Theorems · Theorem · category theory

CategoryTheory.Abelian.PreservesCoimage.hom_coimageImageComparison

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
  {D : Type u₂} [inst_2 : CategoryTheory.Category.{v₂, u₂} D] [inst_3 : CategoryTheory.Limits.HasZeroMorphisms D]
  (F : CategoryTheory.Functor C D) [inst_4 : F.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y)
  [inst_5 : CategoryTheory.Limits.HasKernel f] [inst_6 : CategoryTheory.Limits.HasCokernel f]
  [inst_7 : CategoryTheory.Limits.HasKernel (CategoryTheory.Limits.cokernel.π f)]
  [inst_8 : CategoryTheory.Limits.HasCokernel (CategoryTheory.Limits.kernel.ι f)]
  [inst_9 : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair f 0) F]
  [inst_10 : CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f 0) F]
  [inst_11 :
    CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.cokernel.π f) 0) F]
  [inst_12 :
    CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.kernel.ι f) 0) F]
  [inst_13 : CategoryTheory.Limits.HasKernel (CategoryTheory.Limits.cokernel.π (F.map f))]
  [inst_14 : CategoryTheory.Limits.HasCokernel (CategoryTheory.Limits.kernel.ι (F.map f))],
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Abelian.PreservesCoimage.iso F f).hom
      (CategoryTheory.Abelian.coimageImageComparison (F.map f)) =
    CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.Abelian.coimageImageComparison f))
      (CategoryTheory.Abelian.PreservesImage.iso F f).hom

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Shapes.AbelianImages

Cited by

0 results in Mathlib

Foundations

Depth 46 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.Limits.HasKernelCategoryTheory.Limits.HasCokernelCategoryTheory.Limits.HasKernelCategoryTheory.Limits.HasCokernelCategoryTheory.Limits.PreservesLimitCategoryTheory.Limits.PreservesColimitCategoryTheory.Limits.PreservesLimitCategoryTheory.Limits.PreservesColimitCategoryTheory.Limits.HasKernelCategoryTheory.Limits.HasCokernel