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Multifibrations. A class of shape fibrations with the path lifting property. (English) Zbl 1079.55503
Summary: In this paper we introduce a class of maps possessing a multivalued homotopy lifting property with respect to every topological space. We call these maps multifibrations and they represent a formally stronger concept than that of shape fibration. Multifibrations have the interesting property of being characterized in a completely intrinsic way by a path lifting property involving only the total and the base space of the fibration. We also show that multifibrations (and also, with some restrictions, shape fibrations) have a lifting property for homotopies of fine multivalued maps. This implies, when the spaces considered are metric compacta, that the possibility of lifting a fine multivalued map is a property of the corresponding strong shape morphism and not of the particular map considered.
MSC:
 55P55 Shape theory 55R05 Fiber spaces in algebraic topology 54C56 Shape theory in general topology
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References:
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