Theorems · Theorem · category theory

CategoryTheory.Abelian.LeftResolution.chainComplexMap_f_succ_succ

∀ {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 Y : A} (f : X ⟶ Y) [inst_2 : ι.Full] [inst_3 : ι.Faithful]
  [inst_4 : CategoryTheory.Limits.HasZeroMorphisms C] [inst_5 : CategoryTheory.Abelian A] (n : ℕ),
  (Λ.chainComplexMap f).f (n + 2) =
    CategoryTheory.CategoryStruct.comp (Λ.chainComplexXIso X n).hom
      (CategoryTheory.CategoryStruct.comp
        (Λ.F.map
          (CategoryTheory.Limits.kernel.map (ι.map ((Λ.chainComplex X).d (n + 1) n))
            (ι.map ((Λ.chainComplex Y).d (n + 1) n)) (ι.map ((Λ.chainComplexMap f).f (n + 1)))
            (ι.map ((Λ.chainComplexMap f).f n)) ⋯))
        (Λ.chainComplexXIso Y n).inv)

Defined in

Mathlib.Algebra.Homology.LeftResolution.Basic

Cited by

3 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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