sheaf (original) (raw)

1 Presheaves

Let X be a topological space and let 𝒜 be a category. A_presheaf_ on X with values in 𝒜 is a contravariant functor F from the category ℬ whose objects are open sets in X and whose morphisms are inclusion mappings of open sets of X, to the category 𝒜.

As this definition may be less than helpful to many readers, we offer the following equivalent (but longer) definition. A presheaf Fon X consists of the following data:

1. 1. An object F⁢(U) in 𝒜, for each open set U⊂X
2. 1. A morphism resV,U:F⁢(V)→F⁢(U) for each pair of open sets U⊂V in X (called the restriction morphism), such that:
   2. (a)
      For every open set U⊂X, the morphism resU,U is the identity morphism.
   3. (b)
      For any open sets U⊂V⊂W in X, the diagram