We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. μ-bicomplete categories. We argue that parity games with a given starting position play the role of terms for the theory of μ-bicomplete categories. We show that the interpretation...

We study conditions under which the Hausdorff quasi-uniformity UH of a quasi-uniform space (X,U) on the set P0(X) of the nonempty subsets of X is bicomplete. Indeed we present an explicit method to construct the bicompletion of the T0-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It is used to find a characterization of those quasi-uniform T0spaces (X,U) for which the Hau...

We present a calculus of proofs, the intended models of which are categories with finite products and coproducts, initial algebras and final coalgebras of functors that are recursively constructible out of these operations, that is, μ-bicomplete categories. The calculus satisfies the cut elimination and its main characteristic is that the underlying graph of a proof is allowed to contain a cert...

For an arbitrary category, we consider the least class of functors containing the projections and closed under finite products, finite coproducts, parameterized initial algebras and parameterized final coalgebras, i.e. the class of functors that are definable by μ-terms. We call the category μ-bicomplete if every μ-term defines a functor. We provide concrete examples of such categories and expl...

A signature gives rise to a language L (Var) by extending with variables x 2 Var and binding constructs x and x, corresponding to least and greatest xed points respectively. The natural denotational models for such languages are bicomplete dcpos as monotone-algebras. We prove that several approximating denotational semantics have the usual compositional semantics as their limit. These results p...

One of the authors introduced in [2] a calculus of circular proofs for studying the computability arising from the following categorical operations: finite products and coproducts, initial algebras, final coalgebras. The calculus of [2] is cut-free; yet, even if sound and complete for provability, it lacks an important property for the semantics of proofs, namely fullness w.r.t. the class of na...

We introduce a new type of Caristi’s mapping on partial metric spaces and show that a partial metric space is complete if and only if every Caristi mapping has a fixed point. From this result we deduce a characterization of bicomplete weightable quasi-metric spaces. Several illustrative examples are given.

This paper is concerned with two problems which, although not apparently closely related, are solved in part by the same methods. The first problem is: given a bicomplete (=comple te and cocomplete) category X and a triple T on X, is X T also bicomplete? The second is: given a category X and a functor R: X---~X, does R generate a free triple? This paper began as an attempt to show that the cate...