Decomposing Sheaves into a direct sum I have two questions about sheaves inspired by representation theory: 


*

*Is there a theorem that tells me when a sheaf of vector spaces can be decomposed into the direct sum of two "smaller" sheaves? 

*Let's call a sheaf of vector spaces "irreducible" if it cannot be decomposed into a direct sum of smaller sheaves. Is there a theory that tells us how sheaves are composed into these smaller "irreducible" components? For example, is such a decomposition unique, are there criteria for telling what components a sheaf has, etc. 
 A: What you call irreducibility is usually called indecomposability; irreducibility is the stronger condition that an object (in an abelian category, say) has no nontrivial subobjects. 
The analogy to the representation theory of, say, finite groups fails in an important respect: categories of sheaves of vector spaces are almost never semisimple. That is, objects almost never decompose into direct sums of irreducible objects. Instead, one must solve extension problems, the solution to which can be expressed in terms of the Ext functor. People usually first encounter this sort of issue in module theory. 
For example, the category of locally constant sheaves of vector spaces on $S^1$ is equivalent to the category of representations of the fundamental group $\mathbb{Z}$. This is not a semisimple category: there is a representation
$$\mathbb{Z} \ni n \mapsto \left[ \begin{array}{cc} 1 & n \\\ 0 & 1 \end{array} \right]$$
which is indecomposable but not irreducible. It is a nontrivial extension of the trivial representation by the trivial representation. 
The idea in the representation theory of finite groups to try to understand all representations in terms of irreducible representations just doesn't work in general. Sheaves of vector spaces is a huge category, and trying to understand every object in it is hopeless. This isn't a category you try to understand all the objects in, it's a convenient starting point from which you can do other things.
