## Fibrations and classifying spaces: Overview and the classical examples.(English)Zbl 0991.55010

The paper under review is the fifth in a series of papers by the author [ibid. 34, No. 2, 127-151 (1993; Zbl 0782.55006); 39, No. 2, 83-116 (1998; Zbl 0906.55010), No. 3, 181-203 (1998; Zbl 0917.55009), No. 4, 271-286 (1998; Zbl 0926.55011)]. The objects of study of this series are so-called universal $${\mathcal F}$$-fibrations. Roughly speaking, an $${\mathcal F}$$-fibration should be thought of as a fibration where the fibers are required to be objects in a specified category of spaces $${\mathcal F}$$; and a map of $${\mathcal F}$$-fibrations should be thought of as a map that fiberwise is given by a map in $${\mathcal F}$$. In the case where the category $${\mathcal F}$$ is given by free transitive $$G$$-spaces and $$G$$-maps for a topological group $$G$$, the associated $${\mathcal F}$$-fibrations correspond to principal $$G$$-bundles. Other classical notions that can be described in the general set-up of $${\mathcal F}$$-fibrations are principal fibrations, Hurewicz fibrations and sectioned fibrations.
As it is well known, universal principal $$G$$-bundles were first constructed by Milnor. Dold has shown that a principal $$G$$-bundle is universal if and only if the total space of the principal $$G$$-bundle is contractible, which implies that a universal principal $$G$$-bundle also can be obtained by a geometric bar construction. Universal fibrations for Hurewicz or sectioned fibrations with prescribed (pointed) homotopy type have been constructed by various authors. Typically there are two methods that potentially lead to universal $${\mathcal F}$$-fibrations. The first method uses the geometric bar construction; the second method uses the Brown representability theorem. Both methods have advantages and disadvantages. The main goal of this series of papers is to show that under mild assumptions on the category $${\mathcal F}$$ both methods can be used to construct a universal $${\mathcal F}$$-fibration. In particular, it follows that there is a universal $${\mathcal F}$$-fibration and moreover it follows that this universal $${\mathcal F}$$-fibration will have all the good properties that can be derived from the geometric bar construction approach, as well as all the good properties that can be derived from the Brown representability approach.
Most of the work that goes into the proof of the main result has been carried out in the preceding papers. This paper essentially provides an overview on what has been achieved, and it concentrates on putting the obtained results into context. In particular, the author dicusses the cases where $${\mathcal F}$$-fibrations correspond to principal $$G$$-bundles and principal fibrations as well as the cases where $${\mathcal F}$$-fibrations correspond to Hurewicz and sectioned fibrations with an arbitrary chosen (pointed) fiber homotopy type. He shows that the main result applies in all of the four cases, which then leads to a very general and satisfactory construction of a corresponding universal fibration.

### MSC:

 55R65 Generalizations of fiber spaces and bundles in algebraic topology 55R05 Fiber spaces in algebraic topology 55R10 Fiber bundles in algebraic topology 55R15 Classification of fiber spaces or bundles in algebraic topology

### Keywords:

enriched fibrations; structered fibrations

### Citations:

Zbl 0782.55006; Zbl 0906.55010; Zbl 0917.55009; Zbl 0926.55011
Full Text:

### References:

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