preservesBiproduct_of_preservesProduct

English

A variant asserting the same preservative property under mono/epi assumptions.

Русский

Вариант той же самой сохранности под предпосылками моно/эпиморфизм.

LaTeX

$$$\text{Mono}(\mathrm{biproduct}) \Rightarrow \mathrm{PreservesBiproduct}\ f\ F$$$

Lean4

/-- A functor between preadditive categories that preserves (zero morphisms and) finite products
preserves finite biproducts. -/
theorem preservesBiproduct_of_preservesProduct {f : J → C} [PreservesLimit (Discrete.functor f) F] :
    PreservesBiproduct f F where
  preserves {b}
    hb :=
    let ⟨_⟩ := nonempty_fintype J
    ⟨isBilimitOfIsLimit _ <|
        IsLimit.ofIsoLimit
            ((IsLimit.postcomposeHomEquiv (Discrete.compNatIsoDiscrete _ _) (F.mapCone b.toCone)).symm
              (isLimitOfPreserves F hb.isLimit)) <|
          Cones.ext (Iso.refl _) (by rintro ⟨⟩; simp)⟩

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