Theorems · Definition · category theory

SheafOfModules.pushforwardCompForgetToSheafModuleCat

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        {J : CategoryTheory.GrothendieckTopology C} →
          {K : CategoryTheory.GrothendieckTopology D} →
            {F : CategoryTheory.Functor C D} →
              {S : CategoryTheory.Sheaf J RingCat} →
                {R : CategoryTheory.Sheaf K RingCat} →
                  [inst_2 : F.IsContinuous J K] →
                    (φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R) →
                      (X : Cᵒᵖ) →
                        (hX : CategoryTheory.Limits.IsInitial X) →
                          (hX' : CategoryTheory.Limits.IsInitial (F.op.obj X)) →
                            (SheafOfModules.pushforward φ).comp (SheafOfModules.forgetToSheafModuleCat S X hX) ≅
                              (SheafOfModules.forgetToSheafModuleCat R (F.op.obj X) hX').comp
                                ((CategoryTheory.sheafCompose K
                                      (ModuleCat.restrictScalars (RingCat.Hom.hom (φ.hom.app X)))).comp
                                  (F.sheafPushforwardContinuous (ModuleCat ↑(S.obj.obj X)) J K))

Pushforward commutes with SheafOfModules.forgetToSheafModuleCat

Defined in

Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous

Cited by

0 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsContinuous