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

`
ρ ^{V}_{U} : F(V) → F(U).
`

- Fix a topological space
`Y`. For any other topological space`X`, we can define a presheaf of sets by taking`C(U)`, for all open`U ⊂ X`, to be the set of continuous functions from`U`to`Y`. Here the restriction maps are the usual restrictions of a function from one set to a smaller set contained inside it. - When
`Y=R`or`Y=C`with the usual topologies, then the presheaf defined in the previous example takes its values in the category of commutative rings. (Here additiona and multiplication of funtions is performed pointwise.) - Fix a differentiable manifold
`Y`. For any other differentiable manifold`X`, we can define a presheaf by taking`D(U)`to be the set of differentiable functions from`U`to`Y`. Since`R`can be viewed as a differentiable manifold, we can also get a presheaf of rings . - Give
`R`the usual topology, and let`X`be any topological space. Define a presheaf by taking`CON(U)`to be the (ring of) constant functions on`U`with values in`R`. - Similarly, one can take
`LC(U)`to be the (ring of) locally constant real-valued functions on`U`. - Notice that for each open subset
`U`of a topological space`X`, the set of constant functions on`U`is a (normal) subgroup of the (abelian) group of all locally constant functions. So, as our final example for the moment, we can define`G(U)`to be the abelian group of locally constant real-valued functions on`U`modulo the constant functions. The restriction maps are induced from the usual restrictions of locally constant functions, and are well-defined and functorial because of the universal properties of quotient group homomorphisms.

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
`V _{i}` such that all the restrictions

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Kevin R. CoombesDepartment of Biomedical Informatics

The Ohio State University

Columbus, Ohio 43210