- Theory of frames

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# Theory of frames

Abstract

The study of frames can be traced back to as early as Wallman's work in 1938, in which he initiated the study of topological properties from a lattice-theoretical point of view. C. Ehresmann and J. Benabou firstly regarded complete Heyting algebras as generalised topological spaces in their own right. Such lattices were called 'local lattices'. It was Dowker and Strauss who first used the term 'frame' in their systematic study of such structure. After that many people have made a significant contribution to the study of frames (or locales: the opposite categorical version of frames), such as Isbell, Banaschewski, Joyal, Johnstone, Simmons, etc.

On the other hand, inspired by frames, several types of generalised frames have been introduced and studied in relatively recent times. Three notable examples of generalised frames are σ-frames, κ-frames and preframes. The κ-frames, which were first systematically studied by Madden recently, generalise both frames and σ-frames. While preframes, which have been carefully studied by Johnstone and Vickers, belong to a different type of generalisation.

The emergence of Ζ-continuous posets which unifies various "continuous" structures and discussed most of their basic properties. In 1992, D. Zhao launched a similar programme in attempt to make a uniform approach to various frame-like structure by introducing Ζ-frames. The approach turns out to be very convenient and effective for further categorical treatment.

Category theory is an economical tool that provides a common framework for many branches of mathematics, especially in topology and algebra. In the process of my study of generalised frames, categorical concepts are employed extensively.

My three-years course of study has been constantly motivated by many important papers and publications. The first one, Nuclearity by K.A. Rowe ([19]), is an important paper. The concept of nuclearity aims to characterise finite-dimensionality in symmetric monoidal closed categories. Rowe made a systematic study of nuclearity via many different examples.

The second one is on Nuclearity in the category of complete semilattices by D.A. Higgs and KA. Rowe ([11]). This paper demonstrated that the nuclear objects of the category of complete lattices are precisely the completely distributive lattices (CDL for short). In lattice theory, completely distributive lattices have always attracted special attention. Thus, the CDLs, became one of the most important classes of lattices and have been extensively studied by many authors. This fact, together with many other examples in [19], lead to the following question: Are nuclear objects projective? The first part of my project indicates a positive answer with some minimal assumptions.

The book A compendium of continuous lattices ([10]), written by six expert lattice-theorists (G. Gierz et al.), is an excellent guidebook for me in learning the ropes of continuous lattice theory. Difficult book it is indeed, but it gives a concise and in-depth treatment of continuity in lattice theory. It gives me a very sound foundation that prepares me to understand D.Zhao's approach to generalised frame theory via Ζ-theory.

The doctoral dissertation Generalisation of Continuous Lattices and Frames by D.Zhao ([21]) gives a detailed and clear introduction of Ζ-frames. It opened up a completely new and exciting area of research for me because the concepts and mathematical concepts that arise from Ζ-frames are very rich.

One natural question is whether the concept of nuclearity may be defined for the category of Ζ-frames. The very first step is, of course, to understand how tensor products may be set up in order that we have an autonomous categorical structure.

So the third paper Tensor products and bimorphisms by B.Banaschewski and E. Nelson ([1]) provides very handy information about conditions which will guarantee the existence of tensor multiplication in a concrete category.

Despite the promises that the Ζ-theory seemed to offer, there is one main obstacle that hinders a natural autonomous structure on ZFrm : It is not even clear how the internal hom may be established, let alone the tensor product. However, in the categories of complete join-semilattice, frames and preframes, various constructions have been made to show that they are autonomous categories (see [11], [16] and [17]). This branches off to two alternatives. One of them is to simplify the problem and focus on a less intricate category, namely the category of the Ζ-complete posets and the morphisms that preserve Ζ-sups. Although the internal hom exists, we still cannot enjoy the luxury of having a tensor product.

Another approach, which may be more difficult, is to generalise P. Johnstone's work ([16]). It seems that we can take advantage of the monadic nature of the category of Ζ-frames. Much work, involving Universal Algebra and Proof Theory, remains to be done in this direction.

