Homotopy groups of animae and spaces

Remark 2.4.19. Let \((Y,y_0)\) be any pointed topological space, and let \(X := \Pi _{\infty }(Y,y_0) \in \An _*\). Then for every \(n \geq 0\) there is a natural isomorphism \[ \pi _n(X) \;=\; [S^n,X]_* \;\cong \; [S^n,Y]_* \] with the classical \(n\)-th homotopy group of \((Y,y_0)\). Indeed, Proposition 2.4.12 gives natural isomorphisms \[ \Omega ^n X \cong \Pi _{\infty }(\Omega ^nY). \] The path components of the underlying anima of a topological space agree with its usual path components: after choosing a cell-complex approximation, this follows from Corollary 2.3.7 by mapping out of a point. It follows that \[ \pi _n(X)\cong \pi _0\Pi _{\infty }(\Omega ^nY)\cong \pi _0(\Omega ^nY)=[S^n,Y]_*. \]