Fibrations
In this paper the authors give an account of the main ideas of the theory of fibrations, starting with a presentation of the original results obtained by the pioneers (W. Hurewicz, N. Steenrod and others) and ending with an exposition of the more abstract and more recent developments contained in papers by P. Booth, P. Heath, J. P. May and the authors. In the first section the authors discuss the notions of Hurewicz fibration (fibration, for short) and h-fibration (or weak fibration) in the sense of \textit{A. Dold} [Ann. Math., II. Ser. 78, 223-255 (1963; Zbl 0203.254)]. The second section is dedicated to the study of Serre fibrations (due to J.-P. Serre) and quasifibrations (due to \textit{A. Dold} and \textit{R. Thom} [ibid. 67, 239-281 (1958; Zbl 0091.371)]). Several examples are given. Section three deals with some background material necessary to develop fibrations in a very general sense; in particular, the authors discuss the construction of functorial triples, first introduced by \textit{P. Booth}, \textit{P. Heath} and the second author [Lect. Notes Math. 673, 158-167 (1984; Zbl 0399.55001); 168-184 (1984; Zbl 0392.55006)]; these will be crucial for further developments and in particular, will strongly simplify arguments given in the basic paper by \textit{J. P. May} [Mem. Am. Math. Soc. 155 (1975; Zbl 0321.55033)]. The fourth section deals with fibrations in a highly categorical fashion: the authors introduce the notion of category of fibres, (a category F in which the fibres of considered ''fibrations'' are constrained to lie) and develop the theory of F-fibrations. This is such that, according to the category of fibres, the authors retrieve the usual fibrations that arise in practice, including vector and fiber bundles. In Section 5 the authors construct, under certain conditions, a universal fibration for the category of F-fibrations and show that such a universal fibration gives rise to a classification theorem and a theorem which characterizes universal fibrations in terms of total space of a certain functional triple or via an extension property, first developed by \textit{N. Steenrod} for bundles, The topology of fibre bundles (1951; Zbl 0054.071). Finally, in Section 6, the authors indicate an application of this general theory of fibrations to the study of self-equivalences of fibrations and bundles developed by \textit{P. Booth}, \textit{P. Heath} and the authors [Proc. Lond. Math. Soc., III. Ser. 49, 111-127 (1984; Zbl 0525.55005)].
scientific article; zbMATH DE number 3962755