Theorems · Theorem · category theory

imageToKernel_op

∀ {V : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} V] [inst_1 : CategoryTheory.Abelian V] {X Y Z : V}
  (f : X ⟶ Y) (g : Y ⟶ Z) (w : CategoryTheory.CategoryStruct.comp f g = 0),
  imageToKernel g.op f.op ⋯ =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Limits.imageSubobjectIso g.op ≪≫ (CategoryTheory.imageOpOp g).symm).hom
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Limits.cokernel.desc f (CategoryTheory.Limits.factorThruImage g) ⋯).op
        (CategoryTheory.Limits.kernelSubobjectIso f.op ≪≫ CategoryTheory.kernelOpOp f).inv)

Defined in

Mathlib.Algebra.Homology.Opposite

Cited by

0 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Abelian