quasiIso_map_iff_of_preservesLeftHomology

English

Naturality holds for the primed homology iso under a functor F preserving zero morphisms when mapping φ: S1 → S2.

Русский

Натуральность сохраняется для примарного гомологического изоморфизма под функтором F, сохраняющим нули, при отображении φ: S1 → S2.

LaTeX

$$$\mathrm{homologyMap}'(F.mapShortComplex.map\,\phi) \;\circ\; (S_2.mapHomologyIso'\,F)_{\mathrm{hom}} = (S_1.mapHomologyIso'\,F)_{\mathrm{hom}} \circ F(\mathrm{homologyMap}'(\phi))$$$

Lean4

theorem quasiIso_map_iff_of_preservesLeftHomology [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂]
    [F.ReflectsIsomorphisms] : QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ :=
  by
  have γ : LeftHomologyMapData φ S₁.leftHomologyData S₂.leftHomologyData := default
  rw [γ.quasiIso_iff, (γ.map F).quasiIso_iff, LeftHomologyMapData.map_φH]
  constructor
  · intro
    exact isIso_of_reflects_iso _ F
  · intro
    infer_instance

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