Theorems · Definition · category theory

SheafOfModules.pullbackComp

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        {D' : Type u₃} →
          [inst_2 : CategoryTheory.Category.{v₃, u₃} 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_3 : F.IsContinuous J K] →
                        (φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R) →
                          [inst_4 : (SheafOfModules.pushforward φ).IsRightAdjoint] →
                            {K' : CategoryTheory.GrothendieckTopology D'} →
                              {G : CategoryTheory.Functor D D'} →
                                {R' : CategoryTheory.Sheaf K' RingCat} →
                                  [inst_5 : G.IsContinuous K K'] →
                                    [inst_6 : (F.comp G).IsContinuous J K'] →
                                      (ψ : R ⟶ (G.sheafPushforwardContinuous RingCat K K').obj R') →
                                        [inst_7 : (SheafOfModules.pushforward ψ).IsRightAdjoint] →
                                          (SheafOfModules.pullback φ).comp (SheafOfModules.pullback ψ) ≅
                                            SheafOfModules.pullback
                                              (CategoryTheory.CategoryStruct.comp φ
                                                ((F.sheafPushforwardContinuous RingCat J K).map ψ))

The composition of two pullback functors on sheaves of modules identifies to the pullback for the composition.

Defined in

Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous

Cited by

4 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsContinuousCategoryTheory.Functor.IsRightAdjointCategoryTheory.Functor.IsContinuousCategoryTheory.Functor.IsContinuousCategoryTheory.Functor.IsRightAdjoint