for each open set of , we immediately run into a problem: is not necessarily open. The best we could do is to approximate it by open sets, and even then we will get a presheaf and not a sheaf. Consequently, we define to be the sheaf associated to the presheaf:
(Here is an open subset of and the colimit runs over all open subsets of containing .)
For example, if is just the inclusion of a point of , then is just the stalk of at this point.
While is more complicated to define than , the stalks are easier to compute: given a point , one has .
is an exact functor, as can be seen by the above calculation of the stalks.
is (in general) only right exact. If is exact, f is called flat.
is the left adjoint of the direct image functor. This implies that there are natural unit and counit morphisms and . These morphisms yield a natural adjunction correspondence:
However, the morphisms and are almost never isomorphisms.
For example, if denotes the inclusion of a closed subset, the stalk of at a point is canonically isomorphic to if is in and otherwise. A similar adjunction holds for the case of sheaves of modules, replacing by .