isoCocycles₂_inv_comp_iCocycles

English

The inverse of isoCocycles₂ A composed with iCocycles A 2 equals the i-map composed with the inverse of cochainsIso₂ A. This is a compatibility relation for inverses of the cocycles isomorphisms.

Русский

Обратная изоморфизма isoCocycles₂ A слева от iCocycles A 2 равна iCocycles A 2 слева от обратного ко-cochainsIso₂ A. Совместимость обратных изоморфизмов кокозиал.

LaTeX

$$$(isoCocycles_{2}(A))^{-1} \circ iCocycles(A,2) = (shortComplexH2(A)).moduleCatLeftHomologyData.i \circ (cochainsIso_{2}(A))^{-1}$$$

Lean4

@[reassoc (attr := simp), elementwise (attr := simp)]
theorem isoCocycles₂_inv_comp_iCocycles :
    (isoCocycles₂ A).inv ≫ iCocycles A 2 = (shortComplexH2 A).moduleCatLeftHomologyData.i ≫ (cochainsIso₂ A).inv :=
  (CommSq.horiz_inv ⟨isoCocycles₂_hom_comp_i A⟩).w

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