Theorems · Definition · category theory

HomologicalComplex.homotopyCofiber.descEquiv

{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] →
                  (K : HomologicalComplex C c) →
                    (∀ (j : ι), ∃ i, c.Rel i j) →
                      (α : G ⟶ K) × Homotopy (CategoryTheory.CategoryStruct.comp φ α) 0 ≃
                        (HomologicalComplex.homotopyCofiber φ ⟶ K)

Morphisms homotopyCofiber φ ⟶ K are uniquely determined by a morphism α : G ⟶ K and a homotopy from φ ≫ α to 0.

Defined in

Mathlib.Algebra.Homology.HomotopyCofiber

Cited by

0 results in Mathlib

Foundations

Depth 59 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveHomologicalComplex.HasHomotopyCofiberDecidableRel