  We regard the classification of rational homotopy types as a problem in
algebraic deformation theory: any space with given cohomology is a
perturbation, or deformation, of the "formal" space with that cohomology. The
classifying space is then a "moduli" space --- a certain quotient of an
algebraic variety of perturbations. The description we give of this moduli
space links it with corresponding structures in homotopy theory, especially the
classification of fibres spaces with fixed fibre F in terms of homotopy classes
of maps of the base B into a classifying space constructed from the monoid of
homotopy equivalences of F to itself. We adopt the philosophy, later promoted
by Deligne in response to Goldman and Millson, that any problem in deformation
theory is "controlled" by a differential graded Lie algebra, unique up to
homology equivalence (quasi-isomorphism) of dg Lie algebras. Here we extend
this philosophy further to control by sh-Lie-algebras.
