 Within the algebraic school, varieties have been defined with the understanding that the operations of a signature form a set. Insights from category theory, in particular the expression of a variety as a monad, defined as a monoid object in the category \(C^C\) of endofunctors of a category \(C\) (Set in the case of ordinary universal algebra) indicate that a cleaner and more general notion of variety is obtained when the operations can form a proper class. For example the important classes of complete semilattices, CSLat, and complete atomic  Boolean algebras,  CABA, form varieties only with this broader notion of signature. In the narrow algebraic sense of variety, the dual of a variety can never be a variety, whereas in the broader monadic notion of variety, the variety Set of sets is dual to CABA while CSLat is self-dual.