Cofree coalgebras for signature morphisms

The paper investigates the construction of cofree coalgebras for ‘unsorted signature morphisms'. Thanks to the perfect categorical duality between the traditional concept of equations and the concept of coequations developed in [14] we can fully take profit of the methodological power of Category Theory [2] and follow a clean three step strategy: Firstly, we analyse the traditional Birkhoff construction of free algebras and reformulate it in a systematic categorical way. Then, by dualizing the Birkhoff construction, we obtain, in a second step, corresponding results for cofree coalgebras. And, thirdly, we will interpret the new “abstract” categorical results in terms of more familiar concept. The analysis of a sample cofree construction will provide, finally, some suggestions concerning the potential rôle of cofree coalgebras in System Specifications.