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Mathlib.CategoryTheory.Limits.Preserves.Shapes.AbelianImages

Preservation of coimage-image comparisons #

If a functor preserves kernels and cokernels, then it preserves abelian images, abelian coimages and coimage-image comparisons.

source
def CategoryTheory.Abelian.PreservesImage.iso {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.HasCokernel (F.map f)] [Limits.HasKernel (Limits.cokernel.π (F.map f))] :
F.obj (Abelian.image f) ≅ Abelian.image (F.map f)

If a functor preserves kernels and cokernels, it preserves abelian images.

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      @[simp]
      theorem CategoryTheory.Abelian.PreservesImage.iso_hom_ι {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.HasCokernel (F.map f)] [Limits.HasKernel (Limits.cokernel.π (F.map f))] :
      CategoryStruct.comp (iso F f).hom (image.ι (F.map f)) = F.map (image.ι f)
      source
      @[simp]
      theorem CategoryTheory.Abelian.PreservesImage.iso_hom_ι_assoc {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.HasCokernel (F.map f)] [Limits.HasKernel (Limits.cokernel.π (F.map f))] {Z : D} (h : F.obj Y ⟶ Z) :
      CategoryStruct.comp (iso F f).hom (CategoryStruct.comp (image.ι (F.map f)) h) = CategoryStruct.comp (F.map (image.ι f)) h
      source
      @[simp]
      theorem CategoryTheory.Abelian.PreservesImage.factorThruImage_iso_hom {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.HasCokernel (F.map f)] [Limits.HasKernel (Limits.cokernel.π (F.map f))] :
      CategoryStruct.comp (F.map (Abelian.factorThruImage f)) (iso F f).hom = Abelian.factorThruImage (F.map f)
      source
      @[simp]
      theorem CategoryTheory.Abelian.PreservesImage.factorThruImage_iso_hom_assoc {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.HasCokernel (F.map f)] [Limits.HasKernel (Limits.cokernel.π (F.map f))] {Z : D} (h : Abelian.image (F.map f) ⟶ Z) :
      CategoryStruct.comp (F.map (Abelian.factorThruImage f)) (CategoryStruct.comp (iso F f).hom h) = CategoryStruct.comp (Abelian.factorThruImage (F.map f)) h
      source
      @[simp]
      theorem CategoryTheory.Abelian.PreservesImage.iso_inv_ι {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.HasCokernel (F.map f)] [Limits.HasKernel (Limits.cokernel.π (F.map f))] :
      CategoryStruct.comp (iso F f).inv (F.map (image.ι f)) = image.ι (F.map f)
      source
      @[simp]
      theorem CategoryTheory.Abelian.PreservesImage.iso_inv_ι_assoc {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.HasCokernel (F.map f)] [Limits.HasKernel (Limits.cokernel.π (F.map f))] {Z : D} (h : F.obj Y ⟶ Z) :
      CategoryStruct.comp (iso F f).inv (CategoryStruct.comp (F.map (image.ι f)) h) = CategoryStruct.comp (image.ι (F.map f)) h
      source
      @[simp]
      theorem CategoryTheory.Abelian.PreservesImage.factorThruImage_iso_inv {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.HasCokernel (F.map f)] [Limits.HasKernel (Limits.cokernel.π (F.map f))] :
      CategoryStruct.comp (Abelian.factorThruImage (F.map f)) (iso F f).inv = F.map (Abelian.factorThruImage f)
      source
      @[simp]
      theorem CategoryTheory.Abelian.PreservesImage.factorThruImage_iso_inv_assoc {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.HasCokernel (F.map f)] [Limits.HasKernel (Limits.cokernel.π (F.map f))] {Z : D} (h : F.obj (Abelian.image f) ⟶ Z) :
      CategoryStruct.comp (Abelian.factorThruImage (F.map f)) (CategoryStruct.comp (iso F f).inv h) = CategoryStruct.comp (F.map (Abelian.factorThruImage f)) h
      source
      def CategoryTheory.Abelian.PreservesCoimage.iso {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (F.map f)] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
      F.obj (Abelian.coimage f) ≅ Abelian.coimage (F.map f)

      If a functor preserves kernels and cokernels, it preserves abelian coimages.

