# nLab diffeological space

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Cohesive toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $(\infty,1)$-topos

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion?

# Contents

## Idea

A 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 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 diffeological space is then a set together with a diffeology on it.

Diffeological spaces were originally introduced in (Souriau 79). They have subsequently been developed in the textbook (Iglesias-Zemmour 13)

## Definition

###### Definition

Let $\mathcal{Op}$ denote the site whose objects are the open subsets of the Euclidean spaces $\mathbb{R}^n$ and whose morphisms are smooth maps between these.

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

• $X$ is a set

• and $\mathcal{D} \in Sh(\mathcal{Op})$ is a diffeology on $X$:

• a subsheaf of the sheaf $U \mapsto Hom_{Set}(U,X)$ with $\mathcal{D}(*) = X$

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

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.

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 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 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 spaces 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 smooth sets to be any sheaf of CartSp. A diffeological space is equivalently a concrete sheaf on the concrete site CartSp. (For details see this Prop. at geometry of physics – smooth sets.)

$DiffeologicalSpaces \hookrightarrow Sh(CartSp)$

on all concrete sheaves is not a topos, but is a quasitopos.

This is Prop. 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 $\mathbb{R}^0$.

## Examples

• 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 functions from $U$ to $X$, i.e. the morphisms in Diff:

$X : U \mapsto Hom_{Diff}(U,X) \,.$
• 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 $Y$

$X \times Y : U \mapsto Hom_{DiffSp}(U,X) \times Hom_{DiffSp}(U,Y) \,.$
• 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 $Y$:

$[X,Y] : U \mapsto Hom_{DiffSp}(X \times U, Y) \,.$

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.

## Properties

### Relation to topological and $\Delta$-generated spaces

###### Proposition

(adjunction between topological spaces and diffeological spaces)

There is a pair of adjoint functors

$TopologicalSpaces \underoverset{ \underset{ Cdfflg }{\longrightarrow} }{ \overset{ Dtplg }{\longleftarrow} }{\phantom{AA}\bot\phantom{AA}} DiffeologicalSpaces$

between the categories of TopologicalSpaces and of DiffeologicalSpaces, where

• $Cdfflg$ takes a topological space $X$ to the continuous diffeology, namely the diffeological space on the same underlying set $X_s$ whose plots $U_s \to X_s$ are the continuous functions (from the underlying topological space of the domain $U$).

• $Dtplg$ takes a diffeological space to the diffeological topology (D-topology), namely the topological space with the same underlying set $X_s$ and with the final topology that makes all its plots $U_{s} \to X_{s}$ into continuous functions: called the D-topology.

Hence a subset $O \subset \flat X$ is an open subset in the D-topology precisely if for each plot $f \colon U \to X$ the preimage $f^{-1}(O) \subset U$ is an open subset in the Cartesian space $U$.

Moreover:

1. the fixed points of this adjunction $X \in$TopologicalSpaces (those for which the counit is an isomorphism, hence here: a homeomorphism) are precisely the Delta-generated topological spaces (i.e. D-topological spaces):

$X \;\,\text{is}\;\Delta\text{-generated} \;\;\;\;\; \Leftrightarrow \;\;\;\;\; Dtplg(Cdffg(X)) \underoverset{\simeq}{\;\;\epsilon_X\;\;}{\longrightarrow} X$
2. this is an idempotent adjunction, which exhibits $\Delta$-generated/D-topological spaces as a reflective subcategory inside diffeological spaces and a coreflective subcategory inside all topological spaces:

(1)$TopologicalSpaces \underoverset { \underset{ Cdfflg }{\longrightarrow} } { \overset{ }{\hookleftarrow} } {\phantom{AA}\bot\phantom{AA}} DTopologicalSpaces \underoverset { \underset{ }{\hookrightarrow} } { \overset{ Dtplg }{\longleftarrow} } {\phantom{AA}\bot\phantom{AA}} DiffeologicalSpaces$

Finally, these adjunctions are a sequence of Quillen equivalences with respect to the:

Caution: There was a gap in the original proof that $DTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces$. The gap is claimed to be filled now, see the commented references here.

These adjunctions and their properties are observed in Shimakawa-Yoshida-Haraguchi 10, Prop. 3.1, Prop. 3.2, Lemma 3.3. The model structures and Quillen equivalences are due to Haraguchi 13, Thm. 3.3 (on the left) and Haraguchi-Shimakawa 13, Sec. 7 (on the right, but this may have a gap).

