Theorems · Theorem · category theory

PresheafOfModules.Sheafify.map_smul_eq

∀ {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} (r : ↑(R.obj.obj X)) (m : ↑(A.obj.obj X)) {Y : Cᵒᵖ} (f : X ⟶ Y) (r₀ : ↑(R₀.obj Y)),
  (CategoryTheory.ConcreteCategory.hom (α.app Y)) r₀ = (CategoryTheory.ConcreteCategory.hom (R.obj.map f)) r →
    ∀ (m₀ : ↑(M₀.obj Y)),
      (CategoryTheory.ConcreteCategory.hom (φ.app Y)) m₀ = (CategoryTheory.ConcreteCategory.hom (A.obj.map f)) m →
        (CategoryTheory.ConcreteCategory.hom (A.obj.map f)) (PresheafOfModules.Sheafify.smul α φ r m) =
          (CategoryTheory.ConcreteCategory.hom (φ.app Y)) (r₀ • m₀)

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify

Cited by

7 results in Mathlib

Foundations

Depth 55 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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