This work was motivated
by the goal of removing the hypothesis of simple connectedness from the
rational homotopy theory of D. Sullivan. To a simply connected space X is
associated it’s rational localization ϕ : X → X0, and to the differential graded
algebra XA(X) of rational polynomial forms on X it’s Sullivan minimal model
ψ : XM → XA(X). It is shown that the minimal model M is dual to the
Postnikov tower of X0. Thus M determines the rational homotopy type of
X.
In the present paper we have eliminated the simply connected hypothesis from
the first part of the theory. Working in the category of semi-simplicial complexes, we
show that if X is a one pointed Kan complex, and P is a family of prime integers,
there exists a semi-P-localization f : X → Xp such that f∗: π1(X) → π1(Xp) is an
isomorphism and f∗: πk(X) → πk(Xp) is P-localization of abelian groups, k ≥ 2.
Semi- P-localization is also characterized by a universal mapping property, and the
fact that f induces isomorphisms on twisted coefficient cohomology whenever the
coefficients are in a Z(P)-module.
