David Holgate

Some new characterizations of pointfree pseudocompactness

Abstract By [3], a frame L is pseudocompact iff every ≺≺-sequence in L joining to the top terminates. Here it is shown, for any completely regular L, that pseudocompactness is also equivalent to (i) the analogous condition for ≺-sequences, (ii) the countable almost compactness of L, (iii) the almost compactness of CozL as a σ-frame and (iv) the condition that every countably based proper filter in L clusters. Further we establish the zero-dimensional counterparts of the above, concerning the integer valued notion of pseudocompactness. Finally, we add to this a characterization of pseudocompactness in terms of uniformities.

The Pullback Closure Operator and Generalisations of Perfectness

A categorical closure operator induced via pullback by a pointed endofunctor is introduced. Various notions of a perfect morphism relative to a pointed endofunctor and the induced closure are then considered. The main result explores how these notions are interrelated, linking also with earlier notions of perfectness.

A Link Between Two Connectedness Notions

The composition of two previously introduced Galois connections is used to provide a wider perspective on the Clementino–Tholen connectedness versus separation Galois connection. Moreover, a link between this and Castellinis connectedness–disconnectednes Galois connection is also presented.

Topogenous and Nearness Structures on Categories

We introduce topogenous orders on a general category and demonstrate that they are equivalent to neighbourhood operators and subsume both closure and interior operators. By looking at the basic properties of so-called strict morphisms relative to a topogenous order the ease of working with their axioms and the concurrent generalisation of both interior and closure is evident. In closing we consider Herrlich’s concept of nearness in a category and how it interacts with topogenous orders and syntopogenous structures.

Connected and Disconnected Maps

AbstractA new relation between morphisms in a category is introduced—roughly speaking (accurately in the categories Set and Top), f ∥ g iff morphisms w:dom(f)→dom(g) never map subobjects of fibres of f non-constantly to fibres of g. (In the algebraic setting replace fibre with kernel.) This relation and a slight weakening of it are used to define “connectedness” versus “disconnectedness” for morphisms. This parallels and generalises the classical treatment of connectedness versus disconnectedness for objects in a category (in terms of constant morphisms). The central items of study are pairs

Closure Operator Constructions Depending On One Parameter

({\mathcal F},{\mathcal G})

A Lax Approach to Neighbourhood Operators

of classes of morphisms which are corresponding fixed points of the polarity induced by the ∥-relation. Properties of such pairs are examined and in particular their relation to (pre)factorisation systems is analysed. The main theorems characterise: (a)factorisation systems which factor morphisms through a regular epimorphic “connected” morphism followed by a “disconnected” morphism, and(b)pairs

Approach structures and measures of connectedness

({\mathcal F},{\mathcal G})

Categorical neighborhood operators

consisting of “connected” versus “disconnected” morphisms which induce a (regular) factorisation system.This suggests a generalisation of the pair (Concordant, Dissonant) of classes of continuous maps which was shown by Collins to yield the factorisation system (Concordant quotient, Dissonant) on Top.

Interior and neighbourhood

Let X be an (E , M)-category for sinks. For each subclass N of M, two new Galois connections that generalize the Clementino-Tholen connectedness and separation Galois connections are introduced. In particular, closure operator constructions that generalize the regular and coregular closure operators are given. A big diagram of several different types of Galois connections is built. The usefulness of the dependence on the parameter N is shown in that many previously obtained closure constructions can be deduced from the diagram by choosing N in different ways.