Theorems · Theorem · category theory

imageToKernel_comp_hom_inv_comp

∀ {V : Type u} [inst : CategoryTheory.Category.{v, u} V] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] {A B C : V}
  (f : A ⟶ B) (g : B ⟶ C) [inst_2 : CategoryTheory.Limits.HasEqualizers V] [inst_3 : CategoryTheory.Limits.HasImages V]
  {Z : V} {i : B ≅ Z}
  (w :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f i.hom)
        (CategoryTheory.CategoryStruct.comp i.inv g) =
      0),
  imageToKernel (CategoryTheory.CategoryStruct.comp f i.hom) (CategoryTheory.CategoryStruct.comp i.inv g) w =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.imageSubobjectCompIso f i.hom).hom
      (CategoryTheory.CategoryStruct.comp (imageToKernel f g ⋯)
        (CategoryTheory.Limits.kernelSubobjectIsoComp i.inv g).inv)

Defined in

Mathlib.Algebra.Homology.ImageToKernel

Cited by

0 results in Mathlib

Foundations

Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasEqualizersCategoryTheory.Limits.HasImages