Categorical Structures and Their Applications

Contents

Preface

Introduction

Opening Ceremony of the North-West European Category Seminar 2003

This paper provides an analysis of the formalism presented in Chapters II and III of Boole’s “Laws of Thought”, showing that it amounts to describing a particular kind of pseudocomplemented meet-semilattice, very far from Boolean algebras. In addition, several identities and identical implications are considered which do turn these semilattices into Boolean algebras.

The concept of homeomorphically closed nearness spaces is introduced and used to characterize the strict T₁ extensions of a topological space that have the property that each homeomorphism on the underlying space lifts to a homeomorphism on the extension.

We show that the tensor product of orthomodular posets always exists if the tacit assumption 0 ≠ 1 in the definition of orthomodular posets is dropped. This yields a symmetric monoidal category of orthomodular posets.

In this paper we modify and extend the techniques used by Blumberg in his classic paper of 1922. In particular, we formulate a weakening of the Blumberg property and show that a wide class of not necessarily metrizable spaces has this property.

We show that a whole bunch of well known topological constructs have isomorphic descriptions by means of sets structured by some collection of (generalized) metrics of a certain type together with suitable morphisms. The classification of the categories listed in the table at the end of the paper, is based on different saturation conditions acting on certain collections of generalized metrics. These conditions are composed out of different building blocks. For instance on collections of quasi-pre-metrics we encounter saturation conditions that are combinations of the following properties: prime ideal, ideal, locally qualified saturation, uniform qualified saturation, locally quantified saturation and uniform quantified saturation. Combinations of these enable us to characterize the constructs in the first colum of the summarizing table at the end of the paper: pretopological spaces, preclosure spaces, semi-quasi uniform spaces, pre-approach spaces, quasi -pre metric spaces. The more important subconstructs of these examples, such as topological spaces, completely regular spaces, uniform spaces, proximity spaces, approach spaces and metric spaces, are characterized using essentially the same building blocks of saturation properties, restricted to particular subclasses of generalised metrics.

We give a short, elementary proof of the metrisability of a compact Hausdorff with G_δ diagonal, using uniform space theory and in the spirit of the work of Horst Herrlich.

We describe some features of the (very) large lattice of all closure operators of the category Top of topological spaces and exhibit the central role of the Kuratowski closure, as the only proper hereditary closure operator which behaves well on products. Additivity alone characterizes the Kuratowski closure in some subcategories of Top, such as all connected and locally connected metrizable spaces or all zero-dimensional metrizable spaces.

In this paper the classical notion of modal logic is extended to a more general type related to monoidal predicate logics.

It was shown in [9] that the category SepX of separated affine sets of a hereditary coreflective subcategory X of the category SSet of all affine sets over the two-point set S = {0, 1} admits completions. That is: there exists an epi-embedding reflective subcategory CSepX of SepX such that, if f : X → Y is an epimorphism and an embedding in SepX with Y ∈ CSepX then Y is the CSepX-reflection (= completion) of X. Here, for each hereditary coreflective subcategory X of SSet, an internal characterization of the complete affine sets (also called affine algebraic sets) of SepX is established and several examples are provided. The obvious statements valid, for example, for the completion of metric spaces as “finite implies complete” and “every set admits a complete structure” are shown to be false in this general contest.

We construct a category whose elements compose horizontally, vertically and laterally. The elements are cubical diagrams of Hausdorff spaces with semitracks in their faces and inhabited by 2-tracks. The compositions are associative but the 2 × 2 properties and the 2 × 2 × 2 property are only satisfied in a twisted sense.

This paper is partly a survey on behavior of products in coreflective classes of some topological categories and partly it brings some new results completening those known.

In the present paper we characterize the idempotent endomorphisms of structures. This characterization is used to characterize the retracts of structures in terms of certain equivalence relations on their support. For relation-free structures this result is rephrased to characterize co-retracts of structures in terms of wreath products of structures with admissible categories, a construction introduced in this paper.

