Bibliography: pages 141-144. / Ehresmann in 1959 first articulated the view that a complete lattice with an appropriate distributivity property deserved to be studied as a generalized topological space in its own right. He called the lattice a local lattice. Here is the distributivity property: x ∧ Vxα = V(x∧xα). A map of local lattices should preserve finite meets and arbitrary joins (and hence top and bottom elements). Dowker and Papert introduced the term frame for a local lattice and extended many results of topology to frame theory. At the 1981 international conference on categorical algebra and topology at Cape Town University a suggestion was made that a study of "uniform frames" (whatever they might be) would be an appropriate and useful start to a project concerned with examining, from a lattice theoretical point of view, the many topological structures which have gained acceptance in the topologist's arsenal of useful tools. It was felt that many of the pre-requisites for such a study had been established, and in fact one of the themes of the conference was the growing role of lattice theory in topology. The suggestion was eagerly accepted, and this thesis is the result.

Bibliography: pages 92-97. / Closure operations within objects of various categories have played an important role in the development of Categorical Topology. Notably they have been used to characterise epimorphisms and investigate cowellpoweredness in specific categories, to generalise Hausdorff separation through diagonal theorems, and to extend topological notions such as compactness of objects and perfectness of morphisms to abstract categories. The categorical theory of factorisation structures for families of morphisms which developed in the 1970's laid the foundation for an axiomatic theory of categorical closure operators. This theory drew together many endeavours involving closure operations, and was coalesced in [Dikranjan, Giuli 1987]. The literature on categorical closure operators continues to extend the theory as well as apply it to problems in Category Theory. Central to our thesis is a particular closure operator (in the sense of [Dikranjan, Giuli 1987]) which we name the "pullback closure operator". Its construction is not entirely new, but no author has studied this operator in its own right. We investigate some of the operator's properties, present several examples and then apply it in two areas of Categorical Topology. First we use the pullback closure operator to establish links between two previously disjoint theories of perfect morphisms. One theory, which developed in the 1970's, exploits the orthogonality properties and functor related properties of perfect continuous maps. Another theory, which has developed more recently, generalises the closure and compactness properties of perfect continuous maps. (We should note that this does not include the recent work in [Clementino, Giuli, Tholen 1995] which takes another approach to perfect morphisms via closure operators.) Our investigations centre around finding conditions that are sufficient to ensure that the links between these two theories can be utilised. Our second use of the pullback closure operator is in pursuing the precategorical ideas expressed in [Birkhoff 1937], and some developments of these ideas in [Brummer, Giuli, Herrlich 1992] and [Brummer, Giuli 1992], to build a theory of completion of objects in an abstract category. In this context the pullback closure operator is shown to be appropriate in characterising complete objects, illuminating links with previously studied completion notions and describing epimorphisms in the category in which we are working. (In fact the pullback closure operator can be used to describe epimorphisms in even wider contexts.) Our methodology is what has been termed colloquially as "doing topology in categories". Topological notions and results are expressed in the language of category theory. Using these reformulations, new results are pursued at the level of categories, and are then applied in specific topological or algebraic contexts. Within this, our approach has been to make as few global assumptions as possible. The pullback closure operator is strictly a tool, in the sense that when assumptions are made, they concern the underlying categories, functors and classes of morphisms and objects and not the operator itself.

Includes bibliographical references. / The relationship between transitive uniform spaces and zero-dimensional topological spaces was first established by Banaschewski [1957], and was later investigated by Levine [1969]. The theory of transitive quasi-uniform spaces is treated in [Fletcher and Lindgren 1972], [Brummer 1984] and [Kiinzi 1990, 1992a, 1992b,1993]; a convenient presentation for our purpose is to be found in [Fletcher and Lindgren 1982]. After Reilly [1972] introduced the notion of zero-dimensionality in bitopological spaces, Birsan [1974] and Halpin [1974] studied the relationship between transitive quasi-uniform spaces and zero-dimensional bitopological spaces. In this thesis we define a notion of zero-dimensionality in ordered topological spaces and examine the relationship between transitive quasi-uniform spaces and zero-dimensional ordered topological spaces. To a large extent, our presentation is influenced by the situation in bitopological spaces (cf. [Halpin 1974] and [Birsan 1974]), and uses the commutative diagrams which occur in [Schauerte 1988] and [Brummer 1977, 1982]. We also study strongly zero-dimensional ordered topological spaces and their relation with functorial quasi-uniformities. In this respect, our results are influenced by those of [Fora 1984], [Banaschewski and Brummer 1990] and [Kiinzi 1990] for strongly zero-dimensional bitopological spaces.