###### Proof

We spell out the existence of the idempotent adjunction (1):

First, to see we have an adjunction $Dtplg \dashv Cdfflg$, we check the hom-isomorphism (here).

Let $X \in DiffeologicalSpaces$ and $Y \in TopologicalSpaces$. Write $(-)_s$ for the underlying sets. Then a morphism, hence a continuous function of the form

$f \;\colon\; Dtplg(X) \longrightarrow Y \,,$

is a function $f_s \colon X_s \to Y_s$ of the underlying sets such that for every open subset $A \subset Y_s$ and every smooth function of the form $\phi \colon \mathbb{R}^n \to X$ the preimage $(f_s \circ \phi_s)^{-1}(A) \subset \mathbb{R}^n$ is open. But this means equivalently that for every such $\phi$, $f \circ \phi$ is continuous. This, in turn, means equivalently that the same underlying function $f_s$ constitutes a smooth function $\widetilde f \;\colon\; X \longrightarrow Cdfflg(Y)$.

In summary, we thus have a bijection of hom-sets

$\array{ Hom( Dtplg(X), Y ) &\simeq& Hom(X, Cdfflg(Y)) \\ f_s &\mapsto& (\widetilde f)_s = f_s }$

given simply as the identity on the underlying functions of underlying sets. This makes it immediate that this hom-isomorphism is natural in $X$ and $Y$ and this establishes the adjunction.

Next, to see that the D-topological spaces are the fixed points of this adjunction, we apply the above natural bijection on hom-sets to the case

$\array{ Hom( Dtplg(Cdfflg(Z)), Y ) &\simeq& Hom(Cdfflg(Z), Cdfflg(Y)) \\ (\epsilon_Z)_s &\mapsto& (\mathrm{id})_s }$

to find that the counit of the adjunction

$Dtplg(Cdfflg(X)) \overset{\epsilon_X}{\longrightarrow} X$

is given by the identity function on the underlying sets $(\epsilon_X)_s = id_{(X_s)}$.

Therefore $\eta_X$ is an isomorphism, namely a homeomorphism, precisely if the open subsets of $X_s$ with respect to the topology on $X$ are precisely those with respect to the topology on $Dtplg(Cdfflg(X))$, which means equivalently that the open subsets of $X$ coincide with those whose pre-images under all continuous functions $\phi \colon \mathbb{R}^n \to X$ are open. This means equivalently that $X$ is a D-topological space.

Finally, to see that we have an idempotent adjunction, we check that the comonad

$Dtplg \circ Cdfflg \;\colon\; TopologicalSpaces \to TopologicalSpaces$

is an idempotent comonad, hence that

$Dtplg \circ Cdfflg \overset{ Dtplg \cdot \eta \cdot Cdfflg }{\longrightarrow} Dtplg \circ Cdfflg \circ Dtplg \circ Cdfflg$

is a natural isomorphism. But, as before for the adjunction counit $\epsilon$, we have that also the adjunction unit $\eta$ is the identity function on the underlying sets. Therefore, this being a natural isomorphism is equivalent to the operation of passing to the D-topological refinement of the topology of a topological space being an idempotent operation, which is clearly the case.

Further discussion of the D-topology is in CSW 13.

### Topological homotopy type and diffeological shape

###### Definition

(diffeological singular simplicial set)

Consider the simplicial diffeological space

$\array{ \Delta & \overset{ \Delta^\bullet_{diff} }{ \longrightarrow } & DiffeologicalSpaces \\ [n] &\mapsto& \Delta^n_{diff} \mathrlap{ \coloneqq \big\{ \vec x \in \mathbb{R}^{n+1} \;\vert\; \underset{i}{\sum} x^i = 1 \big\} } }$

which in degree $n$ is the standard extended n-simplex inside Cartesian space $\mathbb{R}^{n+1}$, equipped with its sub-diffeology.

This induces a nerve and realization adjunction between diffeological spaces and simplicial sets:

$DiffeologicalSpaces \underoverset { \underset{Sing_{\mathrlap{diff}}}{\longrightarrow} } { \overset{ \left\vert - \right\vert_{\mathrlap{diff}} }{\longleftarrow} } { \phantom{AA}\bot\phantom{AA} } SimplicialSets \,,$

where the right adjoint is the diffeological singular simplicial set functor $Sing_{diff}$.

