descHomotopy — Mathlib

English

Two descents of a fixed morphism are homotopic; the homotopy is constructed by iterating the descent blocks from zero, one, succ, and beyond.

Русский

Два спуска одномоментного морфизма гомотопны; гомотопия строится по шагам через нулевой, единичный и последующий спуски.

LaTeX

$$descHomotopy(f) = Homotopy.mkCoinductive _ (descHomotopyZeroZero f comm) (descHomotopyZeroOne f comm) (descHomotopyZeroSucc f ...) ...$$

Lean4

/-- Two descents of the same morphism are homotopic. -/
def descHomotopy {Y Z : C} (f : Y ⟶ Z) {I : InjectiveResolution Y} {J : InjectiveResolution Z}
    (g h : I.cocomplex ⟶ J.cocomplex) (g_comm : I.ι ≫ g = (CochainComplex.single₀ C).map f ≫ J.ι)
    (h_comm : I.ι ≫ h = (CochainComplex.single₀ C).map f ≫ J.ι) : Homotopy g h :=
  Homotopy.equivSubZero.invFun (descHomotopyZero _ (by simp [g_comm, h_comm]))

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