Homotopy group objects | Higher Topos Theory notes

Definition 3.23.

Let $X \in T$ and let $n \geq 0$ . Define the $$n$$ -th homotopy object $\pi_n(X)$ as the following static ( $$0$$ -truncated) object of $T_{/X}$ :

\pi_n(X) := \tau_0\bigl(X^{S^n} \to X\bigr) \qin (T_{/X})_{\leq 0} \quad \subseteq \quad T_{/X}.

Here $X^{S^n}$ denotes the cotensoring of $$X$$ by $$S^n$$ , cf. Remark 3.3. Since $$S^n$$ is an $$E_n$$ -cogroup in $\An$ and the map $* \to S^n$ is a map of $$E_n$$ -cogroups, $\pi_n(X)$ is an $$E_n$$ -group. In particular:

$\pi_0(X)$ is a pointed object,
$\pi_1(X)$ is a group,
$\pi_n(X)$ is an abelian group for $n \geq 2$ .

For this reason, we will often somewhat abusively speak of homotopy group objects, even though $\pi_0(X)$ is only a pointed object.