mapOpcyclesIso_hom_naturality — Mathlib

English

The inverse naturality for leftHomologyMapIso: the inverse maps satisfy a dual commuting relation with leftHomologyMap.

Русский

Обратный натурализм для leftHomologyMapIso: обратные отображения удовлетворяют соответствующему диаграммному равенству.

LaTeX

$$$F.map( leftHomologyMap(\varphi) ) \; \circ\; (S_2.mapLeftHomologyIso F).inv = (S_1.mapLeftHomologyIso F).inv \circ\; leftHomologyMap( F.mapShortComplex.map(\varphi) )$$$

Lean4

@[reassoc]
theorem mapOpcyclesIso_hom_naturality [S₁.HasRightHomology] [S₂.HasRightHomology] [F.PreservesRightHomologyOf S₁]
    [F.PreservesRightHomologyOf S₂] :
    opcyclesMap (F.mapShortComplex.map φ) ≫ (S₂.mapOpcyclesIso F).hom =
      (S₁.mapOpcyclesIso F).hom ≫ F.map (opcyclesMap φ) :=
  by
  dsimp only [opcyclesMap, mapOpcyclesIso, RightHomologyData.opcyclesIso, opcyclesMapIso', Iso.refl]
  simp only [RightHomologyData.map_opcyclesMap', Functor.mapShortComplex_obj, ← opcyclesMap'_comp, comp_id, id_comp]

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