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Topological improvements of categories of structured sets. (English) Zbl 0632.54008
A category \({\mathcal A}\) with a forgetful functor \(U: {\mathcal A}\to Set\) is called a construct if every constant map between \({\mathcal A}\)-objects is an \({\mathcal A}\)-morphism. Examples include Top, Unif and Prox. These examples are also topological, i.e., they have sufficient initial and final objects. Unfortunately, many topologists and analysts have found that being topological is not enough; that more “convenience properties” are needed for a satisfactory construct. Two of these properties are “Cartesian closedness” and “heredity”. The author discusses how one might embed constructs into various convenient “hulls” (“completions” in some sense). Many examples are given, and constructions described. Foundational problems are also briefly discussed, and there are many references to the current literature.
Reviewer: P.Bankston

MSC:
54B30 Categorical methods in general topology
18B30 Categories of topological spaces and continuous mappings (MSC2010)
18B15 Embedding theorems, universal categories
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
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