C'era una volta un re seduto in canap`e, che disse alla regina raccontami una storia. La regina cominci`o: "C'era una volta un re seduto in canap`e

This paper is part of a general research programme on the Foundations of Mathematics, Logic and Computer Science, carried out since the early eighties at the the Seminar directed by Ennio De Giorgi at the Scuola Normale Superiore in Pisa. In this context, Foundations are not intended to provide safe and unquestionable grounds to scientific activity, but rather to provide conceptual environments where this activity can be carried out naturally and without artificial constraints. Earlier proposals of such foundational theories appear in [3, 1] (see also [11]). Further investigations, along theses lines, have been carried out by various mathematicians, logicians and computer scientists since Spring 1994, starting from the "Basic Theories" introduced in [4] (see e.g. [17, 6, 5, 7, 12, 13]). The principles which inspire and inform the foundational programme of E. De Giorgi are:

This paper is a contribution to the foundations of coinductive types and coiterative functions, in (Hyper)set-theoretical Categories, in terms of coalgebras. We consider atoms as first class citizens. First of all, we give a sharpening, in the way of cardinality, of Aczel's Special Final Coalgebra Theorem, which allows for good estimates of the cardinality of the final coalgebra. To these end, we introduce the notion of -Y -uniform functor, which subsumes Aczel's original notion. We give also an n-ary version of it, and we show that the resulting class of functors is closed under many interesting operations used in Final Semantics. We define also canonical wellfounded versions of the final coalgebras of functors uniform on maps. This leads to a reduction of coiteration to ordinal induction, giving a possible answer to a question raised by Moss and Danner. Finally, we introduce a generalization of the notion of F -bisimulation inspired by Aczel's notion of precongruence, and we show t...