The close relationship between the theory of fixed points and the theory of coincidences of maps is well known. This presentation is aimed at recording one of the less well documented approaches to fixed point theory as extended to the more general situation of coincidences. The approach referred to is that by way of the Universal Covering Spaces. The existing theory of coincidences is geometrically well realised in this setting and after some consideration, the necessary extensions and generalizations of the techniques as utilized in fixed point theory lead to an appealing conceptual notion of "essentiality of coincidence classes". Many hints have been made in the literature (see [1] and "On the sharpness of the Δ₂ and Δ₁ Nielsen Numbers" by Robin Brooks, J.Reine Angew. Math. 259, (1973), 101-108.) that lifts of mappings and the theory of fibres and related topics lend themselves to coincidence theory. It is the intention of this presentation to follow some of the basic properties through this approach and to show, wherever it is thought desirable, the ties between this and two of the existing approaches - for example, in the definition of the Nielsen Number, which is fundamental to both fixed point theory and coincidence theory.

Bibliography: pages 37-38. / The study of reflections in abstract category theory is widespread, and has often been used to study the concrete notion of "completion of an object" that occurs in. many fields of Mathematics, such as the Cech-Stone compactification of a Tychonoff space ([Cech 37]) or the completion of a uniform space ([Weil 38]). More recent work relating reflections to completions was published by Brummer and Giuli [Brummer Giuli 92], and in this thesis many of their ideas are extended to the more general setting of quasireflections (Bargenda 94]. In particular, one would like to view the well-known concept of an injective hull as a "completion", and this can be accomplished via a Galois correspondence between such hulls on one hand, and quasireflections on the other. Thus the theory of completion of objects can be extended to include many widely studied and significant examples, the most paradigmatic of which is the Mac Neille completion of a partially ordered set [Mac Neille 37]. These ideas are presented in chapters 1 and 2 of the present thesis. Further, the widely accepted characterization of factorization structures for sources in terms of certain colimits (pushouts and cointersections) was successfully extended to a characterization of factorization structures relative to a subcategory in the PhD thesis of Vaclav Vajner ([Vajner 94]). In chapter 3 of this thesis, the characterization is further generalized to include quasifactorization structures relative to a subcategory. This result relates to the results of chapters 1 and 2 via an important result of Bargenda's, which proves a Galois correspondence between quasireflective subcategories and relative quasifactorization structures (proposition 3.7).

Inductive systems and inductive limits have by now become fairly well established in the general theory of topological vector spaces. It is a branch of Functional Analysis which is receiving a reasonable amount of attention by modern mathematicians. It is of course a very interesting subject of its own accord, but is also useful in solving problems and proving theorems which one does not suspect are intimately related to it. As an example we can consider the proof of the non-existence of a countably infinite dimensional metrisable barrelled space.

Chapter 1 deals with basic properties of the category of quasi-uniform spaces and its full subcategory Qut of transitive quasi-uniform spaces. Chapter 2 concerns Fletcher's construction. We extend the class of covers to which this construction may be applied and study the functoriality of the construction. The major result is that every right inverse of the forgetful functor Qut--->Top is obtainable by the extended Fletcher construction. In Chapter 3 we characterize pairwise zero dimensional bitopological spaces as those admitting transitive quasi-uniformities. An initiality characterization of pairwise zero dimensional bitopological spaces suggested by Brümmer leads to a description of the coarsest right inverse of the forgetful functor. In Chapter 4 we discuss countably based transitive quasi-uniformities, in that they relate to quasi-metrization. We elaborate on a result of Fletcher and Lindgren (1972) and obtain a bitopological analogue. In Chapter 5 we bring together a number of topics which relate to our previous chapters and point to further questions.

