Theorems · Definition · category theory

CategoryTheory.Abelian.LeftResolution.chainComplexXOneIso

{A : Type u_1} →
  {C : Type u_2} →
    [inst : CategoryTheory.Category.{v_1, u_2} C] →
      [inst_1 : CategoryTheory.Category.{v_2, u_1} A] →
        {ι : CategoryTheory.Functor C A} →
          (Λ : CategoryTheory.Abelian.LeftResolution ι) →
            (X : A) →
              [inst_2 : ι.Full] →
                [inst_3 : ι.Faithful] →
                  [inst_4 : CategoryTheory.Limits.HasZeroMorphisms C] →
                    [inst_5 : CategoryTheory.Abelian A] →
                      (Λ.chainComplex X).X 1 ≅ Λ.F.obj (CategoryTheory.Limits.kernel (Λ.π.app X))

Given Λ : LeftResolution ι, the chain complex Λ.chainComplex X identifies in degree 1 to Λ.F.obj (kernel (Λ.π.app X)).

Defined in

Mathlib.Algebra.Homology.LeftResolution.Basic

Cited by

7 results in Mathlib

Foundations

Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.FullCategoryTheory.Functor.FaithfulCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Abelian