Rational spaces in resolving classes

A class of pointed spaces is called a resolving class if it is closed under weak equivalences and pointed homotopy limits. Let R(A)R(A) denote the smallest resolving class containing a space A. We say X is A-resolvable if X   is in R(A)R(A), which induces a partial order on the category pointed spaces. We develop an algebraic criterion for determining if X is A-resolvable when X and A are rational spaces. The goal of this work is to develop some understanding of the structure of this resolvability relation, and in particular to use our algebraic criterion to help us better understand the rational case.