We revisit the known problem whether each compact topology is contained in a maximal compact topology and collect some partial answers to this question. For instance we show that each compact topology is contained in a compact topology in which convergent sequences have unique limits. We also answer a question of D.E. Cameron by showing that each sequentially compact topology is contained in a maximal sequentially compact topology. We finally observe that each sober compact T₁-topology is contained in a maximal compact topology and that each sober compact T₁-topology which is locally compact or sequential is the infimum of a family of maximal compact topologies.

Two interesting generalizations of supertopologies in the sense of Doitchinov are considered. The first one of these leads us to generalized nearness spaces, and consequently topological extensions are being examined. Secondly we obtain equiconvergence spaces which fill the gap between uniform convergence spaces and the non-symmetrical Kent-convergence spaces resulting as a convenient topological category which is cartesian closed and extensional.

Usually, when Galois connections are generalized categorically, they are generalized to adjoint situations. Unfortunately, in this generalization many interesting properties are lost. Recently, Galois connections have been generalized in a concrete category setting to form Galois correspondences. Fortunately, this generalization preserves many Galois connection properties. Also, recently, a counterpart to Galois connections has been defined and studied. This counterpart is Lagois connections, and in this paper, we generalize Lagois connections to Lagois correspondences.

We consider concrete categories with free objects, surjective epimorphisms and coproducts and give two as far as we know new characterizations of projective objects. For this we use a concept of projectivity which restricts the lifting property to certain families of epimorphisms and even to one special epimorphism. The respective equivalences are formulated in categories which fulfill a condition (*).

The category CoalgΣ of coalgebras with respect to a (bounded) signature Σ is known to be locally finitely presentable (see [1]). We strenghten this result by showing that CoalgΣ even is a presheaf category. Moreover, we give a presentation of this category as the category of all algebras of some (many-sorted) signature (without any equations).

Since hyperspaces of complete (separated) uniform spaces are not complete in general, it is highly remarkable that in the more general context of semiuniform convergence spaces even a hyperspace completion exists which preserves several invariants, e.g. precompactness (and thus compactness), connectedness (and uniform connectedness), the property of being a filter space (or a semiuniform space), etc. This completion is used to characterize the subspaces of the compact spaces in the realm of semiuniform convergence spaces axiomatically. The complete hyperspace structure is coarser than the usual uniform hyperspace structure in case uniform spaces are considered.

Convex effect algebras have been introduced by S. Gudder and S. Pulmannová in connection with quantum mechanics, enriching the notion of an effect algebra. It is shown that convex effect algebras can be equivalently described as partially ordered, preseparated positively convex modules satisfying an order convexity condition and being equipped with an affine automorphism of order 2 satisfying a characteristic equation.

Fibrewise topology deals with so called total spaces X over a fixed base space T with respect to a continuous map p : X → T, the projection. When T is a one-point space the theory specializes to that of ordinary topology. There are fibrewise versions of many familiar topological concepts like separation, compactness, local compactness, even quasi local compactness, i.e. every neighbourhood U of a point contains a neighbourhood V that is relatively compact in U. This paper offers a notion of fibrewise sobriety and a fibrewise version of the Hofmann-Lawson-Theorem, stating that every quasi locally compact sober space is locally compact. Moreover, Niefield’s result on exponentiability of projections in the category of all sober spaces over a fixed sober space T generalizes to fibrewise sobriety.

We are looking at the sizes of partitions of topological spaces and show: No quasilindelöf space can be partitioned into more than 2^N₀ sets P_i ∈ P, where the character of P_i in X is countable and any two distinct P_i, P_j ∈ P can be separated by open sets with disjoint closure. No H-closed space can be partitioned into more than 2^N₀ sets P_i ∈ P, where the character of P_i in X is countable and any two distinct P_i, P_j ∈ P can be separated by disjoint open sets.

An L-valed category is a category-like conglomerate in which “potential objects” and “potential morphisms” could be respectively objects and morphisms only to a certain degree; this degree is represented by an element of a fixed GL-monoid L = (L,≤, ∧, ∨, *). In this paper, after brief discussing basic concepts related to L-valued categories, we cosider a series of examples of L-valued categories mainly related to algebra and topology.

Horst Herrlich’s publications

Bernhard Banaschewski: Two Poems