sheaf
1 Presheaves
Let be a topological space and let be a category
. A
presheaf
on with values in is a contravariant functor
from the category whose objects are open sets in and whose morphisms
are inclusion mappings of open sets of , to
the category .
As this definition may be less than helpful to many readers, we offer
the following equivalent (but longer) definition. A presheaf
on consists of the following data:
-
1.
An object in , for each open set
-
2.
A morphism for each pair of open sets in (called the restriction
morphism), such that:
-
(a)
For every open set , the morphism is the identity morphism.
-
(b)
For any open sets in , the diagram
-
(a)