structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
A diffeological spaces 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 Euclidean spaces. Therefore in the sequel we shall often restrict our attention to CartSp.
One may define a “very general smooth space” to be any sheaf of CartSp and identify the sheaf topos $Sh(CartSp)$ as the category of very general smooth spaces.
A diffeological space precisely a concrete sheaf on the concrete site CartSp. 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.
The obvious functor from the category Diff of smooth manifolds to the category 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 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 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 $\geq 2$.
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), 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 aspects of the full sheaf topos $Sh(CartSp)$ on the site CartSp – the topos of smooth spaces – and of how diffeological spaces are embedded into this. In summary, we have that $Sh(CartSp)$ is a cohesive topos and that $DiffeologicalSpace \hookrightarrow Sh(CartSp)$ is the canonical sub-quasitopos of concrete sheaves inside it.
The full sheaf topos $Sh(CartSp)$ on CartSp is a locally connected topos in that the terminal global section geometric morphism to Set is an essential geometric morphism:
The extra left adjoint $\Pi_0 : Sh(CartSp) \to Set$ sends diffeological spaces to the set of path-connected components of their underlying topological spaces.
The sheaf topos $Sh(CartSp)$ on CartSp is a locally connected topos.
The following argument works for every site $C$ which is such that constant presheaves on $C$ are already sheaves.
Notice that this is the case for $C = CartSp$ because every Cartesian space is connected: for $S \in Set$ a compatible family of elements of $Const S$ on a cover $\{U_i \to \mathbb{R}^n\}$ of some $\mathbb{R}^n$ is an element of $S$ on each patch, such that their restriction maps to intersections of patches coincide. But the restriction maps are all identities, so this says that all these elements coincide. Therefore the set of compatible families is just the set $S$ itself, hence the presheaf $Const S$ is a sheaf.
So with $L : PSh(C) \to Sh(C)$ the sheafification functor we have that $L Const S \simeq Const S$.
Whenever this is the case the left adjoint to the constant presheaf functor, which always exists for presheaves and is given by the colimit functor, is also left adjoint on the level of sheaves, because for each $X \in Sh(C)$ and $S \in Set$ we have natural bijections
Write $\Pi_0 := \lim_\to : Sh(CartSp) \to Set$ for the left adjoint to $LConst : Set \stackrel{Const}{\to} PSh(C) \stackrel{L}{\to} Sh(C)$.
For $X \in Sh(C)$ a diffeological space, $\Pi_0(X)$ is the set of path-connected components of the topological space underlying $X$.
By the co-Yoneda lemma we may write
and since $\Pi_0$ commutes with colimits we have
But also by the co-Yoneda lemma we have that the colimit over any representable is the singleton set, hence our expression
is the colimit over the category of plots of $X$ of the functor that is constant on the point. This colimit is the coproduct of points over the connected components of the diagram category.
The connected components of the category of plots $y/X$ are the path-connected (or “plot-connected”) components of the underlying topological space of $X$.
The sheaf topos $Sh(CartSp)$ on CartSp is actually a connected topos.
Since $CartSp$ is a connected category it is immediate that $Const \colon Set \to PSh(CartSp)$ is a full and faithful functor. By the above this equals $L Const$, which is hence also full and faithful.
By the discussion at connected topos we could equivalently convince ourselves that $\Pi_0$ preserves the terminal object. The terminal object of $Sh(CartSp)$ is $y(\mathbb{R}^0)$, hence representable. By the above, $\Pi_0$ sends all representable objects to the singleton set, which is the terminal object of $Set$.
The sheaf topos $Sh(CartSp)$ is also a local topos
The site CartSp is a local site: it has a terminal object and the only covering sieve of this object is the trivial one. This implies the claim, by the discussion at local site.
Concretely, the extra right adjoint $CoDisc$ takes a set $S$ to the presheaf given by the assigmnent
that takes a Cartesian space $U$ to the set of functions from its underlying set of points to $S$. This is clearly a sheaf (a function of sets from $U$ to $S$ is clearly fixed by all its restrictions to a collections of subsets of $U$ whose unition is $U$.)
Geometrically, the object $CoDisc S \in Sh(CartSp)$ is the diffeological space codiscrete (indiscrte) smooth structure.
Every local topos comes with its notion of concrete sheaves that form a sub-quasitopos. For the local topos $Sh(CartSp)$ these are precisely the 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.
The sheaf topos $Sh(CartSp)$ is even a cohesive topos in which the axiom pieces have points holds.
The site CartSp is a cohesive site (see there for detail). This implies the statement.
This implies that $Sh(CartSp)$ is a locally connected topos, connected topos, local topos. It means in addition that it is also a strongly connected topos.
This means that there is a homotopy category or concordance category of smooth spaces, with the same objects as $Sh(CartSp)$, but with hom-sets given by
where $[X,Y]_{Sh(CartSp)}$ is the internal hom in the cartesian closed category $Sh(CartSp)$.
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 spaces. This is a general context for differential geometry. From there one can pass further to higher differential geometry: the topos of smooth spaces in turn embeds
into the (∞,1)-topos Smooth∞Grpd of “higher smooth spaces” –smooth ∞-groupoids – as precisely the 0-truncated objects.
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
he 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 smooth spaces? stabilized in
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
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 discusson of the relations of these and other variants of the definition is in
The article
amplifies the point that diffeological spaces are concrete sheaves.
A textbook about differential geometry formulated in terms of diffeological spaces is
The term “diffeological space” originates in the work of this author.
The thesis
contains some useful material that hasn’t yet made it into the book.
The embedding 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 embedding of Fréchet manifolds into diffeological spaces is discussed in
and reviewed in section 3 of
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
Some homotopy theory modeled on diffeological spaces instead of on topological spaces is discussed in
Discussion in the context of applications to continuum mechanics is in
Expository survey includes