Theorems · Theorem · category theory

PresheafOfModules.Sheafify.map_smul

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {J : CategoryTheory.GrothendieckTopology C}
  {R₀ : CategoryTheory.Functor Cᵒᵖ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α : R₀ ⟶ R.obj)
  [inst_1 : CategoryTheory.Presheaf.IsLocallyInjective J α] [inst_2 : CategoryTheory.Presheaf.IsLocallySurjective J α]
  {M₀ : PresheafOfModules R₀} {A : CategoryTheory.Sheaf J AddCommGrpCat} (φ : M₀.presheaf ⟶ A.obj)
  [inst_3 : CategoryTheory.Presheaf.IsLocallyInjective J φ] [inst_4 : CategoryTheory.Presheaf.IsLocallySurjective J φ]
  (X : Cᵒᵖ) {Y : Cᵒᵖ} (π : X ⟶ Y) (r : ↑(R.obj.obj X)) (m : ↑(A.obj.obj X)),
  (CategoryTheory.ConcreteCategory.hom (A.obj.map π)) (PresheafOfModules.Sheafify.smul α φ r m) =
    PresheafOfModules.Sheafify.smul α φ ((CategoryTheory.ConcreteCategory.hom (R.obj.map π)) r)
      ((CategoryTheory.ConcreteCategory.hom (A.obj.map π)) m)

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify

Cited by

0 results in Mathlib

Foundations

Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Presheaf.IsLocallyInjectiveCategoryTheory.Presheaf.IsLocallySurjectiveCategoryTheory.Presheaf.IsLocallyInjectiveCategoryTheory.Presheaf.IsLocallySurjective