Geodesic
Multifibrations. A class of shape fibrations with the path lifting property
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 1, pp. 29-38 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 54C56, 55P55, 55R05
Keywords: shape fibration; multivalued map; path lifting property; strong shape 
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Giraldo, Antonio; Sanjurjo, Jose M. R. Multifibrations. A class of shape fibrations with the path lifting property. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 1, pp. 29-38. http://geodesic.mathdoc.fr/item/CMJ_2001_51_1_a2/

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