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Hereditary topological constructs. (English) Zbl 0662.18003
General topology and its relations to modern analysis and algebra VI, Proc. 6th Symp., Prague/Czech. 1986, Res. Expo. Math. 16, 249-262 (1988).
[For the entire collection see Zbl 0632.00016.]
O. Wyler [Lect. Notes Math. 540, 699-719 (1976; Zbl 0354.54001)] discussed the problem of embedding a concrete category with certain topological properties into a quasitopos (in the sense of J. Penon [C. R. Acad. Sci., Paris, Sér. A 276, 237-240 (1973; Zbl 0268.18013)]). This paper gives new answers for concrete topological categories with small fibres and constant maps being morphisms: such a category is a quasitopos iff it is hereditary (i.e., its quotients and coproducts are) and cartesian closed. Hereditarily topological categories are described externally as injective objects in a quasi-category of concrete categories, and the existence of injective hulls is proved.
Reviewer: W.Tholen

MSC:
18B30 Categories of topological spaces and continuous mappings (MSC2010)
18D99 Categorical structures
54B30 Categorical methods in general topology
18B15 Embedding theorems, universal categories