Gerhard Preuss

Categorical methods in computer science with aspects from topology

A categorical concept of constraints for algebraic specifications.- The integration of logical and algebraic types.- Colimits as parameterized data types.- Empty carriers: The categorical burden on logic.- Monads, matrices and generalized dynamic algebra.- Foundations are rich institutions, but institutions are poor foundations.- Development of algebraic specifications with constraints.- Revised ACT ONE: Categorical constructions for an algebraic specification language.- Objects, object types, and object identification.- Categories for the development of algebraic module specifications.- Parameterized data type and process specifications using projection algebras.- Parameterized algebraic domain equations.- Semantical constructions for categories of behavioural specifications.- Relation-sorted algebraic specifications with built-in coercers: Parameterization and parameter passing.- On declarations.- Cauchy sequences in quasi-uniform spaces: Categorical aspects.- The construct PRO of projection spaces: its internal structure.- Categories and mathematical morphology.- Topological spaces for cpos.- On the topological structures of nets.- Description of the topological universe hull.- On residuated approximations.- On convergence of filters and ultrafilters to subsets.

Some algebraically topological aspects in the realm of convenient topology

A cohomology theory for filter spaces is developed in such a way that a suitable variant of the Eilenberg-Steenrod axioms is satisfied. Furthermore, path connectedness and fundamental groups are generalized to limit spaces.

LOCAL COMPACTNESS IN SEMI-UNIFORM CONVERGENCE SPACES

Abstract Local compactness is studied in the highly convenient setting of semi-uniform convergence spaces which form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniform limit spaces (and thus of uniform spaces). It turns out that it leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the local compact spaces are exactly the compactly generated spaces. Furthermore, a one-point Hausdorff compactification for noncompact locally compact Hausdorff convergence spaces is considered.1

Hausdorffs Studien über Kurven, Bögen und Peano-Kontinua

Zahlreiche Ergebnisse, insbesondere der komplexen Analysis und der Flachentheorie, legen den Gedanken nahe, das Konzept einer Kurve stelle einen Grundbegriff der Topologie dar. Leider ist dem nicht so.

Hausdorffs Studien zu Fundamentalkonstruktionen der Topologie

Beim Studium abstrakter Strukturen (wie z.B. topologischer Raume, Gruppen, Moduln) spielen gewisse kanonische Konstruktionen, insbesondere die von Limiten (wie Produkten und Egalisatoren bzw. Teiloder Unterobjekten) und Colimiten (wie Coprodukten bzw. Summen und Coegalisatoren bzw. Quotienten) eine zentrale Rolle. Denn ein wesentliches Merkmal struktureller Untersuchungen in der Mathematik besteht darin, das versucht wird, komplizierte mathematische Gebilde vermoge besonders ubersichtlicher Konstruktionsverfahren aus besonders einfach gebauten Gebilden zusammenzusetzen. Gelingt dieses, so wird das Studium komplizierter Gebilde wesentlich erleichtert; denn man kann sich vielfach darauf beschranken, zunachst die besonders einfachen Bausteine zu untersuchen und danach zu analysieren, welche Eigenschaften von Gebilden bei der Zusammensetzung derselben mittels standardisierter Konstruktionsverfahren erhalten bleiben. Als wichtigste Konstruktionsverfahren in der Topologie mus man die Bildung von Teilraumen, Quotienten, Produkten und Summen ansehen.1

Relations between Semiuniform Convergence Spaces and Merotopic Spaces (including Nearness Spaces)

M. Katětov [80] originally introduced filter spaces in the realm of his merotopic spaces (studied in the same paper) and called them filter-merotopic spaces. We start the present chapter with this alternative description of filter spaces which have been introduced in chapter 1 and which have also been described in the framework of semiuniform convergence spaces in chapter 2. In other words, a filter space may be regarded as a (filter-)merotopic space or as a Fil-determined semiuniform convergence space. Furthermore, the construct Fil is bicoreflectively embedded in the construct Mer of merotopic spaces, whereas it is bireflectively and bicoreflectively embedded into SUConv. As already mentioned in the introduction of this book, the formation of subspaces in Top (or Tops) is not satisfactory. The reason becomes clear, when subspaces of symmetric topological spaces are formed in SUConv: they are not topological in general unless they are closed. Since symmetric topological spaces may be regarded as complete filter spaces, subspaces of them, formed in SUConv, are filter spaces (regarded as semiuniform convergence spaces). Thus, in order to answer the question how subspaces (in SUConv) of symmetric topological spaces, called subtopological spaces, can be characterized axiomatically, we may focus our interest to Fil. Such an axiomatic characterization in terms of filters is given in the second part of this chapter. Another characterization due to H.L. Bentley [10] is found, when the description of filter spaces in the realm of merotopic spaces is used, namely a filter space is subtopological iff its corresponding merotopic space is a nearness space. Nearness spaces have been introduced and studied first by H. Herrlich [62].

