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.

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