map_leftHomologyMap' — Mathlib

English

For S with left homology data, the functor F preserves the left homology iso up to composition with F.mapIso: S.mapLeftHomologyIso F = (hl.map F).leftHomologyIso ≪≫ F.mapIso hl.leftHomologyIso.symm.

Русский

Для S с данными левой гомологии сохранение левосторонних изоморфизмов через F: S.mapLeftHomologyIso F = (hl.map F).leftHomologyIso ≪≫ F.mapIso hl.leftHomologyIso.symm.

LaTeX

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

Lean4

theorem map_leftHomologyMap' :
    F.map (ShortComplex.leftHomologyMap' φ hl₁ hl₂) =
      ShortComplex.leftHomologyMap' (F.mapShortComplex.map φ) (hl₁.map F) (hl₂.map F) :=
  by
  have γ : ShortComplex.LeftHomologyMapData φ hl₁ hl₂ := default
  rw [γ.leftHomologyMap'_eq, (γ.map F).leftHomologyMap'_eq, ShortComplex.LeftHomologyMapData.map_φH]

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