Hidden algebra is a behavioural algebraic specification formalism for objects. It captures their constructional aspect, concerned with the initialisation and evolution of their states, as well as their observational aspect, concerned with the observable b ehaviour of such states. When attention is restricted to the observational aspect, final/cofree constructions provide suitable denotations fo...

We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra w...

We instantiate the general comonad-based construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F . Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structurally smaller than the given argument, but also subsidiary recursions. We develop a Mendler formul...

A coalgebra is introduced here as a model of a certain signature consisting of a type X with various \destructor" function symbols, satisfying certain equations. These destructor function symbols are like methods and attributes in object-oriented programming: they provide access to the type (or state) X. We show that the category of such coalgebras and structure preserving functions is comonadi...