Theorems · Definition · category theory

HomologicalComplex.homotopyCofiber.XIsoBiprod

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      {ι : Type u_2} →
        {c : ComplexShape ι} →
          {F G : HomologicalComplex C c} →
            (φ : F ⟶ G) →
              [inst_2 : HomologicalComplex.HasHomotopyCofiber φ] →
                [inst_3 : DecidableRel c.Rel] →
                  (i j : ι) →
                    c.Rel i j →
                      [inst_4 : CategoryTheory.Limits.HasBinaryBiproduct (F.X j) (G.X i)] →
                        HomologicalComplex.homotopyCofiber.X φ i ≅ F.X j ⊞ G.X i

The canonical isomorphism (homotopyCofiber φ).X i ≅ F.X j ⊞ G.X i when c.Rel i j.

Defined in

Mathlib.Algebra.Homology.HomotopyCofiber

Cited by

22 results in Mathlib

Foundations

Depth 23 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveHomologicalComplex.HasHomotopyCofiberDecidableRelCategoryTheory.Limits.HasBinaryBiproduct