Abstract:
Reflective subcategories originated as a formal mathematical concept in the 1960's. Perhaps the first abstract definition of reflectivity can be attributed to P. Freyd who, in [Freyd 1960] and [Freyd 1964], gave a definition in terms of reflection arrows. Already in [Isbelll964] the general definition of a reflective subcategory was applied to some concrete situations, and used to formulate one of the first problems concerning reflectivity, namely, whether the intersection of (full, isomorphism-closed) reflective subcategories of the category of uniform spaces is again reflective. This problem, together with analogous questions posed in other contexts (e.g., by H. Herrlich for the category of topological spaces), led to the formulation of the reflective hull problem for subcategories in general, namely, whether a given subcategory is contained in a smallest reflective supercategory. Much of the research concerned with reflectivity and the reflective hull problem considers sufficient conditions for a category such that the reflectivity of (certain) subcategories can be described and the existence of reflective hulls can be guaranteed. These conditions are usually given in terms of (co)completeness and (co)wellpoweredness (see, for example, [Tholen 1987), [Kelly 1987]). A primary objective of this thesis is to provide sufficient and necessary conditions, formulated in subcategory-related terms, for the reflectivity of a given subcategory, and for the characterisation of the existence of reflective hulls. Our approach to finding appropriate descriptions of reflective hulls is essentially a constructive one, in the sense that we attempt to "generate" the reflective hull of a given subcategory (and hence give a concrete description of the hull) by means of certain closure processes applied to the given subcategory. We should also emphasise that our philosophy is not a conservative one in that, apart from applications of our constructions to particular situations, we make as few global assumptions as possible in our considerations. Intuitively, there are several ways in which reflectivity can be viewed as a mathematical concept; the results in this thesis emphasise these points of view. First, reflectivity may be viewed as a completeness property, i.e., as a kind of limit procedure; we study the correspondence between reflective hulls and closures of subcategories under certain types of limits. Reflectivity may also be considered as a cocompleteness property; appropriately we also consider the closure of a given subcategory under certain kinds of colimits and its relation to a possible reflective hull. Both of these constructions are generalisations of natural descriptions of reflection arrows in the special case of partially-ordered classes. Finally, reflectivity can be considered as a (subcategory-related) factorisation property; in this context we consider closures of a subcategory in terms of factorisations relative to the given subcategory, and, related to this, closures under special kinds of colimits relative to the given subcategory. In this thesis we also obtain results concerning the relation between reflectivity and weaker concepts; in particular results concerning intersections of reflective subcategories, and reflective hulls of almost reflective subcategories, are given, and applied to concrete situations, for example, the following problem posed in Rosicky and Tholen 1988: Is the category of complete Boolean algebras an intersection of reflective subcategories of the category of frames?
Reference:
Vajner, V. 1994. Approaches to reflective hulls of subcategories. University of Cape Town.
Bibliography: pages 77-81.