Definition: A cartesian space is a finite dimensional smooth manifold diffeomorphic to \(\mathbb{R}^n\) for some \(n \geq 0\). Given a set \(X\), a parametrization is a set function \(p : U \to X\) where \(U\) is a cartesian space. If \(M\) is a finite dimensional smooth manifold, and \(\mathcal{U} = \{ U_i \subseteq M \}\) is a collection of subsets of \(M\), we say that \(\mathcal{U}\) is a good open cover if each \(U_i\) is a cartesian space, every finite intersection \(U_{i_0} \cap \dots \cap U_{i_n}\) is either empty or a cartesian space, and \(\bigcup_i U_i = M\).
Definition^{1}: Given a set \(X\), a diffeology \(\mathcal{D}_X\) on \(X\) consists of a set of parametrizations \(p: U \to X\) satisfying the following three conditions:
Given two diffeological spaces \((X, \mathcal{D}_X)\) and \((Y, \mathcal{D}_Y)\), a smooth map between them consists of a set function \(f : X \to Y\) such that if \(p : U \to X\) is a plot of \(X\), namely \(p \in \mathcal{D}_X\), then the composite map \(U \xrightarrow{p} X \xrightarrow{f} Y\) is a plot of \(Y\). Let \(\mathsf{Diff}\) denote the category of diffeological spaces with smooth maps.
Notice how simple these definitions are compared to the corresponding definitions of a smooth manifold! Really think about it, the definition of a manifold is really complicated. Its a kind of topological space (Hausdorff, second-countable), and it has all of these charts, which are homeomorphisms from cartesian spaces, and those charts have to be compatible in some way that’s really annoying to write symbolically, and then we consider the biggest collection (maximal atlas) of such charts compatible with the ones you’ve got, and then you call that a smooth structure. Okay, maybe if you are a seasoned differential geometer it isn’t so bad, but I remember spending months understanding the definition of a smooth manifold the first time I learned about it. There are so many nooks and crannies to get stuck on it. The definition of a diffeological space however, by comparison is clean and tidy. Same for the definition of smooth map.
Okay, that’s nice and all, but its just a definition. I can define whatever the hell I want, but its only interesting if it connects with things I care about. Well, suppose that \(M\) is a finite dimensional smooth manifold in classical differential geometry. Then we can consider the set \(\mathcal{D}_M\) of those parametrizations \(p : U \to M\) that are smooth as maps of smooth manifolds, in the sense of classical differential geometry. It turns out that \(\mathcal{D}_M\) satisfies the axioms of a diffeology. In other words, every finite dimensional smooth manifold is canonically a diffeological space. We call this the manifold diffeology on \(M\).
Even more powerfully, a function \(f: M \to N\) between finite dimensional smooth manifolds is smooth in the classical sense if and only if it is a smooth map in the sense of diffeological spaces between \(M\) and \(N\) equipped with their manifold diffeologies! This is proven in the diffeology textbook in Article 4.3, and it implies that assigning the manifold diffeology to a manifold defines a fully faithful functor from the category of finite dimensional smooth manifolds to the category of diffeological spaces.
\[\mathcal{D}_{(-)} : \mathsf{Man} \hookrightarrow \mathsf{Diff}.\]Intuitively, this means that we lose nothing by considering manifolds as diffeological spaces via their manifold diffeology. We could also characterize finite dimensional smooth manifolds as those diffeological spaces which have local diffeomorphisms to a fixed \(\mathbb{R}^n\) and satisfy some additional conditions.
The astute reader will note that this is not the definition of a diffeogical space as given in the “Diffeology textbook”. However, in my paper, I prove that the category of diffeological spaces as given in the Diffeology textbook and the category of diffeological spaces as given in the definition above are equivalent. The above definition is far more convenient to work with for my purposes. ↩
(1988): PIZ writes a preprint showing that there is an obstruction between Cech cohomology and deRham cohomology for diffeological spaces given by the diffeological principal \(\mathbb{R}\)-bundles with connection. It is never published.
(Late 1980s-2000s): PIZ develops a large amount of the theory of diffeological spaces in his thesis and several papers throughout this period. There are a couple of papers from other people, but overall not a large amount of activity.
(2009): Baez and Hoffnung prove that diffeological spaces are precisely concrete sheaves on the site of open subsets of euclidean spaces in their paper “Convenient Categories of Smooth Spaces.” This provides an important bridge connecting diffeology to sheaf theory.
(2013): PIZ publishes the first (and currently only) textbook on diffeology, appropriately named “Diffeology.” This helps attract a new generation of mathematicians to the subject.
(2010s): There is a huge influx of interest in diffeology. Papers are written about diffeology with connections to mathematical physics, Lie groupoid theory, foliations, homotopy theory, and more. Some particularly important papers are written in this period.
(2021): Out of the pandemic sprouts the Monthly Global Diffeology Seminar and a website hub diffeology.net for everything diffeology.
