Theorems · Theorem · category theory

imageToKernel_arrow_apply

∀ {V : Type u} [inst : CategoryTheory.Category.{v, u} V] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] {A B C : V}
  (f : A ⟶ B) [inst_2 : CategoryTheory.Limits.HasImage f] (g : B ⟶ C) [inst_3 : CategoryTheory.Limits.HasKernel g]
  (w : CategoryTheory.CategoryStruct.comp f g = 0) {F : V → V → Type uF} {carrier : V → Type w}
  {instFunLike : (X Y : V) → FunLike (F X Y) (carrier X) (carrier Y)} [inst_4 : CategoryTheory.ConcreteCategory V F]
  (x : carrier (CategoryTheory.Subobject.underlying.obj (CategoryTheory.Limits.imageSubobject f))),
  (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.kernelSubobject g).arrow)
      ((CategoryTheory.ConcreteCategory.hom (imageToKernel f g w)) x) =
    (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.imageSubobject f).arrow) x

Defined in

Mathlib.Algebra.Homology.ImageToKernel

Cited by

0 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasImageCategoryTheory.Limits.HasKernelCategoryTheory.ConcreteCategory