isoCocycles₁ — Mathlib

English

The isomorphism between 1-cocycles and the corresponding module is compatible with the Hom functor.

Русский

Изоморфизм кокцикл1 совместим с производной функцией Hom.

LaTeX

$$(isoCocycles₁ A).hom ≫ shortComplexH1(A).moduleCatLeftHomologyData.i = iCocycles A 1 ≫ (cochainsIso₁ A).hom$$

Lean4

/-- The 1-cocycles of the complex of inhomogeneous cochains of `A` are isomorphic to
`cocycles₁ A`, which is a simpler type. -/
def isoCocycles₁ : cocycles A 1 ≅ ModuleCat.of k (cocycles₁ A) :=
  cyclesMapIso' (isoShortComplexH1 A) _ (shortComplexH1 A).moduleCatLeftHomologyData

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