Let p be a fibration of simply connected CW complexes with base B and fibre F both finite. Let aut_1(p) denote the identity component of the space of all fibre-homotopy self-equivalences of p. Let Baut_1(p) denote the classifying space for this topological monoid. We describe a Quillen model (aka DG Lie algebra model) for Baut_1(p) extending a classical result of Sullivan for the monoid aut_1(X). We use our model to prove classification results for the rational homotopy types represented by Baut_1(p) for fixed F and B and also to obtain conditions under which the monoid aut_1(p) is a double loop-space after rationalization.

This is joint work with Urtzi Buijs.