Proposition 9.2.19. In the situation of Lemma 9.2.18, the canonical morphism \[ \colim _n \Pi _{\infty }(Y_n) \longrightarrow \Pi _{\infty }(Y) \] is an isomorphism of animae.
Proof. We use the Whitehead theorem for animae. The compactness lemma, applied to the point and to the interval, shows that the canonical morphism induces a bijection on path components. Fix a basepoint represented by some \(y\in Y_N\). For every \(q\geq 1\), compactness of \(S^q\) and \(S^q\times [0,1]\) shows that maps \(S^q\to Y\) and homotopies between them factor through finite stages. Consequently, \[ \colim _{n\geq N}\pi _q(Y_n,y) \xrightarrow {\cong } \pi _q(Y,y). \] By Remark 2.4.19, these are the homotopy groups of the corresponding underlying animae. Homotopy groups of pointed animae commute with filtered colimits, so the left-hand side is also the \(q\)-th homotopy group of \(\colim _n\Pi _{\infty }(Y_n)\) at \(y\). The claim now follows from Proposition 2.4.22. □
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