instEssSurjKaroubiToKaroubiOfIsIdempotentComplete

English

If C is idempotent-complete, the functor toKaroubi C is essentially surjective.

Русский

Если C является полно-idempotent, то функтор toKaroubi(C) фактически должна быть сюръективной по существу.

LaTeX

$$$\text{EssSurj} (toKaroubi\ C)$, when $C$ is idempotent-complete.$$

Lean4

instance [IsIdempotentComplete C] : (toKaroubi C).EssSurj :=
  ⟨fun P => by
    rcases IsIdempotentComplete.idempotents_split P.X P.p P.idem with ⟨Y, i, e, ⟨h₁, h₂⟩⟩
    use Y
    exact
      Nonempty.intro
        { hom := ⟨i, by erw [id_comp, ← h₂, ← assoc, h₁, id_comp]⟩
          inv := ⟨e, by erw [comp_id, ← h₂, assoc, h₁, comp_id]⟩ }⟩

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