mapLeftHomologyIso_hom_naturality

English

If two LeftHomologyMapData γ1 and γ2 are equal, then their φK components are equal.

Русский

Если два данных отображения левой гомологии γ1 и γ2 равны, то их компоненты φK также равны.

LaTeX

$$$\gamma_1 = \gamma_2 \Rightarrow \gamma_1.φK = \gamma_2.φK$$$

Lean4

@[reassoc]
theorem mapLeftHomologyIso_hom_naturality [S₁.HasLeftHomology] [S₂.HasLeftHomology] [F.PreservesLeftHomologyOf S₁]
    [F.PreservesLeftHomologyOf S₂] :
    leftHomologyMap (F.mapShortComplex.map φ) ≫ (S₂.mapLeftHomologyIso F).hom =
      (S₁.mapLeftHomologyIso F).hom ≫ F.map (leftHomologyMap φ) :=
  by
  dsimp only [leftHomologyMap, mapLeftHomologyIso, LeftHomologyData.leftHomologyIso, leftHomologyMapIso', Iso.refl]
  simp only [LeftHomologyData.map_leftHomologyMap', Functor.mapShortComplex_obj, ← leftHomologyMap'_comp, comp_id,
    id_comp]

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