## 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

Zbl 0392.55006
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