Homotopy groups of an anima

Definition 2.4.18 (Homotopy groups of an anima). Let \(X \in \An _*\) be a pointed anima and \(n \geq 0\). The \(n\)-th homotopy group of \(X\) is defined as \[ \pi _n(X) := [S^n,X]_*. \] We also write \(\pi _n(X,x)\) if we wish to make explicit the basepoint \(x\) of \(X\).

Since \(S^n = \Sigma ^n(S^0)\), the adjunction Lemma 2.4.8 implies that \[ \pi _n(X) \cong \pi _0(\Omega ^n X), \] where the functor \(\Omega ^n\) is defined inductively as the \(n\)-fold iteration of \(\Omega \).