Abstract:
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.
Reference:
Vajner, V. 1989. A Wyler-type approach to categorical topology. University of Cape Town.
Bibliography: pages 74-77.