Let X be a simplicial set, G a simplicial group and W ¯ G WG the classifying complex of G . Then it is well known [ 1 ], [ 3 ] that the principal fibrations with base X and group G are classified by the components of the function complex ( W ¯ G ) X {( WG)^X} . The aim of the present note is to prove the following complement to this result (1.2): Let p be a principal fibration with base X and group G, and let aut p be its simplicial group of automorphisms (which keep the base fixed). Then W ¯ ( aut p ) W({aut}\,p) has the homotopy type of the component of ( W ¯ G ) X {( WG)^X} which (see above) corresponds to p . A similar result holds for ordinary fibrations.
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Dror et al. (1980) studied this question.