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Hereditary topological categories and topological universes. (English) Zbl 0621.54007
Topological constructs are hereditary (i.e. have hereditary quotients and coproducts) if and only if partial morphisms are representable (by suitable one-point extensions). The author thoroughly investigates one- point extensions to provide useful criteria which help to decide whether a given topological construct is hereditary or not. He then applies his results to the investigation of bi(co)reflective subcategories of hereditary topological constructs. He supplies necessary and sufficient conditions for bi(co)reflective full subconstructs of hereditary topological constructs to be hereditary. In the reflective case, preservation of certain subobjects by the reflector is a sufficient and - in case the subcategory is finally dense - also a necessary condition.
Reviewer: H.Herrlich

54B30 Categorical methods in general topology
18B25 Topoi
18A22 Special properties of functors (faithful, full, etc.)
54B10 Product spaces in general topology
18D99 Categorical structures
54A05 Topological spaces and generalizations (closure spaces, etc.)
Full Text: DOI
[1] Adémek J., Comment. Math. Univ. Carolinae 27 pp 235– (1986)
[2] Bentley H. L., Comment. Math. Univ. Carolinae 17 pp 207– (1976)
[3] Bourdaud G., Cahiers Topol. Géom. Diff. 16 pp 107– (1975)
[4] DOI: 10.1016/0022-4049(72)90021-7 · Zbl 0236.18004
[5] DOI: 10.1007/BFb0063100
[6] Dubuc E. J., Applications of Sheaves (Proc. Durham 1977) pp 239– (1977)
[7] Herrlich H., Math. Colloq. Univ. Cape Town 9 pp 1– (1974)
[8] Herrlich H., General Topology and its Relations to Modern Analysis and Algebra V (Proc. Prague 1981) pp 279– (1982)
[9] DOI: 10.1016/0166-8641(83)90057-3 · Zbl 0538.18004
[10] Herrlich H., Topological improvements of categories of structured sets · Zbl 0632.54008
[11] Herrlich H., Hereditary topological constructs (1986) · Zbl 0662.18003
[12] DOI: 10.1080/16073606.1979.9631571 · Zbl 0407.54006
[13] Kent D. C., Fund. Math. 2 pp 125– (1964)
[14] DOI: 10.4153/CJM-1975-139-9 · Zbl 0294.18002
[15] Nel L. D., Contemporary Math. 30 pp 244– (1984)
[16] Nel L. D., Categorical Topology (Proc. Toledo 1983) pp 408– (1984)
[17] Nel L. D., Enriched locally convex structures · Zbl 0608.46045
[18] Penon J., Acad. Sc. Paris, Sér. A 276 pp 237– (1973)
[19] Penon J., Cahiers Topol. Géom. Diff. 18 pp 181– (1977)
[20] Schwarz F., Categorical Topology (Proc. Berlin 1978). Lecture Notes Math. 719 pp 345– (1979)
[21] DOI: 10.1080/16073606.1982.9632270 · Zbl 0521.54005
[22] DOI: 10.1080/16073606.1983.9632302 · Zbl 0521.54006
[23] Schwarz F., Funktionenräume und exponentiale Objekte in punktetrennend initialen Kategorien (1983)
[24] Schwarz F., Sigma Series Pure Math: 5, in: Categorical Topology (Proc. Toledo 1983) pp 505– (1984)
[25] Schwarz F., Heredity in categories of convergence spaces
[26] DOI: 10.1007/BFb0080884
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