Solid modelling and computational geometry are based on classical topology and geometry in which the basic predicates and operations, such as membership, subset inclusion, union and intersection, are not continuous and therefore not computable. But a sound computational framework for solids and geometry can only be built in a framework with computable predicates and operations. In practice, correctness of algorithms in computational geometry is usually proved using the unrealistic Real RAM machine model of computation, which allows comparison of real numbers, with the undesirable result that correct algorithms, when implemented, turn into unreliable programs. Here, we use a domaintheoretic approach to recursive analysis to develop the basis of an eective and realistic framework for solid modelling. This framework is equipped with a well-dened and realistic notion of computability which reects the observable properties of real solids. The basic predicates and operations o...

It was already known that the category of T 0 topological spaces is not itself cartesian closed, but can be embedded into the cartesian closed categories FIL of filter spaces and EQU of equilogical spaces where the latter embeds into the cartesian closed category ASSM of assemblies over algebraic lattices. Here, we first clarify the notion of filter space---there are at least three versions FIL a ' FIL b ' FIL c in the literature. We establish adjunctions between FIL a and ASSM and between FIL c and ASSM, and show that FIL b and FIL c are equivalent to reflective full subcategories of ASSM. The corresponding categories FIL b 0 and FIL c 0 of T 0 spaces are equivalent to full subcategories of EQU. Keywords: Categorical models and logics, domain theory and applications Author's address: Reinhold Heckmann, FB 14 -- Informatik, Universitat des Saarlandes, Postfach 151150, D-66041 Saarbrucken, Germany Phone: +49 681 302 2454 Fax: +49 681 302 3065 e-mail: heckmann@cs.un...

The category TOP of topological spaces is not cartesian closed, but can be embedded into the cartesian closed category ASSM of assemblies over algebraic lattices, which is a generalisation of Scott's category EQU of equilogical spaces. In this paper, we identify cartesian closed subcategories of assemblies which correspond to well-known separation properties of topology: T 0 , T 1 , Hausdorff, completely Hausdorff, totally disconnected, completely regular, zero-dimensional. 1 Introduction It is well-known that the categories TOP of topological spaces and TOP 0 of T 0 topological spaces are not cartesian closed, i.e., do not admit a function space construction such that typed -calculus could be interpreted in the category. In December 1996, Dana Scott proposed to remedy the situation by extending TOP 0 to the cartesian closed category EQU of equilogical spaces [12, 1, 4]. There are several equivalent descriptions of EQU; one of them is by modest sets over algebraic lattices [1]....

Equilogical spaces form a cartesian closed complete category that contains all T_0 spaces. We first present the cartesian closed complete subcategory of ordered equilogical spaces that still contains all T_0 spaces, but excludes certain pathological equilogical spaces. Then, we further restrict to the cartesian closed complete subcategory of equilogical d-spaces that still contains all dcpo's and all T_1 spaces. Its main advantage is that equilogical d-spaces with least element admit a least fixed point operator.

Abstract. We explain in detail why the notion of list-arithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel’s incompleteness results. We motivate this definition for three reasons: first, Joyal’s arithmetic universes are listarithmetic pretopoi; second, the initial arithmetic universe among Joyal’s constructions is equivalent to the initial list-arithmetic pretopos; third, any list-arithmetic pretopos enjoys the existence of free internal categories and diagrams as required to prove Gödel’s incompleteness. In doing our proofs we make an extensive use of the internal type theory of the categorical structures involved in Joyal’s constructions. The definition of list-arithmetic pretopos is equivalent to the general one that I came to know in a recent talk by André Joyal. 1.