synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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)
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.)
The full subcategory
on all concrete sheaves is not a topos, but is a quasitopos.
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 $\mathbb{R}^0$.
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:
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 $Y$
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 $Y$:
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.
(adjunction between topological spaces and diffeological spaces)
There is a pair of adjoint functors
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:
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):
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:
Finally, these adjunctions are a sequence of Quillen equivalences with respect to the:
classical model structure on topological spaces | model structure on D-topological spaces | model structure on diffeological spaces |
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.
Essentially these adjunctions and their properties are observed in Shimakawa, Yoshida & Haraguchi 2010, Prop. 3.1, Prop. 3.2, Lem. 3.3, see also Christensen, Sinnamon & Wu 2014, Sec. 3.2. 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).
We spell out the existence of the idempotent adjunction (2):
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
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
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
to find that the counit of the adjunction
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, it is sufficient to check (by this Prop.) that the comonad
is an idempotent comonad, hence that
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.
(diffeological singular simplicial set)
Consider the simplicial diffeological space
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:
where the right adjoint is the diffeological singular simplicial set functor $Sing_{diff}$.
(e.g. Christensen-Wu 13, Def. 4.3)
(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:
(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 naturally$\,$weak homotopy equivalent to the homotopy type of $X$ presented by the ordinary singular simplicial set:
(diffeological internal hom on D-topological spaces has correct cohesive shape)
For $X, A \,\in\, DTopSp \hookrightarrow TopSp$ a pair of D-topological spaces, their internal hom formed in diffeological spaces has diffeological singular simplicial complex $Sing_{diff}(-)$ (3) weakly homotopy equivalent to the ordinary singular simplicial set $Sing(-)$ of the ordinary mapping space $Maps_{Top}$ with its compact-open topology:
By SYH 10 we have the following morphism:
where $\mathbf{smap}$ is some topologization of the set of maps (defined on their p. 6 ) of which all we need to know is that:
(shown on the right of (4)) its image under $Cdfflg$ is isomorphic to the internal hom $Maps_{dfflg}$ in diffeological spaces, according to their Prop. 4.7 (p. 7),
(shown on the left of (4)) it is weak homotopy equivalent, via some map $\phi$ according to their Prop. 5.4 (p. 9) to the compact-open topology.
Hence the claim follows by using 2-out-of-3 in the naturality square of the natural weak homotopy equivalence from Prop.
applied to (4).
The obvious functor from the category SmoothManifolds of smooth manifolds to the category DiffeologicalSpaces of diffeological spaces is a full and faithful functor
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.
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
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 $\geq 2$.
(manifolds with boundaries and corners form full subcategory of diffeological spaces)
The evident functor
from the category of smooth manifolds with boundaries and corners to that of diffeological spaces is fully faithful, hence is a full subcategory-embedding.
(Igresias-Zemmour 13, 4.16, Gürer & Iglesias-Zemmour 19)
Also Banach manifolds embed fully faithfully into the category of diffeological spaces. In (Hain) this is discussed in terms of Chen smooth spaces.
We discuss a natural embedding of Fréchet manifolds into the category of diffeological spaces.
Define a functor
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 $U \to X$.
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.
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 $C^\infty(X,Y)_{Fr}$:
This appears as (Waldorf 09, lemma A.1.7).
$\,$
We discuss how diffeological spaces are equivalently the concrete objects in the cohesive topos of smooth sets (see there).
(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
The concrete sheaves for the local topos $Sh(CartSp)$ are by definition those objects $X$ for which the $(\Gamma \dashv CoDisc)$-unit
is a monomorphism. Monomorphisms of sheaves are tested objectwise, so that means equivalently that for every $U \in CartSp$ we have that
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.
The category of diffeological spaces is a quasitopos.
This follows from the discussion at Locality.
This has some immediate general abstract consequences
The category of diffeological spaces is
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
into the (∞,1)-topos Smooth∞Grpd of “higher smooth sets” –smooth ∞-groupoids – as precisely the 0-truncated objects.
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.
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):
Chen considered (apart from iterated integrals) effectively presheaves on a site of convex subsets of Cartesian spaces. In
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
Patrick Iglesias-Zemmour, Diffeologies, talk at New Spaces for Mathematics and Physics, IHP Paris 2015 (video recording)
Patrick Iglesias-Zemmour, An introduction to diffeology, lecture at Modern Mathematics Methods in Physics: Diffeology, Categories and Toposes and Non-commutative Geometry Summer School, 2018, to appear in New Spaces for Mathematics and Physics (pdf)
The article
proved that diffeological spaces are concrete sheaves forming a quasi-topos.
A discussion of the relations of variants of the definition is in
More pointers and monthly seminars at:
Application to classifying spaces and universal connections:
The full subcategory-inclusion of Banach manifolds into the category of diffeological spaces is due to
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
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
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
Kazuhisa Shimakawa, K. Yoshida, Tadayuki Haraguchi, Homology and cohomology via enriched bifunctors, Kyushu Journal of Mathematics, 2018 Volume 72 Issue 2 Pages 239-252 (arXiv:1010.3336, doi:10.2206/kyushujm.72.239)
J. Daniel Christensen, Gord Sinnamon, Enxin Wu, The $D$-topology for diffeological spaces, Pacific Journal of Mathematics 272 (1), 87-110, 2014 (arXiv:1302.2935, doi:10.2140/pjm.2014.272.87)
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
On orbifolds regarded as naive local quotient spaces (instead of homotopy quotients/Lie groupoids/differentiable stacks) but as such formed in diffeological spaces:
Patrick Iglesias-Zemmour, Yael Karshon, Moshe Zadka, Orbifolds as diffeologies, Transactions of the American Mathematical Society 362 (2010), 2811-2831 (arXiv:math/0501093)
Jordan Watts, The Differential Structure of an Orbifold, Rocky Mountain Journal of Mathematics, Vol. 47, No. 1 (2017), pp. 289-327 (arXiv:1503.01740)
On this approach seen in the broader context of higher differential geometry: