Theorems · Theorem · category theory

CategoryTheory.Abelian.coimageImageComparisonFunctor_map

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
  [inst_2 : CategoryTheory.Limits.HasKernels C] [inst_3 : CategoryTheory.Limits.HasCokernels C]
  {f g : CategoryTheory.Arrow C} (η : f ⟶ g),
  CategoryTheory.Abelian.coimageImageComparisonFunctor.map η =
    CategoryTheory.Arrow.homMk
      (CategoryTheory.Limits.cokernel.map (CategoryTheory.Limits.kernel.ι f.hom) (CategoryTheory.Limits.kernel.ι g.hom)
        (CategoryTheory.Limits.kernel.map f.hom g.hom (CategoryTheory.Arrow.Hom.left η)
          (CategoryTheory.Arrow.Hom.right η) ⋯)
        (CategoryTheory.Arrow.Hom.left η) ⋯)
      (CategoryTheory.Limits.kernel.map (CategoryTheory.Limits.cokernel.π f.hom)
        (CategoryTheory.Limits.cokernel.π g.hom) (CategoryTheory.Arrow.Hom.right η)
        (CategoryTheory.Limits.cokernel.map f.hom g.hom (CategoryTheory.Arrow.Hom.left η)
          (CategoryTheory.Arrow.Hom.right η) ⋯)
        ⋯)
      ⋯

Defined in

Mathlib.CategoryTheory.Abelian.Images

Cited by

0 results in Mathlib

Foundations

Depth 41 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasKernelsCategoryTheory.Limits.HasCokernels