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Affine sets: the structure of complete objects and duality. (English) Zbl 1171.54011

In the first section of this paper basic facts about categories of affine sets over an algebraic theory are given. Then a theorem for the existence of completions in this category is proven. Following this idea the authors provide an internal characterization of their complete objects. In this connection the “Zariski closure” comes into play with a number of additional useful properties. As a consequence the existence of completions in several topological categories are recovered together with description of their complete objects. At last a general theorem of “Duality” leads us to many well-known “internal” dualities (e.g. Stone duality, Tarski-duality).

MSC:

54B30 Categorical methods in general topology
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
06D50 Lattices and duality
55M05 Duality in algebraic topology
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