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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 $\mathcal{C}$?

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"sheaf of sections of a fibre bundle $B$ over $X$ whose fibres $B_x$ are objects of $C$" Note that this makes $B$ a '$C$-object' in the category of spaces over $X$, when '$C$-object' is suitably interpreted. From your examples, you are considering algebras for finite-product theories, which make perfect sense in the slice category $Top/X$. Then as Steven points out in his answer, global sections form a $C$-object because of (some abstract nonsense about preservation of finite limits). – David Roberts Feb 5 '13 at 2:46
up vote 5 down vote accepted

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 ${\cal C}$.

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