(2020s): With the start of the new decade comes a newly energized and diverse community of diffeologists. Some notable papers are:
However, Iglesias-Zemmour writes in his Introduction to Diffeology (where I obtained much of this history) that diffeological spaces were “built on the model” of Chen spaces. Later work by Stacey and Baez-Hoffnung showed how Chen spaces are directly comparable to diffeological spaces. ↩
Another model structure was proven to exist on diffeological spaces in “A model structure on diffeological spaces, I”, though it took many years to develop and Kihara’s help in amending previous proofs. ↩
I can’t help but mention that at the time of writing, there are four different versions of Cech cohomology for diffeological spaces in the literature. Indeed, there is PIZ’s, defined in his paper above. There is Krepski, Watts and Wolbert’s diffeological Cech cohomology given in “Sheaves, principal bundles, and Čech cohomology for diffeological spaces”. There is my version of Cech cohomology for diffeological spaces defined in “Diffeological Principal Bundles and Principal Infinity Bundles” which is called \(\infty\)-stack cohomology. Finally there is Ahmadi’s version of diffeological Cech cohomology given in “Diffeological Cech Cohomology.” It is currently an open question as to whether any of these cohomologies agree for all diffeological spaces. ↩
Okay, so that’s actually two questions, and I will handle the second one (“why should I care?”) first. There are actually many different ways to approach this question, and rather than list a bunch of them, I’ll pick one for this post.
So imagine you are a differential geometer. Your research revolves around finite dimensional smooth manifolds, well behaved mathematical objects with a long history (something like 200 years) of study. You know that whatever tools you need, there will be something in the enormous toolbox of classical differential geometry to suit your needs. Hell, partitions of unity probably get you most of the way.
One day you come across some kind of smooth space, maybe a group of diffeomorphisms, or possibly an orbifold of some kind, which is not a finite dimensional smooth manifold! You want to prove some theorems about this misbehaving space, but when you flip through your textbooks on classical differential geometry, there’s nothing! All of its theorems are about finite dimensional smooth manifolds. You glance over at the dusty tome on your shelf about Banach manifolds, with its hundreds of pages of hardcore functional analysis, and cower in fear.
Okay, maybe that’s a little dramatic. But its something that happens all the time in differential geometry. The truth is, modern differential geometry wishes to study all kinds of crazy spaces that are not finite dimensional smooth manifolds. Yet, the way these crazy spaces are usually dealt with is in an ad hoc manner, and sometimes require lots of functional analysis. When a new kind of crazy space comes along, and we wish to study it, none of the tools we’ve developed thus far will help us. We need to redefine all the concepts of differential geometry we’d like to use on this new kind of crazy space.
Diffeology is a new, modern framework for differential geometry that says “enough is enough, we need a big box of smooth spaces that includes manifolds and all kinds of crazy spaces us modern differential geometers care about, and we need a uniform way of reasoning about these spaces, and extending our old theorems to them.” This might sound like wishful thinking. What mathematics will guide us in this new world, far out from our cozy land of Euclidean spaces and tori? That tool is Category Theory.
You’ve probably heard about category theory already if you are looking at this blog, so I won’t bother trying to explain what it is, as any attempts by myself to do so would probably be embarassing anyway. Why category theory matters here is because there is a category staring us in the face. That category is \(\mathsf{Man}\), the category whose objects are finite dimensional smooth manifolds and whose morphisms are smooth functions between them.
This category sucks!
Indeed, the category \(\mathsf{Man}\) has finite products, transverse pullbacks and a few colimits (but noticeably not any coproducts between manifolds of different dimension!). This is not enough for the working category theorist or really for the working geometer. We want to glue things together willy-nilly, like for instance, this is my favorite example, take two copies of the real line, and glue them together at the origin. We’ll call this space the Axes.
This is not a manifold! If it were, then it would be the colimit of the following diagram in \(\mathsf{Man}\).
\[0 \rightrightarrows \mathbb{R} \sqcup \mathbb{R},\]where each map is the inclusion of the origin into either the left or right copy of \(\mathbb{R}\). But c’mon now, this is a simple space! We should be able to have the usual concepts of differential geometry (tangent spaces, bundles, deRham cohomology, etc.) apply to this thing!
Diffeology is an attempt to do just this. It provides us with a category, \(\mathsf{Diff}\), whose objects are diffeological spaces, which is complete, cocomplete, and locally presentable. If you don’t know what that means exactly, don’t worry about it. The point is that it is a wayyy better category than \(\mathsf{Man}\), it lets us glue stuff together however we like. Our friend the Axes is an object in this category, as are all finite dimensional smooth manifolds, and thus the diagram above can be considered in \(\mathsf{Diff}\), and its colimit is precisely Axes!
This was a super rough, motivational post. I’ll be back next time to delve into some more motivation and some of the history of diffeological spaces.
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