The sixth is the paper On projective z-frames by D.Zhao ([23]) which characterises the E-projective objects in the category zFrm in adjunction to the category of semilattices. This leads to the study of E-projective objects in the category of frames in adjunction to the category of Ζ-frames. While working furiously at this problem, I ventured into the topic of generalised Scott-topology.

On the other hand, inspired by frames, several types of generalised frames have been introduced and studied in relatively recent times. Three notable examples of generalised frames are σ-frames, κ-frames and preframes. The κ-frames, which were first systematically studied by Madden recently, generalise both frames and σ-frames. While preframes, which have been carefully studied by Johnstone and Vickers, belong to a different type of generalisation.

The emergence of Ζ-continuous posets which unifies various "continuous" structures and discussed most of their basic properties. In 1992, D. Zhao launched a similar programme in attempt to make a uniform approach to various frame-like structure by introducing Ζ-frames. The approach turns out to be very convenient and effective for further categorical treatment.

Category theory is an economical tool that provides a common framework for many branches of mathematics, especially in topology and algebra. In the process of my study of generalised frames, categorical concepts are employed extensively.

My three-years course of study has been constantly motivated by many important papers and publications. The first one, Nuclearity by K.A. Rowe ([19]), is an important paper. The concept of nuclearity aims to characterise finite-dimensionality in symmetric monoidal closed categories. Rowe made a systematic study of nuclearity via many different examples.

The second one is on Nuclearity in the category of complete semilattices by D.A. Higgs and KA. Rowe ([11]). This paper demonstrated that the nuclear objects of the category of complete lattices are precisely the completely distributive lattices (CDL for short). In lattice theory, completely distributive lattices have always attracted special attention. Thus, the CDLs, became one of the most important classes of lattices and have been extensively studied by many authors. This fact, together with many other examples in [19], lead to the following question: Are nuclear objects projective? The first part of my project indicates a positive answer with some minimal assumptions.

The book A compendium of continuous lattices ([10]), written by six expert lattice-theorists (G. Gierz et al.), is an excellent guidebook for me in learning the ropes of continuous lattice theory. Difficult book it is indeed, but it gives a concise and in-depth treatment of continuity in lattice theory. It gives me a very sound foundation that prepares me to understand D.Zhao's approach to generalised frame theory via Ζ-theory.

The doctoral dissertation Generalisation of Continuous Lattices and Frames by D.Zhao ([21]) gives a detailed and clear introduction of Ζ-frames. It opened up a completely new and exciting area of research for me because the concepts and mathematical concepts that arise from Ζ-frames are very rich.

One natural question is whether the concept of nuclearity may be defined for the category of Ζ-frames. The very first step is, of course, to understand how tensor products may be set up in order that we have an autonomous categorical structure.

So the third paper Tensor products and bimorphisms by B.Banaschewski and E. Nelson ([1]) provides very handy information about conditions which will guarantee the existence of tensor multiplication in a concrete category.

Despite the promises that the Ζ-theory seemed to offer, there is one main obstacle that hinders a natural autonomous structure on ZFrm : It is not even clear how the internal hom may be established, let alone the tensor product. However, in the categories of complete join-semilattice, frames and preframes, various constructions have been made to show that they are autonomous categories (see [11], [16] and [17]). This branches off to two alternatives. One of them is to simplify the problem and focus on a less intricate category, namely the category of the Ζ-complete posets and the morphisms that preserve Ζ-sups. Although the internal hom exists, we still cannot enjoy the luxury of having a tensor product.

Another approach, which may be more difficult, is to generalise P. Johnstone's work ([16]). It seems that we can take advantage of the monadic nature of the category of Ζ-frames. Much work, involving Universal Algebra and Proof Theory, remains to be done in this direction.

The sixth is the paper On projective z-frames by D.Zhao ([23]) which characterises the E-projective objects in the category zFrm in adjunction to the category of semilattices. This leads to the study of E-projective objects in the category of frames in adjunction to the category of Ζ-frames. While working furiously at this problem, I ventured into the topic of generalised Scott-topology.

Date Issued

2001

Call Number

QA171.5 Ho

Date Submitted

2001