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          @[simp]
          theorem CategoryTheory.Abelian.PreservesCoimage.iso_hom_π {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (F.map f)] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
          CategoryStruct.comp (F.map (coimage.π f)) (iso F f).hom = coimage.π (F.map f)
          source
          @[simp]
          theorem CategoryTheory.Abelian.PreservesCoimage.iso_hom_π_assoc {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (F.map f)] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] {Z : D} (h : Abelian.coimage (F.map f) ⟶ Z) :
          CategoryStruct.comp (F.map (coimage.π f)) (CategoryStruct.comp (iso F f).hom h) = CategoryStruct.comp (coimage.π (F.map f)) h
          source
          @[simp]
          theorem CategoryTheory.Abelian.PreservesCoimage.factorThruCoimage_iso_inv {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (F.map f)] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
          CategoryStruct.comp (iso F f).inv (F.map (Abelian.factorThruCoimage f)) = Abelian.factorThruCoimage (F.map f)
          source
          @[simp]
          theorem CategoryTheory.Abelian.PreservesCoimage.factorThruCoimage_iso_inv_assoc {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (F.map f)] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] {Z : D} (h : F.obj Y ⟶ Z) :
          CategoryStruct.comp (iso F f).inv (CategoryStruct.comp (F.map (Abelian.factorThruCoimage f)) h) = CategoryStruct.comp (Abelian.factorThruCoimage (F.map f)) h
          source
          @[simp]
          theorem CategoryTheory.Abelian.PreservesCoimage.factorThruCoimage_iso_hom {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (F.map f)] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
          CategoryStruct.comp (iso F f).hom (Abelian.factorThruCoimage (F.map f)) = F.map (Abelian.factorThruCoimage f)
          source
          @[simp]
          theorem CategoryTheory.Abelian.PreservesCoimage.factorThruCoimage_iso_hom_assoc {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (F.map f)] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] {Z : D} (h : F.obj Y ⟶ Z) :
          CategoryStruct.comp (iso F f).hom (CategoryStruct.comp (Abelian.factorThruCoimage (F.map f)) h) = CategoryStruct.comp (F.map (Abelian.factorThruCoimage f)) h
          source
          @[simp]
          theorem CategoryTheory.Abelian.PreservesCoimage.iso_inv_π {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (F.map f)] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
          CategoryStruct.comp (coimage.π (F.map f)) (iso F f).inv = F.map (coimage.π f)
          source
          @[simp]
          theorem CategoryTheory.Abelian.PreservesCoimage.iso_inv_π_assoc {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (F.map f)] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] {Z : D} (h : F.obj (Abelian.coimage f) ⟶ Z) :
          CategoryStruct.comp (coimage.π (F.map f)) (CategoryStruct.comp (iso F f).inv h) = CategoryStruct.comp (F.map (coimage.π f)) h
          source
          theorem CategoryTheory.Abelian.PreservesCoimage.hom_coimageImageComparison {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (Limits.cokernel.π (F.map f))] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
          CategoryStruct.comp (iso F f).hom (coimageImageComparison (F.map f)) = CategoryStruct.comp (F.map (coimageImageComparison f)) (PreservesImage.iso F f).hom
          source
          def CategoryTheory.Abelian.PreservesCoimageImageComparison.iso {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (Limits.cokernel.π (F.map f))] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
          Arrow.mk (F.map (coimageImageComparison f)) ≅ Arrow.mk (coimageImageComparison (F.map f))

          If a functor preserves kernels and cokernels, it preserves coimage-image comparisons.

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              @[simp]
              theorem CategoryTheory.Abelian.PreservesCoimageImageComparison.iso_inv_left {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (Limits.cokernel.π (F.map f))] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
              (iso F f).inv.left = (PreservesCoimage.iso F f).inv
              source
              @[simp]
              theorem CategoryTheory.Abelian.PreservesCoimageImageComparison.iso_hom_left {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (Limits.cokernel.π (F.map f))] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
              (iso F f).hom.left = (PreservesCoimage.iso F f).hom
              source
              @[simp]
              theorem CategoryTheory.Abelian.PreservesCoimageImageComparison.iso_hom_right {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (Limits.cokernel.π (F.map f))] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
              (iso F f).hom.right = (PreservesImage.iso F f).hom
              source
              @[simp]
              theorem CategoryTheory.Abelian.PreservesCoimageImageComparison.iso_inv_right {C : Type u₁} [Category.{v₁, u₁} C] [Limits.HasZeroMorphisms C] {D : Type u₂} [Category.{v₂, u₂} D] [Limits.HasZeroMorphisms D] (F : Functor C D) [F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y) [Limits.HasKernel f] [Limits.HasCokernel f] [Limits.HasKernel (Limits.cokernel.π f)] [Limits.HasCokernel (Limits.kernel.ι f)] [Limits.PreservesLimit (Limits.parallelPair f 0) F] [Limits.PreservesColimit (Limits.parallelPair f 0) F] [Limits.PreservesLimit (Limits.parallelPair (Limits.cokernel.π f) 0) F] [Limits.PreservesColimit (Limits.parallelPair (Limits.kernel.ι f) 0) F] [Limits.HasKernel (Limits.cokernel.π (F.map f))] [Limits.HasCokernel (Limits.kernel.ι (F.map f))] :
              (iso F f).inv.right = (PreservesImage.iso F f).inv