$ CW_0$ -complexes are analogs of $ CW$ -complexes, in which the “building blocks” are the rational disks $ D^{n+1}_0$ whose boundaries are given by $ \partial D^{n+1}_0= S^n_0$ , where $ S^n_0$ is a rationalization of the $ n$ -sphere. In partcular, $ CW_0$ -complexes are rational spaces. One can read more about this construction in Hess – Rational homotopy theory: a brief introduction, Def 1.3 or in FHT – Rational Homotopy Theory, Ch 9.

Basically, a simply connected $ CW_0$ -complex $ X$ is a space endowed by a filtration $ $ X(1)\subseteq X(2)\subseteq\cdots\subseteq X =\bigcup X(i)$ $ such that there exists a pushout square $ \require{AMScd}$ \begin{CD} \bigsqcup_{j\in J_n} S^n_0 @>>> X(n)\ @V V V @VV V\ \bigsqcup_{j \in J_n}D^{n+1}_0 @>>> X(n+1) \end{CD}

My question is if there is any analog of the cellular approximation theorem saying that a map $ f\colon X\to Y$ between two $ CW_0$ -complexes is homotopic to to a map $ f’$ that respects the filtration (i.e $ f'(X(n))\subseteq Y(n)$ ).

This question makes also sense in a more general form. Given a set of primes $ \mathcal P$ , we may replace the $ CW_0$ -complexes by $ CW_\mathcal P$ -complexes, whose building blocks are $ \mathcal P$ -local disks (whose boundaries are $ \mathcal P$ -local spheres), etc, and ask the same question.