Proposition 2.4.22 (Whitehead theorem for animae). A morphism \(f\colon X \to Y\) of animae is an isomorphism if and only if it induces a bijection \(\pi _0(X) \xrightarrow {\cong } \pi _0(Y)\) on path components and for every \(x \in X\) it induces an isomorphism \(\pi _n(X,x) \xrightarrow {\cong } \pi _n(Y,f(x))\) on higher homotopy groups for all \(n \geq 1\).
Proof. By Corollary 2.3.7, the map \(f\) is the image under \(\Pi _{\infty }\colon \Cell \to \An \) of a continuous map between cell complexes. In light of Remark 2.4.19, the claim is thus an instance of the usual Whitehead theorem for cell complexes, recorded in Theorem 2.3.3. □
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