain}{Hain}). The fact that maps between Fr\'e{}chet spaces are smooth if and only if they send smooth curves to smooth curves was proved earlier in (\hyperlink{Frolicher}{Fr\"o{}licher 81, th\'e{}or\`e{}me 1})

The statement is also implied by (\hyperlink{KrieglMichor}{Kriegl-Michor 97, cor. 3.14}) which states that functions between [[locally convex vector spaces]] are diffeologically smooth precisely if they send smooth [[curves]] to smooth curves. This is not true if one uses [[Michal-Bastiani smooth map|Michal-Bastiani smoothness]] (\hyperlink{Glockner06}{Gl\"o{}ckner 06}), in which case one merely has a [[faithful functor]] $lctvs \to DiffeologicalSpaces$. Notice that the choice of topology in (\hyperlink{KrieglMichor}{Kriegl-Michor 97}) is such that this equivalence of notions reduces to the above just for Fr\'e{}chet manifolds.

\begin{prop}
\label{}\hypertarget{}{}
Let $X, Y \in SmoothManifold$ with $X$ a [[compact manifold]].

Then under this embedding, the diffeological mapping space structure $C^\infty(X,Y)_{diff}$ on the mapping space coincides with the Fr\'e{}chet manifold structure $C^\infty(X,Y)_{Fr}$:

\begin{displaymath}
\iota(C^\infty(X,Y)_{Fr})
  \simeq
  C^\infty(X,Y)_{diff}
  \,.
\end{displaymath}
\end{prop}
This appears as (\hyperlink{Waldorf09}{Waldorf 09, lemma A.1.7}).

$\,$

\hypertarget{EmbeddingOfDiffeologicalSpacesIntoTheSheafTopos}{}\subsubsection*{{Embedding of diffeological spaces into smooth sets}}\label{EmbeddingOfDiffeologicalSpacesIntoTheSheafTopos}

We discuss how diffeological spaces are equivalently the [[concrete objects]] in the [[cohesive topos]] of [[smooth sets]] (see \href{smooth+set#Cohesion}{there}).

\begin{prop}
\label{DiffeologicalSpacesAreTheConcreteSmoothSets}\hypertarget{DiffeologicalSpacesAreTheConcreteSmoothSets}{}
\textbf{(diffeological spaces are the concrete smooth sets)}

The [[full subcategory]] on the [[concrete objects]] in the [[topos]] $SmoothSet \coloneqq Sh(Cart)$ of [[smooth sets]] is [[equivalence of categories|equivalent]] to the category of diffeological spaces

\end{prop}
\begin{proof}
The [[concrete sheaves]] for the [[local topos]] $Sh(CartSp)$ are by definition those objects $X$ for which the $(\Gamma \dashv CoDisc)$-[[unit of an adjunction|unit]]

\begin{displaymath}
X \to CoDisc \Gamma X
\end{displaymath}
is a [[monomorphism]]. Monomorphisms of sheaves are tested objectwise, so that means equivalently that for every $U \in CartSp$ we have that

\begin{displaymath}
X(U) \simeq Hom_{Sh}(U,X) \to Hom_{Sh}(U, Codisc \Gamma X)
  \simeq Hom_{Set}(\Gamma U, \Gamma X)
\end{displaymath}
is a monomorphism. This is precisely the condition on a sheaf to be a diffeological space.

\end{proof}
For a fully detailed proof see \href{geometry+of+physics+--+smooth+sets#DiffeologicalSpacesAreTheConcreteSmoothSets}{this Prop.} at \emph{[[geometry of physics -- smooth sets]]}.

\begin{cor}
\label{}\hypertarget{}{}
The category of diffeological spaces is a [[quasitopos]].

\end{cor}
\begin{proof}
This follows from the discussion at \hyperlink{Locality}{Locality}.

\end{proof}
This has some immediate general abstract consequences

\begin{cor}
\label{}\hypertarget{}{}
The category of diffeological spaces is

\begin{itemize}%
\item a [[cartesian closed category]]


\item a [[closed monoidal category]].



\end{itemize}
\end{cor}
\hypertarget{EmbeddingOfDiffeologicalSpacesIntoHigherDifferentialGeometry}{}\subsubsection*{{Embedding of diffeological spaces into higher differential geometry}}\label{EmbeddingOfDiffeologicalSpacesIntoHigherDifferentialGeometry}

In the last section we saw the embedding of diffeological spaces as precisely the [[concrete objects]] is the [[sheaf topos]] $Sh(CartSp) \simeq Sh(SmthMfd)$ of [[smooth sets]]. This is a general context for [[differential geometry]]. From there one can pass further to [[higher differential geometry]]: the topos of smooth sets in turn embeds

\begin{displaymath}
Sh(CartSp) \hookrightarrow
  Smooth \infty Grpd \coloneqq Sh_\infty(CartSp)
\end{displaymath}
into the [[(∞,1)-topos]] [[Smooth∞Grpd]] of ``higher [[smooth sets]]'' --[[smooth ∞-groupoids]] -- as precisely the [[truncated object in an (∞,1)-category|0-truncated objects]].

