Terminal coalgebras for endofunctors on sets

This paper shows that the main results of Aczel and Mendler on the existence of terminal coalgebras for an endofunctor on the category of sets do not, for the main part of the results require looking at functors on the category of (possibly proper) classes. We will see here that the main results are valid for sets up to some regular cardinal. Should that cardinal be inaccessible, then Aczel and Mendler’s results are derived. In addition we discuss the canonical map from the initial algebra for an endofunctor on sets to the terminal coalgebra and show that in many cases it embeds the former as a dense subset of the latter in a certain natural topology. By way of example, we calculate the terminal coalgebra for various simple endofunctors.