Godement resolution

The Godement resolution of a sheaf is a construction in homological algebra that allows one to view global, cohomological information about the sheaf in terms of local information coming from its stalks. It is useful for computing sheaf cohomology. It was discovered by Roger Godement.
Godement construction.
Given a topological space "X" (more generally, a topos X with enough points), and a sheaf "F" on X, the Godement construction for "F" gives a sheaf formula_1 constructed as follows. For each point formula_2, let formula_3 denote the stalk of "F" at "x". Given an open set formula_4, define
An open subset formula_6 clearly induces a restriction map formula_7, so formula_1 is a presheaf. One checks the sheaf axiom easily. One also proves easily that formula_1 is flabby, meaning each restriction map is surjective. The map formula_10 can be turned into a functor because a map between two sheaves induces maps between their stalks. Finally, there is a canonical map of sheaves formula_11 that sends each section to the 'product' of its germs. This canonical map is a natural transformation between the identity functor and formula_10.
Another way to view formula_10 is as follows. Let formula_14 be the set "X" with the discrete topology. Let formula_15 be the continuous map induced by the identity. It induces adjoint direct and inverse image functors formula_16 and formula_17. Then formula_18, and the unit of this adjunction is the natural transformation described above.
Because of this adjunction, there is an associated monad on the category of sheaves on "X". Using this monad there is a way to turn a sheaf "F" into a coaugmented cosimplicial sheaf. This coaugmented cosimplicial sheaf gives rise to an augmented cochain complex that is defined to be the Godement resolution of "F".
In more down-to-earth terms, let formula_19, and let formula_20 denote the canonical map. For each formula_21, let formula_22 denote formula_23, and let formula_24 denote the canonical map. The resulting resolution is a flabby resolution of "F", and its cohomology is the sheaf cohomology of "F".