I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a nutshell, is there a natural condition to impose on a structure sheaf $\mathcal{C}^k\_M$ of a topological space $M$ that can stand-in for the requirement that $M$ be second-countable Hausdorff?


### Background


Most textbooks introduce differentiable manifolds via [atlases and charts](http://en.wikipedia.org/wiki/Differentiable_manifold#Definition). This has the advantage of being concrete, and the disadvantage of involving an arbitrary choice of atlas, which obscures the basic property that a differential manifold "looks the same at all points" (for $M$ connected, without boundary: diffeomorphism group acts transitively). And isn't introducing local coordinates "an act of violence"?


I saw a much nicer definition of differentiable manifolds on Wikipedia, which I don't know a good textbook reference for. This definition proceeds via [sheaves of local rings](http://en.wikipedia.org/wiki/Differentiable_manifold#Sheaves_of_local_rings). The Wikipedia definition stated:



> 
> A differentiable manifold (of class $C\_k$) consists of a pair $(M, \mathcal{O}\_M)$ where $M$ is a topological space, and $\mathcal{O}\_M$ is a sheaf of local $\mathbb{R}$-algebras defined on $M$, such that the locally ringed space $(M,\mathcal{O}\_M)$ is locally isomorphic to $(\mathbb{R}^n, \mathcal{O})$.   
> [$\mathcal{O}(U)=C^k(U,\mathbb{R})$ is the structure sheaf on $\mathbb{R}^n$.]
> 


Beautiful, really! Entirely coordinate free. But isn't there a General Topology condition missing?


I [confirmed on math.SE](https://math.stackexchange.com/questions/107319/sanity-check-about-wikipedia-definition-of-differentiable-manifold-as-a-locally) (to make sure that I wasn't hallucinating) that this definition is indeed missing the condition that $M$ be second-countable Hausdorff. That indeed turned out to be the case, so I edited the Wikipedia definition to require $M$ to be second-countable Hausdorff.  



### Why am I still not happy?


The deep reason that we require a differentiable manifold to be paracompact, as per [Georges Elencwajg's extremely informative answer](https://math.stackexchange.com/questions/98105/why-are-smooth-manifolds-defined-to-be-paracompact/98124#98124), is that paracompactness makes sheaves of $C\_M^k$-modules (maybe $k=\infty$) [acyclic](http://en.wikipedia.org/wiki/Injective_sheaf#Acyclic_sheaves). This is a purely sheaf-theoretic property (a condition on the structure sheaf of $M$ rather than on $M$ itself), which quickly implies good things like that every subbundle of a vector bundle on $M$ be a direct summand. Is this in fact enough?


If it were enough to require that $\mathcal{O}\_M$ be [acyclic](http://en.wikipedia.org/wiki/Injective_sheaf#Acyclic_sheaves), or maybe [fine](http://en.wikipedia.org/wiki/Injective_sheaf#Fine_sheaves), then the nicest, most flexible (and, in a strange sense, most enlightening) definition of differentiable manifold, would be:



> 
> **Definition**: A differentiable manifold (of class $C\_k$) consists of a pair $(M, \mathcal{O}\_M)$ where $M$ is a topological space, and $\mathcal{O}\_M$ is an **acyclic** sheaf of local $\mathbb{R}$-algebras defined on $M$, such that the locally ringed space $(M,\mathcal{O}\_M)$ is locally isomorphic to $(\mathbb{R}^n, \mathcal{O})$. 
> 


Maybe the word **acyclic** should be **fine**. Maybe soft and acyclic. Maybe a bit more, but still something that can be stated in terms of the structure sheaf.



> 
> **Question**: Can I put a natural sheaf-theoretic condition on $\mathcal{O}\_M$ (acyclic? fine?) which ensures that $M$ (a topological space) must be a second-countable Hausdorff space? If not, would such a condition at least ensure that $M$ be a generalized differentiable manifold in some kind of useful sense? 
> 




---


**Update**: This question really bothers me, so I've started a bounty. I'd like to narrow it down a little in order to make it easier to answer:


Does **acyclicity** (or a slightly stronger condition) on $\mathcal{O}\_M$ of a topological (Hausdorff?) space imply **paracompactness** (or a slightly weaker but still useful condition)? 


Hausdorff bothers me as well, of course; but a sheafy characterization of paracompactness somehow seems like it has the potential to be lovely and really enlightening. 



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.


