Fibrations and geometric realizations

Abstract

There is a folk theorem associated to the construction of classifying spaces for topological groups which says that a map of simplicial spaces which is a fibration in every degree has a fibration as its geometric realization. Peter May [GILS] has given a useful form of this which involves quasifibrations. This result has seemed to me a very interesting one, as it relates two rather opposite types of operations. On the one hand, it involves the geometric realization, which is defined by mapping into other things, and on the other hand, it involves fibrations, which are characterized by the properties of maps of other things into them. Any theorem which mixes "left" and "right" mapping properties should be expected to be difficult to prove. Since what is really wanted in applications is a homotopy theoretic result, the possibilities for complication are almost infinite.