\hypertarget{distribution_theory}{}\subsubsection*{{Distribution theory}}\label{distribution_theory}

Since a space of [[smooth functions]] on a [[smooth manifold]] is canonically a diffeological space, it is natural to consider the \emph{smooth} [[linear functionals]] on such [[mapping spaces]]. These turn out to be equivalent to the [[continuous linear functionals]], hence to [[distributional densities]]. See at \emph{[[distributions are the smooth linear functionals]]} for details.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[smooth set]]


\item [[diffeological groupoid]], [[diffeological ∞-groupoid]]


\item [[connectology]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The basic idea of understanding a smooth space as a [[concrete sheaf]] on a site of smooth test spaces originates in work of Chen. In

\begin{itemize}%
\item [[Kuo Tsai Chen]], \emph{Iterated integrals of differential forms and loop space homology}, Ann. Math. 97 (1973), 217--246.

\end{itemize}
he considered (apart from [[iterated integrals]]) effectively [[presheaves]] on a site of [[convex subsets]] of [[Cartesian spaces]]. In

\begin{itemize}%
\item [[Kuo Tsai Chen]], Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 83--98.

\end{itemize}
roughly the [[sheaf]] condition was added (without using any of this sheaf-theoretic terminology). The definition of \emph{[[Chen smooth spaces]]} stabilized in

\begin{itemize}%
\item [[Kuo Tsai Chen]], \emph{Iterated path integrals} , Bull. Amer. Math. Soc. 83, (1977), 831--879.

\end{itemize}
and served as the basis of a celebrated theorem on the [[de Rham cohomology]] of [[loop spaces]].

The variant of this idea with the site of convex subsets replaced by that of open subsets (and hence equivalently by the site [[CartSp]]${}_{smooth}$) appeared in

The [[diffeological space]]-structure is at least implicit in

\begin{itemize}%
\item [[Jean-Marie Souriau]], \emph{Groupes diff\'e{}rentiels}, in \emph{Differential Geometrical Methods in Mathematical Physics} (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91--128. (\href{http://www.ams.org/mathscinet-getitem?mr=607688}{MathScinet})

\end{itemize}
motivated from the desire to realize the infinite dimensional groups that appear in [[geometric quantization]], such that (Hamiltonian) [[diffeomorphism group]] and their [[group extensions]] by [[quantomorphism groups]] as [[diffeological groups]].

A detailed discussion of the relations of these and other variants of the definition is in

\begin{itemize}%
\item [[Andrew Stacey]], \emph{Comparative Smootheology}, Theory and Applications of Categories, Vol. 25, 2011, No. 4, pp 64-117. (\href{http://www.tac.mta.ca/tac/volumes/25/4/25-04abs.html}{tac})

\end{itemize}
The article

\begin{itemize}%
\item [[John Baez]], [[Alexander Hoffnung]], \emph{Convenient Categories of Smooth Spaces} (\href{http://arxiv.org/abs/0807.1704}{arXiv})

\end{itemize}
amplifies the point that diffeological spaces are [[concrete sheaves]].

A textbook about [[differential geometry]] formulated in terms of diffeological spaces is

\begin{itemize}%
\item [[Patrick Iglesias-Zemmour]], \emph{Diffeology}, Mathematical Surveys and Monographs, AMS (2013) (\href{http://math.huji.ac.il/~piz/Site/The%20Book.html}{web}, \href{http://www.ams.org/bookstore-getitem/item=SURV-185}{publisher})

\end{itemize}
The term ``diffeological space'' originates here. The thesis

\begin{itemize}%
\item [[Patrick Iglesias-Zemmour]], \emph{Fibrations diff\'e{}ologiques et Homotopie}, PhD thesis (1985) (\href{http://math.huji.ac.il/~piz/documents/TheseEtatPI.pdf}{pdf})

\end{itemize}
contains some useful material that hasn't yet made it into the book.

Exposition and lecture notes are in

\begin{itemize}%
\item [[Patrick Iglesias-Zemmour]], \emph{Diffeologies}, talk at \emph{[[New Spaces for Mathematics and Physics]]}, IHP Paris 2015 (\href{https://www.youtube.com/watch?v=4sZDmiVOhaA}{video recording})


\item [[Patrick Iglesias-Zemmour]], \emph{An introduction to diffeology}, lecture at \emph{\href{http://www.nesinkoyleri.org/eng/events-detail.php?egitimkod=203}{Modern Mathematics Methods in Physics: Diffeology, Categories and Toposes and Non-commutative Geometry Summer School}}, 2018, to appear in \emph{[[New Spaces for Mathematics and Physics]]} (\href{http://math.huji.ac.il/~piz/documents/AITD.pdf}{pdf})



\end{itemize}
Discussion in the context of applications to [[continuum mechanics]] is in

\begin{itemize}%
\item [[William Lawvere]], [[Stephen Schanuel]] (eds.), \emph{[[Categories in Continuum Physics]]}, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 986

\end{itemize}
\hypertarget{full_subcategories}{}\subsubsection*{{Full subcategories}}\label{full_subcategories}

