Is there a sheaf theoretical characterization of a differentiable manifold?

Definition: ''A smooth manifold is a locally ringed space $(M;C^{\infty})$ which satisfies the conditions:

1. Each $x \in M$ admits a neighborhood $U$, such that $(U,C^{\infty})$ is isomorphic to $(\mathbb{R}^n,C^{\infty})$ as a locally ringed space.
2. The global sections of $C^{\infty}(M)$ separate points.
3. The structure sheaf $C^{\infty}$ is fine as a sheaf of modules over itself.
4. $C^{\infty}(M)$ has at most countably many indecomposable idempotents.''

Explanations:

1) is evident.

2) means that for $x \neq y \in M$, there exists $f \in C^{\infty}(M)$ with $f(x)=0$, $f(y) \neq 0$. This ensures the Hausdorff condition once we know that elements of $C^{\infty}(M)$ give rise to continuous maps $M \to \mathbb{R}$. This is as follows: $f \in C^{\infty}(M)$ given, $x \in M$. Pick a chart $h:U \to \mathbb{R}^n$; under this chart, $f|_U$ corresponds to a smooth function on $\mathbb{R}^n$, whose value at $h(x)$ does not depend on the choice of the chart. Call this value $f(x)$. Checking the continuity of $x \mapsto f(x)$ can be done in charts.

3.) By this I mean that for each open cover $(U_i)$, there is a partition of unity $\lambda_i$ with the usual properties and that $\lambda_i$ is a map of $C^{\infty} (M)$-modules. A standard argument shows that $\lambda_i$ is given by multiplication with a smooth function. Therefore, the underlying space $M$ is paracompact.

4.) An idempotent $p\neq 0 $ in a commutative ring $A$ is called indecomposable if

$$ p=q +r; r^2 =r; q^2 =q , q \neq 0 \Rightarrow p = q $$

holds. Indecomposable idempotents in $C^{\infty}(M)$ correspond to connected components. Therefore condition 4 means that $M$ has only countably many connected components.

These conditions together imply that $M$ is Hausdorff and second countable, because a locally euclidean, connected and paracompact Hausdorff space is second countable, see Gauld, "Topological properties of manifolds", Theorem 7 (see http://www.jstor.org/stable/2319220 ). Paracompactness alone does not guarantee second countability, see $\mathbb{R}$ with the discrete topology.

However, I think that sheaf theory and locally ringed spaces are the wrong software for differential geometry and differential topology.

For another perspective, think about definitions of "complex manifold" or "real analytic manifold". Normally you use atlases for this, imposing the Hausdorff and paracompactness conditions separately. You can restate the atlas part of the definition in sheaf language, but you can't hope to get the rest of it that way, can you? I mean, you can't get the paracompactness from properties like acyclicity or flasqueness, because you don't have those properties.

Of course, you can always fall back on "a complex manifold is a real manifold plus the following extra structure", but that seems to go against the spirit of the question.

On a related note, of course the "basic" property of transitivity of the automorphism group fails in the complex-analytic case. (Query: what about the real-analytic case?)

EDIT From the comments it is clear that I haven't expressed myself clearly. Let me try again.

The usual definition of "smooth manifold" says (1) the space is equipped with an atlas in which all the charts are pairwise smoothly compatible, or rather an equivalence class of such atlases, or if you prefer a maximal such atlas, (2) the space is paracompact, (3) the space is Hausdorff.

Daniel likes the option of replacing (1) by the logically equivalent:

(1)' the space is equipped with a sheaf of algebras such that every point has a nbhd such that ...

He is wondering if he can simultaneously replace (2) by some extra requirement on the sheaf (something implying acyclicity) to get either an equivalent definition or a slightly weaker but still useful notion, and he offers as evidence the idea that the usefulness of (2) can be expressed in sheaf language. I am not offering an opinion about that.

I am just pointing out the following: If you did succeed in reworking the definition of smooth manifold in this esthetically pleasing way, then you might want to do the same for the definition of complex analytic manifold. But you'd get stuck on the fact that the structure sheaf is not acyclic.

So maybe you would fall back on describing a complex manifold as a smooth manifold plus extra structure.

But wait, what about topological manifolds? Don't we like them to be paracompact, too? For similar reasons as in the real case. Does it seem esthetically right to emphasize the ring of continuous functions when dealing with topological manifolds?

By the way, why do we want complex manifolds to be paracompact? Probably to allow the occasional use of $C^\infty$ methods.

Oh, and what about piecewise linear manifolds? Here (and also, by the way, for $C^k$ manifolds for $k\ge 1$ finite) there is no sheaf of rings involved. But there is the option of using charts, where compatibility of charts depends on the notion of PL homeomorphism between open subsets of $\mathbb R^n$). If you don't also assume paracompactness and Hausdorff then you won't have triangulations. Is there some sheafy way to discuss this?

OK, I've rambled, and probably replaced one unclear thing by several other unclear things.

Dear Daniel, you should take a look into the nice book "Global Calculus" of S. Ramanan.

From the Review of the book by John Miller: "This is a decidedly individual course on analysis and geometry on manifolds. The book begins by introducing smooth manifolds as spaces equipped with sheaves of differentiable functions. Although this approach is unusual in a textbook, it works rather smoothly. ...."