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Characterizations of \({\mathcal F}\)-fibrations. (English) Zbl 0542.55014

We assume that a category of fibres \({\mathcal F}\) is given. This is just a suitable category of spaces in which the fibres of fibrations are constrained to lie. An \({\mathcal F}\)-space is a map p: \(E\to B\) such that each fibre \(E_ b=p^{-1}(b)\) is an object of \({\mathcal F}\). An \({\mathcal F}\)-space p: \(E\to B\) is an \({\mathcal F}\)-fibration if it has the F-covering homotopy property. Proposition 2. An \({\mathcal F}\)-space p: \(E\to B\) is an \({\mathcal F}\)-fibration if, and only if, p*p is a fibration. This sharpens results of P. I. Booth, P.R. Heath and R. A. Piccinini [Lect. Notes Math. 673, 168-184 (1978; Zbl 0392.55006)] on the characterization of universal fibrations.

MSC:

55R05 Fiber spaces in algebraic topology

Citations:

Zbl 0392.55006
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