QFactorsThroughHomotopy_of_exists_prev

English

Quasi-isomorphisms in the homotopy category equal the image under the quotient map of quasi-isomorphisms in the underlying homological complex category.

Русский

Квазиизоморфизмы в гомотопической категории равны образу под тождественно-приведенной картой quotient квазиизоморфизмов в исходной категории гомологического комплекса.

LaTeX

$$$ \operatorname{QuasiIso}_{Homotopy}(C,c) = (\operatorname{QuasiIso}(C,c)).map( (\text{quotient } C c) ). $$$

Lean4

theorem QFactorsThroughHomotopy_of_exists_prev [CategoryWithHomology C] : c.QFactorsThroughHomotopy C where
  areEqualizedByLocalization {K L f g}
    h := by
    have : DecidableRel c.Rel := by classical infer_instance
    exact
      h.map_eq_of_inverts_homotopyEquivalences hc _
        (MorphismProperty.IsInvertedBy.of_le _ _ _ (Localization.inverts _ (HomologicalComplex.quasiIso C _))
          (homotopyEquivalences_le_quasiIso C _))

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