English
The pushforward of a sheaf along a morphism is a sheaf.
Русский
Перенос или выталкивание sheaf вдоль морфизма сохраняет свойство sheaf.
LaTeX
$$$\\text{pushforward protein} \\ \text{IsSheaf}$$$
Lean4
/-- If `G : C ⥤ D` is a functor which reflects isomorphisms and preserves limits
(we assume all limits exist in `C`),
then checking the sheaf condition for a presheaf `F : Presheaf C X`
is equivalent to checking the sheaf condition for `F ⋙ G`.
The important special case is when
`C` is a concrete category with a forgetful functor
that preserves limits and reflects isomorphisms.
Then to check the sheaf condition it suffices to check it on the underlying sheaf of types.
Another useful example is the forgetful functor `TopCommRingCat ⥤ TopCat`. -/
@[stacks 0073 "In fact we prove a stronger version with arbitrary target category."]
theorem isSheaf_iff_isSheaf_comp' {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D] (G : C ⥤ D)
[G.ReflectsIsomorphisms] [HasLimitsOfSize.{v, v} C] [PreservesLimitsOfSize.{v, v} G] {X : TopCat.{v}}
(F : Presheaf C X) : Presheaf.IsSheaf F ↔ Presheaf.IsSheaf (F ⋙ G) :=
Presheaf.isSheaf_iff_isSheaf_comp _ F G
