Let $R$ be a commutative ring. We can define an $R$-module (alternatively to the usual definition) to be an abelian group $M$ together with a ring morphism $\varphi_M: R \to \text{End}(M)$ (sending $x$ to the multiplication-with-$x$ map).
Now let $(X, {\cal{O}}_X)$ be a ringed space. I want to define the notion of ${\cal O}_X$-module. I started by the following "natural" approach:
An ${\cal O}_X$-module ${\cal F}$ is a sheaf of abelian groups on $X$ with ring morphisms $\varphi_{{\cal F}(U)} : {\cal O}_X(U) \to \text{End}({\cal F}(U))$ such that [we need to define module compatibility now] for $U \subseteq V$ the following diagram $$ \require{AMScd} \begin{CD} {\cal O}_X(V) @>\varphi_{{\cal F}(V)}>> \text{End}({\cal F}(V)) \\ @V\text{res}VV @VV\Psi V \\ {\cal O}_X(U) @>\varphi_{{\cal F}(U)}>> \text{End}({\cal F}(U)) \end{CD} $$ commutes, where $\Psi$ is induced by the restriction on ${\cal F}$ ... ???!? [realizes that this doesn't work].
As far as I can see there is no induced map $\Psi$, which is why this definition doesn't work.
I find this weird since my definition of module is pretty categorical and so I thought a categorical definition of ${\cal O}_X$-module is easily done. So my questions are:
- Did I overlook something or can my definition be fixed somehow?
- Rather philosophical: Why doesn't this work?