preservesBinaryBiproduct_of_preservesBinaryCoproduct

English

If a functor preserves binary coproducts and zero morphisms, then it preserves binary biproducts.

Русский

Если функтор сохраняет бинарные копроизведения и нулевые морфизмы, то он сохраняет бинарные би-производные.

LaTeX

$$$$[\\PreservesBinaryBiproducts\ F] \\Rightarrow\\ PreservesBinaryBiproducts F.$$$$

Lean4

/-- A functor between preadditive categories that preserves (zero morphisms and) binary coproducts
preserves binary biproducts. -/
theorem preservesBinaryBiproduct_of_preservesBinaryCoproduct {X Y : C} [PreservesColimit (pair X Y) F] :
    PreservesBinaryBiproduct X Y F where
  preserves {b}
    hb :=
    ⟨isBinaryBilimitOfIsColimit _ <|
        IsColimit.ofIsoColimit
            ((IsColimit.precomposeInvEquiv (diagramIsoPair _) (F.mapCocone b.toCocone)).symm
              (isColimitOfPreserves F hb.isColimit)) <|
          Cocones.ext (Iso.refl _) fun j => by rcases j with ⟨⟨⟩⟩ <;> simp⟩

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