Cellular approximation for animae with CW-structure

Remark 3.3.6 (Cellular approximation). We will freely use the cellular approximation theorem: every pointed map between pointed cell complexes is homotopic to a cellular one. Together with the comparison of classical and intrinsic homotopy groups in Remark 2.4.19, it has the following consequence for an anima \(X\) with a CW-structure, which is all we will need. The inclusion of the \(n\)-skeleton induces a map \[ [S^k,X^n]_* \to [S^k,X]_* \] that is a bijection for \(k < n\) and a surjection for \(k = n\); the same holds for the inclusion \(X^n \to X^m\) of skeleta with \(m > n\). More generally, if \(W\) is obtained from an anima \(Z\) by attaching cells of dimension \({}\geq m\), then \([S^k,Z]_* \to [S^k,W]_*\) is a bijection for \(k < m-1\) and a surjection for \(k = m-1\). Equivalently, every pointed map \(S^k \to W\) is homotopic to one factoring through \(Z\) once \(k \leq m-1\), and two such maps that become homotopic in \(W\) already become homotopic after attaching the cells of dimension \(m\).