Let X be a simplicial set, G a simplicial group and 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 ( 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) has the homotopy type of the component of ( 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.