What are the "correct" modules over locally ringed spaces? $$\begin{array}{ccccc} \text{schemes} & \longrightarrow & \text{locally ringed spaces} & \longrightarrow & \text{ringed spaces} \\ | && | && | \\ \text{quasi-coherent sheaves} & \longrightarrow & \text{?} & \longrightarrow & \text{module sheaves}\end{array}$$
What do you suggest for $?$, fitting into this picture?
This is a soft question, but perhaps I can make it more precise: I would like to know if there is any reasonable substack $\mathsf{LMod}$ of $\mathsf{Mod} : \mathsf{LRS} \to \mathsf{SymMonCat}^{\mathrm{op}}$ such that $\mathsf{LMod}(X)$ preserves "much" of the structure of $X$ (in particular we cannot just use the forgetful functor $\mathsf{LRS} \to \mathsf{RS}$). I would like to see something different from quasi-coherent or coherent sheaves, which really used the local rings. For example, when $x \in \overline{\{y\}}$, one can require that the canonical homomorphism $M_x \otimes_{\mathcal{O}_{X,x}} \mathcal{O}_{X,y} \to M_y$ is an isomorphism modulo $(\mathfrak{m}_y)^n$, but this is a little bit weak.
 A: For every open $U \subset X$ and every $f \in \mathcal{O}_X(U)$ we can consider the open set $U_f$ of points $x$ of $U$ where $f$ does not map to the maximal ideal of $\mathcal{O}_{X, x} = \mathcal{O}_{U, x}$.
In the schemes case, if $U$ is quasi-compact and quasi-separated, then a quasi-coherent sheaf $\mathcal{F}$ has the property that $\mathcal{F}(U_f) = \mathcal{F}(U)_f$, see for example Lemma Tag 01P7. If $U$ is general, then I think we can still conclude that


*

*if $s \in \mathcal{F}(U)$ restricts to zero over $U_f$, then for every $x \in U$ there exists an open neighbourhood $x \in W \subset U$ and integer $n \geq 0$ such that $f^ns|_W = 0$

*if $t \in \mathcal{F}(U_f)$, then for every $x \in U$ there exists an open neighbourhood $x \in W \subset U$, a section $s \in \mathcal{F}(W)$, and an integer $n \geq 0$ such that $s|_{W_f} = f^nt|_{W_f}$.
Namely, we can just take $W$ to be an affine open neighbourhood of $x$ and apply the previous result. (There are variants of 1 and 2 using open coverings so that you can generalize to locally ringed topoi if you like.)
So a possibility would be to consider sheaves of $\mathcal{O}_X$-modules on a locally ringed space $(X, \mathcal{O}_X)$ which satisfy conditions 1 and 2.
On a scheme you'd recover the usual quasi-coherent modules (didn't check all details but it seems obvious -- please correct me if I am wrong). For a general locally ringed space I think you get a different notion than the usual quasi-coherent sheaves (for example for the real line endowed with continuous functions, it appears that the structure sheaf doesn't satisfy condition 2). In fact, I am not at all sure this is something interesting for any type of locally ringed space (or topos) different from a scheme or algebraic space or algebraic stack.
Actually, I think there are many natural properties one can impose on $\mathcal{O}_X$-modules which, when $X$ is a scheme, give the class of quasi-coherent modules. One is the condition above. Another one is that $X$ should have a covering $X = \bigcup U_i$ such that $\mathcal{F}|_{U_i}$ is associated to a $\mathcal{O}_X(U_i)$-module (as in Lemma Tag 01BH). Finally, there is the definition of quasi-coherent modules. But presumably there are many others.
