Advantages of diffeological spaces over general sheaves

Question

I have been playing with/thinking about diffeological spaces a bit recently, and I would like understand something rather crucial before going further. First a little background:

Diffeological spaces are a Cartesian-closed, complete, and cocomplete category containing all infinite dimensional manifolds, and in fact even form a quasi-topos.

Diffeological spaces, concisely, are nothing more than concrete sheaves on the site of Cartesian manifolds (manifolds of the form $\mathbb{R}^n$):

http://ncatlab.org/nlab/show/concrete+sheaf

However, the category of ALL sheaves on Cartesian manifolds, categorically is even nicer, since it is a genuine topos.

$\textbf{My question is:}$ What can you do with diffeological spaces that you cannot do with general sheaves? Or, more generally, what are the advantages of diffeological spaces over general sheaves?

All of the generalizations of differential geometry concepts to diffeological spaces I have seen so far, actually carry over to genuine topos of sheaves (though sometimes with a little more work).

I'm aware that you gain the ability to work with a set with extra structure and talk about its points etc, but, what does this gain you? It seems that you can always use Grothendieck's functor of points approach instead.

Is it that limits and colimits are more like their counterparts for manifolds?

Answer 1

I want to give two simple observations about diffeological spaces that might provide a partial answer to your question.

First, we have the following inclusions of full subcategories $$\mathit{Mfd} \subset \mathit{Diff} \subset \mathit{Sh} \subset \mathit{PSh}$$ where $\mathit{Mfd}$ is the category of smooth finite dimensional manifolds, $\mathit{Diff}$ are diffeological spaces (i.e. concrete sheaves on cartesian spaces), $\mathit{Sh}$ are sheaves on cartesian spaces and $\mathit{PSh}$ are presheaves on cartesian spaces. The last two inclusions are reflexive.

Lets us first have a closer look at the inclusion $\mathit{Sh} \subset \mathit{PSh}$. Following the same vein of argument as above, there is a priori no reason to work with $\mathit{Sh}$ instead of $\mathit{PSh}$ since both categories are equally nice (topoi) and the definition of a presheaf is clearly simpler than that of a sheaf. But there are some colimits in $\mathit{Mfd}$ that we really like, namely the coequalizer diagram correspoding to an open cover $(U_\alpha)$ of a manifold $M$. Under the inclusion of $\mathit{Mfd}$ into $\mathit{PSh}$ this is not a coequalizer anymore, in other words: If we glue open sets in $\mathit{PSh}$ together we do not get the same thing that we get when glueing together as manifolds. This defect is exactly cured by the sheaf property. That means restricting to the smaller subcategory $Sh \subset \mathit{PSh}$ the colimits change such that gluing of open sets behaves as nice as in manifolds. The punchline is that the restriction to $\mathit{Sh}$ provides the category with the "right" coequalizers of open sets.

Now lets turn towards the inclusion $\mathit{Diff} \subset \mathit{Sh}$. The situation is exactly the same as before. Limits in $\mathit{Diff}$ are computed as Limits in $\mathit{Sh}$ (and hence also $\mathit{PSh}$) but colimits are different in general (one has to apply the concretization functor). This is what happens categorically. Now it turns out that there are colimits in manifolds that become colimits in diffeological spaces but not colimits in sheaves. Here an example would be very nice. Unfortunately I have not been able the remember the example I had for this behaviour. Even so, from abstract reasoning it is clear that the colimits in the two categories have to differ.

Hence one could argue that diffeological spaces have the right "geometric" colimits and sheaves do not. The price is of course that we exclude some interesing "spaces" like the sheaf of diffential forms and loose the property that the category is a topos.

Second, if we want to "make" geometry over diffeological spaces it turns out that there are two possible definition of principal bundles:

• a bundle over a diffeological space $M$ is a morphism to the stack of bundles over finite dimensional manifolds. This means that we have a family of bundles over each plot together with coherent isomorphisms. Note that this type of bundle is determined by its pullback to finite dimensional spaces. This is equivalent to have a diffeological space $P \to M$ together with a free transitive on fibers action such that the quotient map $P \to M$ is a surjective subduction (i.e. becomes a submersion on each plot). To get those type of bundles we have to equip diffeological spaces with the Grothendieck Topology of subductions.

• a bundle over a diffeological space $M$ is a space $P \to M$ with a free, transitive on fibers, action such that it is locally trivial, where locally refers to the underlying topological space of $M$. This is the type of bundle which people consider in the world of $\infty$-dimensional manifolds. To get this we have to take the grothendieck topology of morphisms that are surjective and admits local (in the topology) sections. Hence therefore we really need the underlying topological space.

I do not prefer one of the two possible Grothendieck Topologies, but the second one is closer to what people have done in the $\infty$-dimensional setting. And one can show that the universal bundle $EG \to BG$ for a compact Lie-group is of this type (of course one has to find diffeological models of $BG$ and $EG$).

