English
The unit of the sheafification adjunction sends each presheaf section to its germs in the corresponding stalks, forming a natural transformation from F to F.sheafify.
Русский
Единица сонноадъюнкции шейфайфикации отправляет секцию прешейфа в ее жалюзи (гермы) в соответствующих стволиках, образуя естественное преобразование.
LaTeX
$$$$toSheafify : F \to F.sheafify.1\;\text{defined by }\;\text{app } U\ f = \langle x \mapsto germ\_U f(x), \text{sheafifyOf} \rangle$$$$
Lean4
/-- The morphism from a presheaf to its sheafification,
sending each section to its germs.
(This forms the unit of the adjunction.)
-/
def toSheafify : F ⟶ F.sheafify.1
where
app U f := ⟨fun x => F.germ _ x x.2 f, PrelocalPredicate.sheafifyOf ⟨f, fun x => rfl⟩⟩
naturality U U'
f := by
ext x
apply Subtype.ext
ext ⟨u, m⟩
exact germ_res_apply F f.unop u m x
