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- why are sheafs defined using abelian groups?

Let M be a manifold. The space of cotangent spaces to M is called the cotangent bundle T*M. a function on M can be lifted into the cotangent bundle. On the manifold T*M we can define a 1-form θ which describes the natural lifts and so on. A vector field on M is a section of the tangent bundle. The space of sections of T*M is denoted Γ(T*M) and is an example of a sheaf. The set of k-forms at the point p is denoted ∧

^{k}T*M_{p}. the sheaf of p-forms on M is denoted Ω^{p}, which is equal to Γ(∧^{p}T*M). Restrictions of the sections Γ to an open set U has the natural structure of a vector space. If we want to talk about a sheaf of groups, or a sheaf of rings or a sheaf of some other type of objects, we have to assign abelian groups to the restrictions Γ(U) rather than vector spaces. how is this idea motivated or understood?
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