The first topology has an obvious analogue on the category $\mathit{Sh}$ of all sheaves but the second crucially uses the underlying topological space of a diffeological space.

Answer 2

I'm going to pre-empt my esteemed colleague here and say that to some people, the purpose of diffeological spaces is as a stepping stone between manifolds and the category of sheaves on manifolds (or on cartesian spaces, it's the same thing). So, to these people, you've stumbled on the main point: we really ought to be working with sheaves all along.

The problem is that there are some ornery people who really like manifolds as they are, but sometimes have to work with things that are almost but not quite completely unlike manifolds. For these people, the further away from true manifolds they get, the more uncomfortable they feel. One of the biggest steps for such people is losing the underlying set. So diffeological spaces are a category in which those people can have most of the benefits of sheaves without having to discard their comfort blanket of something that still resembles manifolds in some way.

So diffeological spaces are a convenient (yes, I use the word deliberately!) half-way house whereby those who have Seen The Light can still talk to those still quivering under their comfort blankets.

To name names, people in the first category include Urs Schreiber and John Baez (indeed, I think that John makes that point somewhere on the n-Cafe). People in the second category include me!

Indeed, I would say that diffeological spaces are closer to the One True Category of Smooth Spaces than sheaves on cartesian spaces. Frolicher spaces seem to irretrievably have underlying sets - I and a few others have wondered from time to time if there is a way to remove that property but it seems tied up with what they are.

---

(Added in edit): I don't know why, but I didn't spot first time around the last line:

Is it that limits and colimits are more like their counterparts for manifolds?

which is odd, because that's the subject of a little theorem I've proved which can be found on the nLab:

http://ncatlab.org/nlab/show/topological+notions+of+Fr%C3%B6licher+spaces#hausdorff

Essentially, if you want to preserve those limits and colimits that already exist in the category of manifolds, then you need to work in the category of Hausdorff Frölicher spaces. When you enlarge that category (say to Frölicher spaces or to Diffeological spaces, or to sheaves) then you add in stuff "in the gaps" and create new limits or colimits that disagree with the ones that you had before (in these cases, it's almost always colimits, but if you take the "maps out" view, it will be limits). So that question isn't really a sensible one to ask of Diffeological spaces as you've already lost some colimits. I suppose you can try to do a bit of damage limitation ...

Answer 3

I think the argument for diffeological spaces is just that it eliminates certain kinds of pathological constructions that are possible with general sheaves, without costing you anything. General sheaves allow constructions that geometrically are sick and wrong. For example, you can define a sheaf that geometrically consists of two lines, so that every single point of the two lines are identified, but the two lines are still distinct. The "concrete" condition prevents that kind of pathology.

Answer 4

I'm not used to read this website, so I post my remark very late, after having randomly googled this page. I think that one of the reason why diffeological spaces are better than general sheaves is the possibility to consider infinitesimal Fermat extensions:

1. Giordano P. " Fermat reals: nilpotent infinitesimals and infinite dimensional spaces" . Book in preparation, see http://arxiv.org/abs/0907.1872, July 2009.

2. Giordano P. "The ring of Fermat reals", Advances in Mathematics 225 (2010), pp. 2050-2075. https://doi.org/10.1016/j.aim.2010.04.010

3. Giordano P. "Infinitesimals without logic", Russian Journal of Mathematical Physics, 17(2), pp.159-191, 2010 https://doi.org/10.1134/S1061920810020032

4. Giordano P., Kunzinger M. "Topological and algebraic structures on the ring of Fermat reals". Submitted to Israel Journal of Mathematics on April 2011. See http://arxiv.org/abs/1104.1492

5. Giordano P. "Fermat-Reyes method in the ring of Fermat reals". To appear in Advances in Mathematics, 2011. https://doi.org/10.1016/j.aim.2011.06.008

6. Giordano P. "Infinite dimensional spaces and cartesian closedness". Journal of Mathematical Physics, Analysis, Geometry, 2011. http://mi.mathnet.ru/eng/jmag/v7/i3/p225

This is a new theory, and I post this answer also because I think it is not known. The basis of the theory is a surprisingly simple extension of the real field containing nilpotent infinitesimals. We start from the class of little-oh polynomials, i.e. functions $x:\mathbb{R}_{\ge 0}\rightarrow \mathbb{R}$ that can be written as $x(t)=r+\sum_{i=1}^{k}\alpha_{i}\cdot t^{a_{i}}+o(t)$ as $t\to 0^+$, where all the coefficients and powers are reals. Then, we introduce the equivalence relation between little-oh polynomials $x\sim y$ iff $x(t)=y(t)+o(t) \text{ as }t \to 0^{+}$. The ring of Fermat reals $ {}^\bullet\mathbb{R}$ is the corresponding quotient set. The theory of Fermat reals has been developed trying always to obtain a good dialectic between formal mathematics and intuitive interpretation. Even if there are several theories of infinitesimals, only a couple of them always have this intuitive interpretation, and this contradicts the idea that (rigorous) infinitesimals are a strong support to guess some mathematical truths. Of course, Fermat reals take strong inspiration from smooth infinitesimal analysis, even if, at the end, it is a radically different theory. In fact, in the corresponding ring of scalars, which extends the classical reals, we have nilpotent infinitesimals of every order, infinitesimal Taylor's formulas (analogous of the Kock-Lawvere axiom), powers, roots of (nilpotent!) infinitesimals, logarithms, a total order relation, and the ring is also geometrically representable, so that we can finally state that infinitesimals are no longer ghosts of departed quantities.

