# Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)

$$CW_0$$-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.