Let $\mathcal{H}$ be a Grothendieck $(\infty,1)$-topos. According to this page in nlab, for any $X \in \mathcal{H}$, the suspension object $\Sigma X$ is homotopy equivalent to the smash product $B \mathbb{Z} \wedge X$, where $B \mathbb{Z}$ is the "classifying space of the discrete group of integers." Furthermore, for any pointed object $X \in \mathcal{H}_*$ and any group object $G \in Grp(\mathcal{H})$, the article says we can "form the tensor product $X \otimes G \in Grp(\mathcal{H})$."

My problem is: none of this terminology is explained, nor does the page provide a reference. Specifically, what is $\mathbb{Z}$ in an arbitrary $\infty$-topos? What is the smash product $\wedge$? What is the tensor product $\otimes$? My best guess is that $\otimes$ refers to the unique tensor structure on $\mathcal{H}_*$ such that the map $\mathcal{H} \to \mathcal{H}_*$ is symmetric monoidal (here $\mathcal{H}$ is given the Cartesian monoidal structure), but this is only a guess.

Is there a reference where all these notions are defined?