SPLIT STRUCTURES To our friend Aurelio Carboni for his 60 th birthday

In the early 1990’s the authors proved that the full subcategory of ‘suplattices’ determined by the constructively completely distributive (CCD) lattices is equivalent to the idempotent splitting completion of the bicategory of sets and relations. Having many corollaries, this was an extremely useful result. Moreover, as the authors soon suspected, it specializes a much more general result. Let D be a monad on a category C in which idempotents split. Write kar(CD) for the idempotent splitting completion of the Kleisli category. Write spl(CD) for the category whose objects are pairs ((L, s), t), where (L, s) is an object of CD, the Eilenberg-Moore category, and t : (L, s) (DL,mL) is a homomorphism that splits s : (DL,mL) (L, s), with spl(CD)(((L, s), t), ((L′, s′), t′)) = CD((L, s)(L′, s′)). The main result is that kar(CD) ∼= spl(C). We also show how this implies the CCD lattice characterization theorem and consider a more general context.