This paper establishes that the
homotopy category of rational differential graded commutative coalgebras is
equivalent to the homotopy category of rational differential graded Lie algebras which
have a nilpotent completion as homology. This generalizes a result which Quillen
proved in the simply connected case. When combined with Sullivan’s work on
rational homotopy theory, our result shows that the homotopy category of rational
differential graded Lie algebras with nilpotent finite type homology is equivalent to
the rational homotopy category of nilpotent topological spaces with finite type
rational homology.
Our results include the construction of minimal Lie algebra models for
simply connected spaces, and we show that the rational homotopy groups of a
simply connected CW complex may be calculated from a free Lie algebra
generated by the cells with a differential given on generators by the attaching
maps.
