Quasireflections and quasifactorizations
Master Thesis
1996
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University of Cape Town
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Abstract
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).
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Bibliography: pages 37-38.
Reference:
Henning, P. 1996. Quasireflections and quasifactorizations. University of Cape Town.