noncomputable def TopCat.Sheaf.sectionsFunctor (C : Type u) [CategoryTheory.Category.{v, u} C] {X : TopCat} (U : TopologicalSpace.Opens ↑X) : CategoryTheory.Functor (Sheaf C X) C

The sections functor Γ(U, –) : Sheaf C X → C sending a sheaf to its sections on a fixed open U, constructed by forgetting to a presheaf and evaluating at U.

theorem noetherianSpace_sections_preservesFilteredColimits (C : Type u) [CategoryTheory.Category.{v, u} C] {X : TopCat} [TopologicalSpace.NoetherianSpace ↑X] (U : TopologicalSpace.Opens ↑X) : CategoryTheory.Limits.PreservesFilteredColimits (TopCat.Sheaf.sectionsFunctor C U)

Key Noetherian fact (used for higher direct images in Lec 11): on a Noetherian topological space, the sections functor Γ(U, –) commutes with filtered colimits of sheaves.