The [[full subcategory]]-inclusion of [[Banach manifolds]] into the category of diffeological spaces is due to

\begin{itemize}%
\item [[Richard Hain]], \emph{A characterization of smooth functions defined on a Banach space}, Proc. Amer. Math. Soc. 77 (1979), 63-67 (\href{http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/home.html}{web}, \href{http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/S0002-9939-1979-0539632-8.pdf}{pdf})

\end{itemize}
The (non-full) embedding of [[locally convex vector spaces]] and [[Michal-Bastiani smooth maps]] into diffeological spaces is discussed around corollary 3.14 in

\begin{itemize}%
\item [[Andreas Kriegl]], [[Peter Michor]]: \emph{[[The convenient setting of global analysis]]}, AMS (1997)

\end{itemize}
That there are diffeologically-smooth maps between locally convex vector spaces that are not continuous, and a fortiori not smooth in the sense of Michal-Bastiani is given, for instance, in

\begin{itemize}%
\item Helge Gl\"o{}ckner, \emph{Discontinuous non-linear mappings on locally convex direct limits}, Publ. Math. Debrecen 68 (2006) 1-13, \href{http://arxiv.org/abs/math/0503387}{arXiv:math/0503387}.

\end{itemize}
The [[full subcategory]]-inclusion of [[Fréchet manifolds]] into diffeological spaces is discussed in

\begin{itemize}%
\item M. V. Losik, \emph{Fr\'e{}chet manifolds as diffeological spaces}, Soviet. Math. 5 (1992)

\end{itemize}
and reviewed in section 3 of

\begin{itemize}%
\item M. V. Losik, \emph{Categorical Differential Geometry} Cah. Topol. G\'e{}om. Diff\'e{}r. Cat\'e{}g., 35(4):274--290, 1994.

\end{itemize}
The proof can in fact be deduced from th\'e{}or\`e{}me 1 of

\begin{itemize}%
\item [[Alfred Frölicher]], \emph{Applications lisses entre espaces et vari\'e{}t\'e{}s de Fr\'e{}chet}, C. R. Acad. Sci. Paris S\'e{}r. I Math. \textbf{293} (1981), no. 2, 125--127. \href{http://gallica.bnf.fr/ark:/12148/bpt6k5533894s/f31.image}{BnF}

\end{itemize}
The preservation of [[mapping spaces]] under this embedding is due to

\begin{itemize}%
\item [[Konrad Waldorf]] \emph{Transgression to Loop Spaces and its Inverse I}, Cah. Topol. Geom. Differ. Categ., 2012, Vol. LIII, 162-210 (\href{http://arxiv.org/abs/0911.3212}{arXiv:0911.3212})

\end{itemize}
The largest [[topological space|topology]] on the set which underlies a diffeological space with respect to which all plots are [[continuous functions]] (the ``[[D-topology]]'') is studied in

\begin{itemize}%
\item [[Dan Christensen|J. Daniel Christensen]], Gord Sinnamon, Enxin Wu, \emph{The $D$-topology for diffeological spaces} (\href{http://arxiv.org/abs/1302.2935}{arXiv:1302.2935})

\end{itemize}
Some [[homotopy theory]] modeled on diffeological spaces instead of on [[topological spaces]] is discussed in

\begin{itemize}%
\item [[Dan Christensen|J. Daniel Christensen]], Enxin Wu, \emph{The homotopy theory of diffeological spaces, I. Fibrant and cofibrant objects} (\href{http://arxiv.org/abs/1311.6394}{arXiv:1311.6394})

\end{itemize}
The [[full subcategory]]-inclusion of [[manifolds with boundaries and corners]] is discussed in

\begin{itemize}%
\item [[Serap Gürer]], [[Patrick Iglesias-Zemmour]], \emph{Differential forms on manifolds with boundary and corners}, Indagationes Mathematicae, Volume 30, Issue 5, September 2019, Pages 920-929 (\href{https://doi.org/10.1016/j.indag.2019.07.004}{doi:10.1016/j.indag.2019.07.004})

\end{itemize}
\hypertarget{ReferencesForOrbifolds}{}\subsubsection*{{For orbifolds}}\label{ReferencesForOrbifolds}

On [[orbifolds]] regarded as naive local quotient diffeological spaces:

\begin{itemize}%
\item [[Patrick Iglesias-Zemmour]], [[Yael Karshon]], Moshe Zadka, \emph{Orbifolds as diffeologies}, Transactions of the American Mathematical Society 362 (2010), 2811-2831 (\href{https://arxiv.org/abs/math/0501093}{arXiv:math/0501093})


\item [[Jordan Watts]], \emph{The Differential Structure of an Orbifold}, Rocky Mountain Journal of Mathematics, Vol. 47, No. 1 (2017), pp. 289-327 (\href{https://arxiv.org/abs/1503.01740}{arXiv:1503.01740})



\end{itemize}
[[!redirects diffeological space]] [[!redirects diffeological spaces]]

[[!redirects diffeological structure]] [[!redirects diffeological structures]]

[[!redirects diffeology]] [[!redirects diffeologies]]

[[!redirects Chen smooth space]] [[!redirects Chen smooth spaces]]



\end{document}