Sheaves and gratings

Question

A grating is a notion in algebraic topology from the 1940 introduced by Alexander. Cartan extended it as follows.

A grating (carapace in french) is defined by a topological space $X$, a module (or a differential ring) $A$ and for each $x \in X$, a surjective morphism $\varphi_x$ from $A$ to some quotient $A_x$ such that

• for all $\alpha \in A$ and $x \in X$ such that $\varphi_x(\alpha) = 0$, we have $\varphi_y(\alpha) = 0$ for all $y$ close to $x$.
• If $\varphi_x(\alpha) = 0$ for all $x\in X$, then $\alpha = 0$.

In his seminar Carapaces from the 1950, Cartan showed that gratings correspond to (pre-)sheaves on $X$ and this is the reason why, I guess, the notion of grating is not used anymore.

The sheaf $\mathcal F$ associated to a grating $(A, \varphi_x)$ is built by taking the $A_x$ as the stalks. This, I understand. The grating associated to a sheaf is built from the module of sections. Cartan does not give any other details in the seminar. I don't understand what $A$ should be, hence my question:

What is the module $A$ associated to the sheaf $\mathcal F$?

The module $A_x$ must be the stalk of the sheaf. So $A$ can't be the module of global sections, since it does not always surject on the stalks. My guess would be something like the inverse limit of all the ${\mathcal F}(U)$, where $U$ is an arbitrary open subset of $X$. But I have never seen this limit used before in that context.

Answer 1

I think you may have misunderstood what Cartan was doing. In modern language he defined functors $\Gamma$ from sheaves to gratings and $\mathcal{F}$ from gratings to sheaves but he did not claim that these are mutually inverse equivalences of categories (even allowing for the apparent anachronism). He did write down a natural transformation from the identity functor on gratings to the functor $\Gamma \circ F$ explained at the end of section 4 of the seminar you link to.

He called objects $A$ where the map $A\to \Gamma\circ F(A)$ is an isomorphism complete gratings and argues that $\Gamma$ of any sheaf is a complete grating. I guess we should understand this as meaning that $\Gamma$ is right adjoint to $F$ but I haven't worked through the details to check this is true.

To directly answer your specific question, I think the module $A$ underlying $\Gamma(\mathcal{F})$ is in fact the space of global sections $\mathcal{F}(X)$ and your argument that it cannot be so fails since we don't have $A_x=\mathcal{F}_x$ in general. Rather, I think if you work through Cartan's argument, $A_x$ can be defined to be the image of $\mathcal{F}(X)$ in $\mathcal{F}_x$.