Bibliography: pages 74-77. / Chapter 0 contains a summary of well-known terminology which will subsequently be used in the thesis. In Chapter 1 we begin by describing how topological categories may be viewed as categories of models corresponding to theories into the category of complete lattices. This leads naturally to the study of categories of T-models corresponding to theories into categories other than the category of complete lattices. It is shown, for example, that a concrete category corresponds to a poset-valued theory just in the case that it is (co)fibration complete. This shows that a concrete category is of the form Mod(T) only for poset-valued theories T. We make some technical observations regarding the correspondence between transformations and concrete functors. In particular, the fact that natural transformations between theories are in a bijective correspondence to finality preserving concrete functors between their respective categories of models will be of importance in Chapter 2. A theoretic interpretation is given of those categories which are (co)reflective modifications of certain concrete categories. Chapter 2 deals with the theoretic interpretation of certain topological completions of concrete categories. These are described in abstract theoretic terms using the correspondence between transformations and concrete functors. We also consider how concrete categories are embedded into (co)fibration complete categories. These "weak" completions have the nice property that they are always legitimate. For an arbitrary concrete category, the relationship between its topological completions and the various order-theoretic completions of its fibres is rather weak. However, if one assumes some additional structure properties, such as (co)fibration completeness, then the concepts of a categorical completion and an order-theoretic completion are more closely related, as shown by the result that for certain kinds of cofibrations, taking the universal order-theoretic completion of each fibre even yields the universal final topological completion. Chapter 3 is entirely concerned with the main goal of this thesis. We study so-called "convenient" topological categories, i.e., topological categories with additional structure. The purpose is to characterise each such type of category as a category of T-models for some theory T which satisfies a special "preservation" property with respect to pullbacks. The cartesian closed topological categories are characterised as those categories of T-models where the associated theory T sends a pointwise pullback of any regular sink into product covering family of diagrams. The concretely cartesian closed topological categories are characterised as those for which the associated theory T sends the pointwise pullback of an arbitrary sink into a product covering family. We also characterise the concretely cartesian closed categories by means of a certain natural transformation, given by the product of two structures. Perhaps the most satisfactory result of this Chapter is the characterisation of the universally topological categories. The theories corresponding to these categories may be described in two ways : firstly, they are shown to be frame-valued, send pullbacks into covering diagrams, and send morphisms into downset-preserving, cover-reflecting maps; secondly, they are shown to send the pointwise pullback of any sink into an order-covering diagram. Similarly, the concrete quasitopoi may be characterised by those theories which send the pullback of any regular sink into an order-covering family of diagrams. Finally, we consider hereditary topological categories. These are characterised as categories of T-models for which the theory T preserves terminal objects and sends the pointwise pullback of an arbitrary sink along an embedding into a weakly covering diagram family. In this context, a notion of strong heredity is introduced and characterised by a frame-valued theory sending pullbacks along monomorphisms into order-covering diagrams.

Indiana University-Purdue University Indianapolis (IUPUI) / The passenger safety is one of the most important factors in the automotive industries. At the same time, in order to improve the overall efficiency of passenger cars, lightweight structures are preferred while designing the vehicle structures. Among various structural optimization techniques, topology optimization techniques are usually preferred to address the issue of crashworthiness. The hybrid cellular automaton (HCA) is a truly nonlinear explicit topology design method developed for obtaining conceptual designs of crashworthy vehicle components. In comparison to linear implicit methods, such as equivalent static loads, and partially nonlinear implicit methods, the HCA method fully captures all the relevant aspect of a fully nonlinear, transient dynamic crash simulation. Traditionally, the focus of the HCA method has been on designing load paths in the crash component that increase the uniform internal energy absorption ability; thus far, other relevant crashworthiness indicators such as peak crushing force and displacement have been less studied. The objective of this research is to extend the HCA method to synthesize load paths to obtain the different acceleration-displacement profiles, which allow reduced peak crushing force as well as reduced penetration during a crash event. To achieve this goal, this work introduces the concept of achieving uniform energy distribution at variable design simulation times. In the proposed work, the design time is used as a new design parameter in topology optimization. The desired volume fraction of the final design and the design time provided two dimensional design space for topology optimization, which is followed by the formulation of design of experiments (DOEs). The nonlinear analyses of the corresponding DOEs are performed using nonlinear explicit code LS-DYNA, which is followed by topology synthesis in HCA. The performance of the resulting structures showed that the short design times lead to design obtained by linear optimizers, while long simulation times lead to designs obtained by the traditional HCA method. To achieve the target crucial crash responses such as maximum acceleration and maximum displacement of the structure under the dynamic load, the geological predictor has been implemented. The concept of design time is further developed to improve structural performance of a vehicle component under the multiple loads using the method of multi-design time. Finally, the design time is implemented to generated merged designs by performing binary operations on topology-optimized designs. Numerical example of the simplified front frame is utilized to demonstrate the capabilities of the proposed approach. / 2019-11-21

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