Reflections and Coreflections

As is well-known topological spaces can be related to each other by means of continuous maps. More generally, objects in a category can be related to each other by means of morphisms. There is an analogous relationship between categories via so-called functors. The classical definition of universal maps in the sense of N. Bourbaki [18] corresponds to a categorical one which utilizes a functor. The existence of all universal maps with respect to a given functor F is related to a pair of adjoint functors (G, F), where G (resp. F) is called a left adjoint (resp. right adoint). The relationships between these functors are described by means of natural transformations u and v (which occur as “maps” between functors). Thus, an adjoint situation (G, F, u, v) is obtained. In the first part of this chapter adjoint situations are studied together with some examples. In the second part an important special case of adjoint situations (G, F, u, v) is investigated, namely the case where F is an inclusion functor I from a subcategory A of a category C to C (the notion of inclusion functor corresponds to the notion of inclusion map in classical mathematics). Then G is called a reflector from C to A and A is called reflective. If the morphisms belonging to all universal maps with respect to I are epimorphisms, extremal epimorphisms or bimorphisms, then G is called an epireflector, extremal epireflector or bireflector respectively and we say epireflective, extremal epireflective or bireflective subcategory rather than reflective subcategory. The famous characterization theorem for epireflective (and extremal epireflective) subcategories is proved and the results are applied to bire-flective subconstructs of topological constructs.

Cartesian Closed Topological Categories

The category Top of topological spaces and continuous maps fails to have some desirable properties, e.g. the product of two quotient maps need not be a quotient map and there is in general no natural function space topology, i.e. Top is not cartesian closed. Because of this fact, which is inconvenient for investigations in algebraic topology (homotopy theory), functional analysis (duality theory) or topological algebra (quotients), Top has been substituted either by well-behaved subcategories or by more convenient supercategories. Unfortunately most of these categories suffer from other deficiencies. Some of them are too small, e.g. the coreflective hull of all (compact) metrizable spaces in Top [whose objects are called sequential spaces] or too big, e.g. the category of quasi-topological spaces introduced by E. Spanier (the quasitopologies on a fixed set in general form a proper class!). Another well-behaved candidate namely the coreflective hull of all compact Hausdorff spaces in Top [whose objects are called compactly generated spaces] has not been described by suitable axioms.

Cohomology and Dimension of Nearness Spaces

It is a well-known fact that cohomology theory leads to better results in dimension theory than homology theory. The beautiful results characterizing finite-dimensional compact metric spaces by means of homology resp. cohomology (cf. Hurewicz and Wallman [48]) may be generalized in a slightly modified form to compact Hausdorff spaces provided Lebesgue’s covering dimension is considered (cf. Nagata [64]). But already for the wider class of paracompact Hausdorff spaces a corresponding homological characterization of covering dimension is not valid. Nevertheless a cohomological characterization of finite-dimensional paracompact Hausdorff spaces is known. In 1952 C.H. Dowker [24] has shown that Cech’s cohomology theory (and homology theory) may be defined for structures which — as we know today — include nearness structures. H.L. Bentley [11] and D. Czarcinski [21] have proved that these theories satisfy a variant of the Eilenberg-Steenrod axioms. During this chapter it is expected that the reader is acquainted with simplicial cohomology and classical Cech cohomology.

Relations Between Special Topological Categories

Besides the categorical approach of chapter I there is another approach in order to handle problems of a “topological” nature namely the conceptual one. The aim of this approach is to find a basic topological concept by means of which any topological concept or idea can be expressed. A fundamental requirement is the following: By means of such a concept one should be able to explain “nearness”. Axiomatizing the concept of nearness between a point x and a set A (usually denoted by x ∈Ā, i.e. x belongs to the closure of A) one can obtain topological spaces. But there are other types of spaces: Proximity spaces for instance are obtained by an axiomatization of the concept of nearness between two sets, and by means of contiguity spaces one can even explain axiomatically nearness between a finite collection of sets. Thus, H. Herrlich filled a gap by defining nearness spaces as an axiomatization of the concept of nearness between arbitrary collections of sets. Though there is a difference of a “topological” nature between removing a point from the usual topological space R of real numbers and removing a closed interval of length one respectively the obtained topological spaces are homeomorphic.