A comprehensive functor of points approach for manifolds

Question

This seems unrealistic, because the topology on a manifold doesn't have anything to do with the properties its structure sheaf, but I figured I might as well ask. This wouldn't be the first time I was pleasantly surprised about something like this. If not, is there any sort of way to attack differential geometry with abstract nonsense?

Even though schemes have singularities, "it's better to work with a nice category of bad objects than a bad category of nice objects". Manifolds seem to be perfect illustration of this fact.

Edit: Apparently my question wasn't clear enough. The actual question here is if we can define manifolds entirely as "functors of points" like we can with schemes (sheaves on the affine zariski site). There is no fully categorical and algebraic description of the category of smooth manifolds. When I say a "comprehensive functor of points approach", I mean a fully categorical description of the category of smooth manifolds.

Answer 1

Here are two things that I think are relevant to the question.

First, I want to support Andrew's suggestion #5: synthetic differential geometry. This definitely constitutes a "yes" to your question

is there any sort of way to attack differential geometry with abstract nonsense?

--- assuming the usual interpretation of "abstract nonsense". It's also a "yes" to your question

Can we describe it as some subcategory of some nice grothendieck topos?

--- assuming that "it" is the category of manifolds and smooth maps. Indeed, you can make it a full subcategory.

Anders Kock has two nice books on synthetic differential geometry. There's also "A Primer of Infinitesimal Analysis" by John Bell, written for a much less sophisticated audience. And there's a brief chapter about it in Colin McLarty's book "Elementary Categories, Elementary Toposes", section 23.3 of which contains an outline of how to embed the category of manifolds into a Grothendieck topos.

Second, it's almost a categorical triviality that there is a full embedding of Mfd into the category Set${}^{U^{op}}$, where $U$ is the category of open subsets of Euclidean space and smooth embeddings between them.

The point is this: $U$ can be regarded as a subcategory of Mfd, and then every object of Mfd is a colimit of objects of $U$. This says, in casual language, that $U$ is a dense subcategory of Mfd. But by a standard result about density, this is equivalent to the statement that the canonical functor Mfd$\to$Set${}^{U^{op}}$ is full and faithful. So, Mfd is equivalent to a full subcategory of Set${}^{U^{op}}$.

There's a more relaxed explanation of that in section 10.2 of my book Higher Operads, Higher Categories, though I'm sure the observation isn't original to me.

Answer 2

I'm having a hard time understanding what the actual question is here, but the little I do get suggests that you start reading as follows:

1. generalised smooth space on the nLab
2. If you are particularly inclined towards sheaves, then read about Chen spaces and diffeological spaces (and be sure to take in Convenient Categories of Smooth Spaces while you are at it).
3. If you are a little more ambivalent about sheaves and just want to embed manifolds in a "nice" category, be sure to take in my personal favourite.
4. If you really want a comparison of the lot, consider wading through the extremely murky Comparative Smootheology.
5. Going further afield, there's synthetic differential geometry.

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Added later:

(Edit inserted here so that the last line of the original post remains the last line of the edited post.)

Two (minor) thoughts after having read Tom's answer:

1. The "site" Tom uses is bigger than necessary. I'm no expert on the categorical side of things, but subject to checking a few details then you can work with just the monoid $C^\infty(\mathbb{R},\mathbb{R})$ viewed as a one-object category. The point is that although it seems as though you need open sets of all dimensions, actually manifolds are determined completely by their smooth curves. So if you want a topos in which manifolds sits, sheaves on $C^\infty(\mathbb{R},\mathbb{R})$ will do. Of course, if you want your category to have other things as well then other sites may be more appropriate. See the extensive discussions on this on the nLab and nCafe.

2. Whenever you embed manifolds in a topos then you are going to break something. There is no way to embed manifolds in a topos and have the subcategory of manifolds behave exactly as the category of manifolds does. In brief, if you want to have a locally cartesian closed category then your embedding cannot preserve colimits. That is, there will be some diagrams in manifolds that have colimits in manifolds but have different colimits in your topos. For more details, see the details in the nLab page on Froelicher spaces.

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Finally, I disagree with the sentiment behind:

"it's better to work with a nice category of bad objects than a bad category of nice objects". Manifolds seem to be perfect illustration of this fact.

I much prefer:

Manifolds are fantastic spaces. It’s a pity that there aren’t more of them.

Answer 3

I certainly wouldn't call it an "attack with abstract nonsense", but the "functor of points" language can also be used in the context of differentiable manifolds. A very good place to start would be Weil's article from 1953: "Théorie des points proches sur les variétés différentiables"

While I imagine it doesn't specifically contain the phrase "functor of points", the idea is precisely the same. Weil's article also sheds some light (for algebraically oriented people like myself) on the "prolongations" studied much earlier by Cartan.

Answer 4

As far as abstract nonsense goes there is Joyce's approach to manifolds, yielding $C^\infty$-schemes and $C^\infty$-stacks.

http://arxiv.org/abs/1001.0023

I don't suppose it has much to do with the functor of points, but it enlarges the category of manifolds in order to gain fibre products, and thus some intersection theory on manifolds.