If $\pi:E\to M$ is a vector bundle then the set of sections $\Gamma(E)$ is naturally a vector space under fibrewise addition and scalar multiplication on the bundle $E$. This holds similarily for bundles of algebras or modules, thought I'm not sure if it holds for bundles of groups (certainly not for principal bundles). The main example I have in mind is the algebra $\mathcal{C}^\infty (M)$ of smooth real-valued functions on a smooth manifold (just considering the ring structure of the reals, not the field structure).


Now, given a sheaf $\mathcal{O}$ with values in some category $\mathcal{C}$, when is $\mathcal{O}\_X$ an object in $\mathcal{C}$? 


As the examples above show (basically abelian groups with extra structures) this is true when $\mathcal{O}\_X$ is the sheaf of sections of a fibre bundle whose fibres are objects of $\mathcal{C}$. Another relevant question would be if this is indeed the case for locally trivial fibrations only. Concretely:


Is it true that the sheaf $\mathcal{O}\_X$ is an object of $\mathcal{C}$ only when $\mathcal{O}\_X$ is the sheaf of sections of a fibre bundle $B$ over $X$ whose fibres $B\_x$ are objects of $\mathcal{C}$?



When is a sheaf of groups (algebras, rings, modules) a group (algebra, ring, module)?
A sheaf ${\cal O}$ is *a fortiori* a presheaf. This means it's a functor that takes the category of open sets in $X$ to your fixed category ${\cal C}$. So, by definition, $\Gamma(X,{\cal O})={\cal O}(X)$ is an object of ${\cal C}$.


