TITLE: Is the sheaf associated to a differential structure of a specific type?
QUESTION [5 upvotes]: On a set $X$, let us define a set $\mathcal{D}$ of functions from $X$ to $\mathbb{R}$. Consider first the initial topology $\tau_\mathcal{D}$ on $X$ with respect to $\mathcal{D}$, i.e. the coarsest topology that makes all functions in $\mathcal{D}$ continuous (with the usual topology on $\mathbb{R}$). Given a $X$ and a set of functions $\mathcal{D}$, for $W \subseteq X $ we say $f:W \to \mathbb{R}$ verifies the property $P_W$ if
$$\forall x \in W, \ \exists (V,g) \in \tau_\mathcal{D} \times \mathcal{D} \text{ such that } V \ni x, \text{ and } f|_V = g|_V. $$
Now, a differential structure on a set $X$ is a set $\mathcal{D}$ of functions from $X$ to $\mathbb{R}$ that verifies the following axioms

$\mathcal{D}$ is a $\mathcal{C}^\infty$-ring: if $u_1,\ldots,u_N \in \mathcal{D}$ and $g \in \mathcal{C}^\infty(\mathbb{R}^N,\mathbb{R})$, then $g \circ (u_1,\ldots,u_N) \in \mathcal{D}$
It is locally determined: If $f:X \to \mathbb{R}$ verifies $P_X$ then $f \in \mathcal{D}$.

We can consider the functor that associates to an open set $U \in \tau_\mathcal{D}$ the set $$\mathbf{C}_\mathcal{D}(U) := \{ f:U \to \mathbb{R} \mid f \text{ verifies the property } P_U \}.$$ The two axioms ensure this is a sheaf of $\mathcal{C}^\infty$-ring.
The set of smooth functions of a paracompact manifold defines a sheaf of $\mathcal{C}^\infty$-ring which is fine, and it is for sure an important feature. Differential structure are meant to mimick these functions, when one does not have a manifold structure. On the other hand, we have by definition $\mathcal{D} = \Gamma(X,\mathbf{C}_\mathcal{D})$, so it seems the associated sheaf is somehow "determined" by its global sections. 
I understand there is a whole typology of sheaves related to the local-VS-global properties: injective, flabby, soft, acyclic, fine. Is $\mathbf{C}_\mathcal{D}$ of one of these types? Can I actually define a differential structure as a sheaf of $\mathcal{C}^\infty$-ring of some well-known type ? Note that it is ok if you need some topological assumption, like $(X, \tau_\mathcal{D})$ is paracompact.

REPLY [2 votes]: I have found  some answer, in a context that is large enough to satisfy me. Any other comment in any other context is still welcome.
In the book, Differential geometry of singular space and reduction of symmetry 2013, Sniatycki proves in section 2.2 the existence of a partition of unity for any locally compact, second countable Hausdorff differential spaces. So in this context, we have indeed a fine sheaf.