###### Remark

(diffeological singular simplicial set as path ∞-groupoid)

Regarding simplicial sets as presenting ∞-groupoids, we may think of $Sing_{diff}(X)$ (Def. ) as the path ∞-groupoid of the diffeological space $X$.

In fact, by the discussion at shape via cohesive path ∞-groupoid we have that $Sing_{diff}$ is equvialent to the shape of diffeological spaces regarded as objects of the cohesive (∞,1)-topos of smooth ∞-groupoids:

$Sing_{diff} \;\simeq\; Shp \circ i \;\;\colon\;\; DiffeologicalSpaces \overset{i}{\hookrightarrow} SmoothGroupoids_{\infty} \overset{Shape}{\longrightarrow} Groupoids_\infty$
###### Proposition

(topological homotopy type is cohesive shape of continuous diffeology)

For every $X \in$ TopologicalSpaces, the cohesive shape/path ∞-groupoid presented by its diffeological singular simplicial set (Def. , Remark ) of its continuous diffeology is equivalent to the homotopy type of $X$ presented by the ordinary singular simplicial set:

$Sing_{diff} \big( Cdfflg(X) \big) \;\simeq_{whe}\; Sing(X) \,.$

### Embedding of smooth manifolds into diffeological spaces

###### Proposition

The obvious functor from the category SmoothManifolds of smooth manifolds to the category DiffeologicalSpaces of diffeological spaces is a full and faithful functor

$SmoothManifolds \hookrightarrow DiffeologicalSpacess \,.$
###### Proof

This is a direct consequence of the fact that CartSp$_{smooth}$ is a dense sub-site of SmoothManifolds and the Yoneda lemma.

It may nevertheless be useful to spell out the elementary proof directly:

To see that the functor is faithful, notice that if $f,g : X \to Y$ are two smooth functions 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 derivatives 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.

###### Remark

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

$Diff \hookrightarrow Sh(CartSp_{dim \leq 1}) \,.$

This fact plays a role in the definition of Frölicher spaces, which are generalized smooth spaces 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 sheaves of interest, such as the sheaves of differential forms of degree $\geq 2$.

### Embedding of smooth manifolds with boundary into diffeological spaces

###### Proposition

(manifolds with boundaries and corners form full subcategory of diffeological spaces)

The evident functor

$SmthMfdWBdrCrn \overset{\phantom{AAAA}}{\hookrightarrow} DiffeologicalSpaces$

from the category of smooth manifolds with boundaries and corners to that of diffeological spaces is fully faithful, hence is a full subcategory-embedding.

### Embedding of Banach manifolds into diffeological spaces

Also Banach manifolds embed fully faithfully into the category of diffeological spaces. In (Hain) this is discussed in terms of Chen smooth spaces.

### Embedding of Fréchet manifolds into diffeological spaces

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

###### Definition

Define a functor

$\iota \colon FrechetManifolds \to DiffeologicalSpaces$

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 $U \to X$.

###### Proposition

The functor $\iota \colon FrechetManifolds \hookrightarrow DiffeologicalSpaces$ is a full and faithful functor.

This appears as (Losik 94, theorem 3.1.1, following Losik 92), as variant of the analogous statement for Banach manifolds in (Hain). The fact that maps between Fréchet spaces are smooth if and only if they send smooth curves to smooth curves was proved earlier in (Frölicher 81, théorème 1)

The statement is also implied by (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 smoothness (Glöckner 06), in which case one merely has a faithful functor $lctvs \to DiffeologicalSpaces$. Notice that the choice of topology in (Kriegl-Michor 97) is such that this equivalence of notions reduces to the above just for Fréchet manifolds.

###### Proposition

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 Fréchet manifold structure $C^\infty(X,Y)_{Fr}$:

$\iota(C^\infty(X,Y)_{Fr}) \simeq C^\infty(X,Y)_{diff} \,.$

This appears as (Waldorf 09, lemma A.1.7).

$\,$

### Embedding of diffeological spaces into smooth sets

We discuss how diffeological spaces are equivalently the concrete objects in the cohesive topos of smooth sets (see there).

###### Proposition

(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 equivalent to the category of diffeological spaces

###### Proof

The concrete sheaves for the local topos $Sh(CartSp)$ are by definition those objects $X$ for which the $(\Gamma \dashv CoDisc)$-unit

$X \to CoDisc \Gamma X$

is a monomorphism. Monomorphisms of sheaves are tested objectwise, so that means equivalently that for every $U \in CartSp$ we have that

$X(U) \simeq Hom_{Sh}(U,X) \to Hom_{Sh}(U, Codisc \Gamma X) \simeq Hom_{Set}(\Gamma U, \Gamma X)$

is a monomorphism. This is precisely the condition on a sheaf to be a diffeological space.

