We present a general theory for the computation of stream transformers of the form F: (R --> B) --> (T --> A), where time T and R, and data A and B, are discrete or continuous. We show how methods for representing topological algebras by algebraic domains
Orthogonality between two stably embedded definable sets is preserved under the addition of constants.
It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial continuous functions. The reason for introducing partial continuous functions is that by passing to partial maps, we are free to consider totalities which are not dense. We show that there is a natural subcategory of the category of representable spaces with morphisms representable maps which is Cartesian closed. Finally, we consider the question of effectivity.
We prove, by a probabilistic argument, that a class of ω-categorical structures, on which algebraic closure defines a pregeometry, has the finite submodel property. This class includes any expansion of a pure set or of a vector space, projective space or affine space over a finite field such that the new relations are sufficiently independent of each other and over the original structure. In particular, the random graph belongs to this class, since it is a sufficiently independent expansion of an infinite set, with no structure. The class also contains structures for which the pregeometry given by algebraic closure is non-trivial.
There is an ``algebraisation'' of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of maps-with-structure, where the extra structure on a map now encodes a choice of liftings with respect to the other class. This extra structure has pleasant consequences: for example, a natural w.f.s. on C induces a canonical natural w.f.s. structure on any functor category [A, C].
In this paper, we define cofibrantly generated natural weak factorisation systems by analogy with cofibrantly generated w.f.s.'s. We then construct them by a method which is reminiscent of Quillen's small object argument but produces factorisations which are much smaller and easier to handle, and show that the resultant natural w.f.s. is, in a suitable sense, freely generated by its generating cofibrations. Finally, we show that the two categories of maps-with-structure for a natural w.f.s. are closed under all the constructions we would expect of them: (co)limits, pushouts / pullbacks, transfinite composition, and so on.
We develop a theory of double clubs which extends Kelly's theory of clubs to the pseudo double categories of Pare and Grandis. We then show that the club for symmetric strict monoidal categories on Cat extends to a `double club' on the pseudo double category of `categories, functors, profunctors and transformations'.
We study a class c of aleph(0)-categorical simple structures such that every M in c has uncomplicated forking behavior and such that definable relations in M which do not cause forking are independent in a sense that is made precise; we call structures in c independent. The SU-rank of such M may be n for any natural number n > 0. The most well-known unstable member of c is the random graph, which has SU-rank one. The main result is that for every strongly independent structure M in e, if a sentence phi is true in M then phi is true in a finite substructure of M. The same conclusion holds for every structure in c with SU-rank one: so in this case the word 'strongly' can be removed. A probability theoretic argument is involved and it requires sufficient independence between relations which do not cause forking. A stable structure M belongs to c if and only if it is aleph(0)-categorical, aleph(0)-stable and every definable strictly minimal Subset of M-eq is indiscernible.
The paper establishes, within constructive mathematics, a full and faithful functor M from the category of locally compact metric spaces and continuous functions into the category of formal topologies (or equivalently locales). The functor preserves finite products, and moreover satisfies f ≤ g if, and only if, M (f) ≤ M (g) for continuous f, g : X → R. This makes it possible to transfer results between Bishop's constructive theory of metric spaces and constructive locale theory.