It is also very interesting to note that its mathematical definition uses only elementary analysis and Landau's little-oh notation, without requiring a background in mathematical logic. In particular, the model is so simple that can be studied directly in classical logic without any need to switch to intuitionistic logic. On the other hand, this extension of the real field is generalizable both to finite and infinite dimensional manifolds (more generally to diffeological spaces). The extension ${}^\bullet(-): \mathcal{C}^\infty \rightarrow {}^\bullet\mathcal{C}^\infty$ (here $\mathcal{C}^\infty$ is the category of diffeological spaces and $ {}^\bullet\mathcal{C}^\infty$ is the category of Fermat spaces, which are defined similarly to diffeological spaces) is functorial and has very good preservation properties: a full transfer theorem for intuitionistically valid sentences is indeed provable (the "true" logic of smooth spaces is always intuitionistic!). Several applications to differential geometry has been already developed: e.g. tangent vectors to any diffeological space $X \in \mathcal{C}^\infty$ can be defined, similarly to SDG, as smooth functions of the form $t:D\rightarrow {}^\bullet X$, where $ D :=\{h\in {}^\bullet\mathbb{R}|h^2=0\}$ is the ideal of first order infinitesimals and where ${}^\bullet X\in {}^\bullet\mathcal{C}^\infty$ is the Fermat space obtained extending $X$ with new infinitesimally closed points. At present, we are developing several notions of differential geometry in this framework and are trying to extend the theory so as to include infinities and generalized functions (distributions).

Answer 5

If you are an algebraist: general sheaves. If you are a geometer: diffeology.

---

I will try to explain what I mean. I may however still modify the following.

The philosophy of the theory of sheaves is summarised by this excerpt taken from François de Marçay in his lecture "Faisceaux":

Comment, à partir d’une collection d’objets qui sont définis seulement sur des petits ouverts, peut-on construire un objet qui est défini sur l’espace tout entier ?

That approach from local to global epitomises the theory of sheaves: knowing an object by knowing its details. The object of study appears as a puzzle of small parts. Indeed a diffeological space is associated with the sheaf of its plots. But is that all a diffeological space is? Is that able to satisfy the geometer?

According to Felix Klein, a geometry is defined in his "Erlangen program" as a group of transformations (the principal group) on some set (manifoldness):

Given a manifoldness and a group of transformations of the same; to investigate the configurations belonging to the manifoldness with regard to such properties as are not altered by the transformations of the group.

Thus, the geometer starts with a manifoldness, however defined or chosen. He then chooses a group of transformations of this set, thus defining a geometry. The examples are numerous: from Euclidean geometry to projective geometry via spheric or hyperbolic geometry and so on. So:

The geometer knows an object not by its details, but in its unity, as a whole. His vision of the object is sustained by the admissible global transformations that define its geometry.

How, then, does "differential geometry" appear in this picture? Is it a geometry in Klein's sense? The short answer:

• Every diffeological space defines a (differential) geometry by the action of its group of diffeomorphisms. What is forbidden in classical differential geometry, because a triangle is not a manifold, is admitted in diffeology, because a triangle is a diffeological space whose diffeomorphisms can exchange only vertices with each other, like edges and preserve the interior.

Hence, diffeology enhances the concept of geometry by making each diffeological space a manifoldness à la Klein with principal group its group of diffeomorphisms. This could be a formal definition of Differential Geometry.

This is why the sheaf algebraic approach to diffeology is a contrario of its geometric approach, even if a diffeological space is a sheaf of plots.

BTW the global geometrical point of view of diffeology does not prevent from extending the group of diffeomorphisms to the germs of local diffeomorphisms to highlight the local structure and the singularities of the object. This is the point of view we used to revisit the theory of stratifications(*) in the paper :

P.I-Z with Serap Gürer. Orbifolds as stratified diffeologies. Differential Geometry and its Applications, Volume 86 (Feb. 2023).

I would add that looking at diffeology only from the point of view of sheaf theory constitutes the passive approach, whereas seeing diffeological spaces from the point of view of geometry, as the loci of the action of their group of diffeomorphisms, constitutes the active approach.

(*) What is defined above, the partition in orbits by the group of diffeomorphisms, is actually what we call the Klein stratification of a diffeological space.

Answer 6

The thing you can do with a diffeological space and that you can't do with a sheaf is to define it by a sentence that starts like this: "It's a set equipped with ..."

You might object that I'm just restating the definition. But I really don't think that there's much more to be said.