  Let p be a fibration of simply connected CW complexes with finite base B and
fibre F. Let aut_1(p) denote the identity component of the space of all
fibre-homotopy self-equivalences of p and Baut_1(p) the classifying space for
this topological monoid. We give a differential graded Lie algebra model for
Baut_1(p). We use this model to give classification results for the rational
homotopy types represented by Baut_1(p) and also to obtain conditions under
which the monoid aut_1(p) is a double loop-space after rationalization.