For a fully detailed proof see this Prop. at geometry of physics – smooth sets.

###### Corollary

The category of diffeological spaces is a quasitopos.

###### Proof

This follows from the discussion at Locality.

This has some immediate general abstract consequences

###### Corollary

The category of diffeological spaces is

### Embedding of diffeological spaces into higher differential geometry

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

$Sh(CartSp) \hookrightarrow Smooth \infty Grpd \coloneqq Sh_\infty(CartSp)$

into the (∞,1)-topos Smooth∞Grpd of “higher smooth sets” –smooth ∞-groupoids – as precisely the 0-truncated objects.

### Distribution theory

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

### General

The basic idea of understanding generalized smooth spaces as concrete sheaves on a site of smooth test spaces originates in work of Kuo Tsai Chen (see also at Chen space):

• Kuo Tsai Chen, Iterated integrals of differential forms and loop space homology, Ann. Math. 97 (1973), 217–246 (jstor:1970846)

Chen considered (apart from iterated integrals) effectively presheaves on a site of convex subsets of Cartesian spaces. In

• Kuo Tsai Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 83–98 (jstor:1997148)

roughly the sheaf condition was added (without using any of this sheaf-theoretic terminology). The definition of Chen spaces stabilized in

and served as the basis of a celebrated theorem on the de Rham cohomology of loop spaces.

A brief review is in

(which, curiously, still does not make the connection to the theory of sheaves).

However, Chen does not require the domains of his plots to be open subsets, which makes Chen spaces be closely related to but different from diffeological spaces (see Stacey 11, p. 32)

The proper concept of diffeological spaces was introduced, under the name difféologie and apparently independently from Chen, in:

• Jean-Marie Souriau, Groupes différentiels, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. (doi:10.1007/BFb0089728, mr:607688)

• Jean-Marie Souriau, Groupes différentiels et physique mathématique, In: Denardo G., Ghirardi G., Weber T. (eds.) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 201. Springer 1984 (doi:10.1007/BFb0016198)

motivated there by diffeological groups arising in geometric quantization.

Following Souriau, a comprehensive textbook account of differential geometry formulated in terms of diffeological spaces (and coining that term) is

following the thesis

which contains some useful material that may not yet have made it into the book.

Further exposition and lecture notes are in

The article

amplifies the point that diffeological spaces are concrete sheaves forming a quasi-topos.

A discussion of the relations of variants of the definition is in

• Andrew Stacey, Comparative Smootheology, Theory and Applications of Categories, Vol. 25, 2011, No. 4, pp 64-117. (tac:25-04)

More pointers and monthly seminars at:

Application to classifying spaces and universal connections:

• Mark Mostow, The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations, J. Differential Geom. Volume 14, Number 2 (1979), 255-293 (euclid:jdg/1214434974)

### Full subcategories

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

• Richard Hain, A characterization of smooth functions defined on a Banach space, Proc. Amer. Math. Soc. 77 (1979), 63-67 (web, pdf)

The (non-full) embedding of locally convex vector spaces and Michal-Bastiani smooth maps into diffeological spaces is discussed around corollary 3.14 in

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

• Helge Glöckner, Discontinuous non-linear mappings on locally convex direct limits, Publ. Math. Debrecen 68 (2006) 1-13, arXiv:math/0503387.

The full subcategory-inclusion of Fréchet manifolds into diffeological spaces is discussed in

and reviewed in

The proof can in fact be deduced from théorème 1 of

• Alfred Frölicher, Applications lisses entre espaces et variétés de Fréchet, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 2, 125–127. BnF

The preservation of mapping spaces under this embedding is due to

The largest topology on the set which underlies a diffeological space with respect to which all plots are continuous functions (the “D-topology”) is studied in

Some homotopy theory modeled on diffeological spaces (instead of on topological spaces) via their cohesive shape is discussed in

The full subcategory-inclusion of manifolds with boundaries and corners is discussed in

### For orbifolds

On orbifolds regarded as naive local quotient spaces (instead of homotopy quotients/Lie groupoids/differentiable stacks) but as such formed in diffeological spaces:

On this approach seen in the broader context of higher differential geometry: