Do the fiber bundles over an Abelian category form an Abelian category?

Question

Assume I have an Abelian category $C$. Is the category of fiber bundles, where the fibers are objects of $C$ Abelian?

Answer 1

It's not at all clear what this should mean in general. To define fiber bundles with fiber $F$ requires that you specify a topological group $G$ acting on $F$, and if we're just given that $F$ is an object in some abelian category then it just has some automorphism group with no obvious topology in play other than the discrete topology. Picking the discrete topology does not, for example, reproduce real vector bundles if we pick the abelian category to be $\text{Vect}(\mathbb{R})$; the construction of vector bundles is sensitive to the topology of $GL_n(\mathbb{R})$ but the bare abelian category structure of $\text{Vect}(\mathbb{R})$ is not.

In any case, the category of vector bundles already fails to be an abelian category, e.g. it does not have kernels; see, for example, this MO question.

In a positive direction, for a topological space $X$ and an abelian category $C$ we can define the category $\text{Sh}(X, C)$ of sheaves of objects of $C$ on $X$, and this is an abelian category; see, for example, this math.SE answer (Edit: or maybe not, the linked answer only cites a claim in Weibel without a proof so who knows).