  The purpose of this paper is to generalise Sullivan's rational homotopy
theory to non-nilpotent spaces, providing an alternative approach to defining
Toen's schematic homotopy types over any field k of characteristic zero. New
features include an explicit description of homotopy groups using the
Maurer-Cartan equations, convergent spectral sequences comparing schematic
homotopy groups with cohomology of the universal semisimple local system, and a
generalisation of the Baues-Lemaire conjecture. For compact Kaehler manifolds,
the schematic homotopy groups can be described explicitly in terms of this
cohomology ring, giving canonical weight decompositions. There are also notions
of minimal models, unpointed homotopy types and algebraic automorphism groups.
For a space with algebraically good fundamental group and higher homotopy
groups of finite rank, the schematic homotopy groups are shown to be
\pi_n(X)\otimes k.
