The pullback closure, perfect morphisms and completions

Doctoral Thesis


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University of Cape Town

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.

Bibliography: pages 92-97.