quasiIsoAt₀_iff — Mathlib

English

For Cochain complexes over Nat with zero-th homology, QuasiIsoAt f 0 is equivalent to the shortComplexQuasiIso of the specialized shortComplexFunctor'.

Русский

Для коCHAIN-комплексах над Nat нулевой степени эквивалентно QuasiIsoAt f 0 короткому комплексному отображению.

LaTeX

$$QuasiIsoAt f 0 \iff ShortComplex.QuasiIso ((HomologicalComplex.shortComplexFunctor' C _ 0 0 1).map f)$$

Lean4

theorem quasiIsoAt₀_iff {K L : CochainComplex C ℕ} (f : K ⟶ L) [K.HasHomology 0] [L.HasHomology 0]
    [(K.sc' 0 0 1).HasHomology] [(L.sc' 0 0 1).HasHomology] :
    QuasiIsoAt f 0 ↔ ShortComplex.QuasiIso ((HomologicalComplex.shortComplexFunctor' C _ 0 0 1).map f) :=
  quasiIsoAt_iff' _ _ _ _ (by simp) (by simp)

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