Both pre-orders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' retraction pairs between them has a limit in the category of complete domains/spaces with retraction pairs as morphisms. The pre-order version was discovered first by Scott in 1969, and is referred to as Scott's inverse limit theorem. The metric version was mainly developed by de Bakker and Zucker and refined and generalized by America and Rutten. The theorem in both its versions provides the main tool for solving recursive domain equations. The proofs of the two versions of the theorem look astonishingly similar, but until now the preconditions for the pre-order and the metric versions have seemed to be fundamentally different. In this thesis we establish a more general theory of domains based on the noti...

We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ω-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, can be used as the basis for a theory of computability, and provides a model of parametric polymorphism.

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

We make an initial step towards a categorical semantics of guarded induction. While ordinary induction is usually modelled in terms of the least fixpoints and the initial algebras, guarded induction is based on the unique fixpoints of certain operations, called guarded, on the final coalgebras. So far, such operations were treated syntactically [3,8,9,23]. We analyse them categorically. Guarded induction appears as couched in coinductively constructed domains, but turns out to be reducible to coinduction only in special cases. The applications of the presented analysis span across the gamut of the applications of guarded induction --- from modelling computation to solving differential equations. A subsequent paper [26] will provide an account of some domain theoretical aspects, which are presently left implicit. "In order to establish that a proposition OE follows from other propositions OE 1 ; : : : ; OE q , it is enough to build a proof term e for it, using not only natural deduction...

The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a non-classical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the successor map) s1 : N ! N, arbitrary operations su : N u ! N of arities u `between 0 and 1'. That is, u is allowed to range over subsets of a singleton set.

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