Our goal in this section is to introduce the machinery necessary to be able to give local definitions of morphisms in a useful way.
Let AB denote the category of abelian groups.
TOP(X)op → AB.
We can give a more concrete description of the data that must be specified in order to define a presheaf F on a topological space X. First, for each open subset U ⊂ X, you must give an abelian group F(U). Next, whenever you have an inclusion U ⊂ V of open subsets of X, you must give a group homomorphism
ρVU : F(V) → F(U).We refer to the map ρVU as a restriction map. The restriction maps must fit together to define a functor. In particular, the composite of two restriction maps must be a restriction map, and the restriction associated to the identity map on an open set must be the identity homomorphism. As a final piece of notation, we usually refer to an element s ∈ F(U) as a section of F over U. We'll try to explain this terminology later.
The presheaf G of locally constant functions modulo constant functions illustrates another potential problem. Take any open set U ⊂ X and any section s ∈ G(U). By choosing U appropriately, we can choose a non-constant section s. However, since this section is locally constant, we can cover U by open subsets Vi such that all the restrictions ρUVi(s) are consant, and hence are given by the zero element of G(Vi). That is, all (nontrivial) sections of G are locally indistinguishable (from each other or from the trivial section).
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