# How are infinite-dimensional manifolds most commonly treated?

I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I am still looking for an answer to (mainly the first part of) my question.

At a summer school I recently attended, infinite-dimensional manifolds popped up. I have never worked with them before (although I'm very familiar with finite-dimensional manifolds). The lecturer at the school did not give any details about the technical realization of infinite-dimensional manifolds, mentioning that there were issues (such as picking a topology) that he would leave out for the sake of clarity, since the relevant results were true independent of the exact technical details. An internet search reveals that Banach manifolds are one way of treating infinite-dimensional manifolds, but there are others.

Are Banach manifolds the most common way of defining infinite-dimensional manifolds, or are there other notions commonly used? Is there a more or less universal consensus about when to use which treatment? What are the most important (dis)advantages of each?

Supposing I want to learn the basics of infinite-dimensional manifolds, are there any well-written introductory texts you would recommend? (on StackExchange, The Convenient Setting of Global Analysis by A. Kriegl and P. Michor was recommended)

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Banach manifolds have found many uses such as, gauge theory (Donaldson theory, Seiberg-Witten theory, Floer theory), symplectic topology (Gromov-Witten theory), to name few I am more familiar with.

One great advantage of Banach manifolds over Frechet manifolds is the implicit function theorem which in the Banach context takes a simpler form, and thus Banach manifolds are easier to recognize.

One disadvantage of Banach manifolds over Frechet manifolds is the fact that natural notions of real analyticity are harder to implement on Banach spaces.

One place to learn about Banach manifolds is Lang's Differential and Riemannian Manifolds

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For many applications, Banach manifolds are not suitable: Groups of Sobolev or $C^k$ diffeomorphisms are only topological groups. If a Banach Lie group acts effectively on a compact (thus finite dimensional) smooth manifold, then it is finite dimensional itself.

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One emerging trend seems to be that the category of sheaves of sets on the site of smooth manifolds (also known as the category of generalized manifolds) is the right category of what one might call smooth sets. (Here we no longer restrict our attention to spaces that look the same at every point, and in fact we have spaces that have no points at all.) In particular, it includes all sorts of infinite-dimensional manifolds, such as Banach and Fréchet manifolds as full subcategories. It also contains many other categories of smooth objects, such as diffeological spaces, as full subcategories. Also, this category can be generalized nicely to higher smooth homotopy types, e.g., smooth stacks = smooth homotopy 1-types, which constantly pop up even if you're studying ordinary differential geometry.

As an example, one can cite the following result. Consider the smooth stack B^∇G of smooth principal G-bundles with connection and the smooth set Ω of differential forms. The set of maps B^∇G→Ω turns out to be canonically isomorphic to the algebra of Ad-invariant polynomials on the Lie algebra of G. Thus one recovers Chern-Weil theory in a very natural way. See the recent paper by Freed and Hopkins for details: http://arxiv.org/abs/1301.5959. I don't think this result can be obtained in any other model of smooth objects, because other models do not allow for spaces like B^∇G and Ω.

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Also important are sheaves of sets on slightly larger sites which include smooth manifolds, such as the category of f.p. $C^\infty$ rings with appropriate topologies, as considered in synthetic differential geometry. This allows one to consider objects of nilpotent infinitesimals as representing jet-space functors; a comprehensive treatment is given in the book by Moerdijk and Reyes. See also the pioneering work of Urs Schreiber and collaborators throughout the nLab on the importance of cohesive sites for higher geometry, as in differential cohomology in $(\infty, 1)$-toposes. –  Todd Trimble Feb 9 '13 at 16:45
Dmitry, while I of course agree that (oo-)sheaves on the site of smooth manifolds is the right context for doing (higher) differential geometry, it is not good to say, as you do, that these are categories of "manifold-like objects" -- unless the term "manifold" loses all its intended meaning. The objects in these sheaf toposes have the only common property that they are characterized by smooth probes, but only very few of them are characterized as locally equivalent to a given model space V. Those that are are the V-manifolds (the "differential geometric V-schemes"). –  Urs Schreiber Feb 9 '13 at 17:25
–  Urs Schreiber Feb 9 '13 at 18:13
@Urs: I guess I should add that the term ‘smooth space’ makes me somewhat uncomfortable, because ‘space’ often means ‘∞-groupoid’ and ‘smooth space’ can be misunderstood as ‘smooth ∞-groupoid’. ‘Smooth set’ avoids this ambiguity. –  Dmitri Pavlov Feb 9 '13 at 18:59
I was waiting for you to say that. In an unpublished version of my previous comment I had a side remark saying "and if we say 'diffeological space' we may also say 'smooth space' ". But either way "smooth set" also redirects to the entry. Personally, I think it is a bad move to think that "space" means "oo-groupoid" by default, because 'topological spaces' represent oo-groupoids. The thing is that its really the homotopy type, not the space, that is the oo-groupoid. All around us 'space' refers to geometric spaces. I think it is a bad move to identify spaces with their homotopy types. –  Urs Schreiber Feb 9 '13 at 19:23

Different notions of manifolds may be useful in different approaches. Maybe more important than finding "universal consensus" on which one is suppoosed to be used where is to have a language to treat the various notions uniformly such as to be able pass between them in a useful way.

One such more general category is that of "diffeological spaces". In

http://ncatlab.org/nlab/show/diffeological+space

is discussed how for instance Frechet manifolds faithfully embed into these.

Diffeological spaces form a "quasi-topos". Following Grothendieck's lead, it is better to go one step further to an actual topos for differential geometry. The topos generalization of diffeological spaces is that of "smooth spaces" (smooth sets/smooth 0-types)

http://ncatlab.org/nlab/show/smooth+spaces

which is the sheaf topos over the category of smooth manifolds (or equivalently just over that of Euclidean spaces with smooth maps between them). Variants of this with a bit more information about the differential aspect of differential geometry include for instance the "Cahier topos"

http://ncatlab.org/nlab/show/Cahiers+topos

See there for pointers for how "convenient vector spaces" and hence the infinite-dimensional manifolds modeled on them ("convenient manifolds") are faithfully embedded into that topos.

In these toposes for instance all mapping spaces exist and can be usefully treated, while they agree with the infinite-dimensional manifold structures on mapping spaces whenever those actually exist. Similar statements hold for all other universal constructions.

Thereby topos theory transforms the question of finding "universal consensus" on which definition is best to a more relevant technical question: which concrete definition happens to constitute a presentation of a universaly existing construction in the topos. Presentations are useful, but are man-made. They may apply or not, may be useful here or there. But the smooth spaces which they present exist universally, robustly and meaniningfully irrespective of such choices.

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It seems there are tons of ways to consider infinite-dimensional manifolds. I wonder how many ways are useful (i.e. come naturally in concrete meaningful applications). Could you say a word on this? –  alvarezpaiva Feb 9 '13 at 19:36
Frechet manifolds are probably the most popular, at least in application to mapping spaces (loop groups, etc). But in loads of cases, more useful than the Frechet manifold structure is actually the diffeological space structure. That's really useful, and in many cases this is what is actually being used. One should ask: "In my application, do I absolutely need to know that my space is locally modeled on a vector space?" Most of differential geometry goes through without this extra information. One big exception where one maybe strictly needs local linear structure is integration. –  Urs Schreiber Feb 9 '13 at 20:59