iso_hom_naturality — Mathlib

English

The descent map desc is defined for any f: X → Y and any pair of injective resolutions, producing an explicit homotopy between desc f J I and other desc maps.

Русский

Спуск-desc задаёт гомотопию между спусками f между резолюциями.

LaTeX

$$desc {X} {Y} (f) {I J} (φ) (ψ) : Homotopy g h$$

Lean4

@[reassoc]
theorem iso_hom_naturality {X Y : C} (f : X ⟶ Y) (I : InjectiveResolution X) (J : InjectiveResolution Y)
    (φ : I.cocomplex ⟶ J.cocomplex) (comm : I.ι.f 0 ≫ φ.f 0 = f ≫ J.ι.f 0) :
    (injectiveResolutions C).map f ≫ J.iso.hom = I.iso.hom ≫ (HomotopyCategory.quotient _ _).map φ :=
  by
  apply HomotopyCategory.eq_of_homotopy
  apply descHomotopy f
  all_goals aesop

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