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+{-# OPTIONS --without-K --safe #-}
+
+open import Categories.Category using (Category)
+open import Level using (Level; suc; _⊔_)
+
+module Category.KaroubiComplete {o ℓ e : Level} (𝒞 : Category o ℓ e) where
+
+open import Categories.Morphism.Idempotent 𝒞 using (IsSplitIdempotent)
+
+open Category 𝒞
+
+-- A Karoubi complete category is one in which all idempotents split
+record KaroubiComplete : Set (suc (o ⊔ ℓ ⊔ e)) where
+
+  field
+    split : {A : Obj} {f : A ⇒ A} → f ∘ f ≈ f → IsSplitIdempotent f