Theorems · Definition · category theory

PresheafOfModules.Sheafify.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) →
            [CategoryTheory.Presheaf.IsLocallyInjective J α] →
              [CategoryTheory.Presheaf.IsLocallySurjective J α] →
                {M₀ : PresheafOfModules R₀} →
                  {A : CategoryTheory.Sheaf J AddCommGrpCat} →
                    (φ : M₀.presheaf ⟶ A.obj) →
                      [CategoryTheory.Presheaf.IsLocallyInjective J φ] →
                        [CategoryTheory.Presheaf.IsLocallySurjective J φ] →
                          {X : Cᵒᵖ} → ↑(R.obj.obj X) → ↑(A.obj.obj X) → ↑(A.obj.obj X)

The scalar multiplication on the sheafification of a presheaf of modules.

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify

Cited by

9 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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