mapLeftHomologyIso_eq — Mathlib

English

For hl (LeftHomologyData) and other data, mapCyclesIso F equals the trans of the mapped left homology iso and F.mapIso: S.mapCyclesIso F = (hl.map F).cyclesIso ≪≫ F.mapIso hl.cyclesIso.symm.

Русский

Для hl и остальных данных отображение циклов левой гомологии равно композиции: S.mapCyclesIso F = (hl.map F).cyclesIso ≪≫ F.mapIso hl.cyclesIso.symm.

LaTeX

$$$S.mapCyclesIso F = (hl.map F).cyclesIso \;\overset{\text{trans}}{=}\; F.mapIso hl.cyclesIso.symm.$$$

Lean4

theorem mapLeftHomologyIso_eq [S.HasLeftHomology] [F.PreservesLeftHomologyOf S] :
    S.mapLeftHomologyIso F = (hl.map F).leftHomologyIso ≪≫ F.mapIso hl.leftHomologyIso.symm :=
  by
  ext
  dsimp [mapLeftHomologyIso, leftHomologyIso]
  simp only [map_leftHomologyMap', ← leftHomologyMap'_comp, Functor.map_id, comp_id, Functor.mapShortComplex_obj]

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