Let $\mathcal{H}$ be a Grothendieck $(\infty,1)$-topos. According to [this page](https://ncatlab.org/nlab/show/suspension+object) 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?



Smash product and the integers in a Grothendieck $(\infty, 1)$-topos
To summarize the comments so far:


* In an arbitrary $(\infty,1)$-topos $\mathcal{E}$, the integers can be defined as the loop space of the circle $S^1\_\mathcal{E}$, which itself is given as the (homotopy) pushout of two copies of the map $\mathrm{pt}\sqcup \mathrm{pt} \to \mathrm{pt}$ in $\mathcal{E}$. Otherwise, one can calculate them as the image of the usual $\mathbb{Z}$ under the inverse image $\mathcal{S}paces \to \mathcal{E}$ of the canonical map to $\mathcal{S}paces$. Note that in this definition, $\mathbf{B}\mathbb{Z}$ *is* $S^1\_\mathcal{E}$, and has a canonical (up to equivalence) basepoint.
* The smash product of a pointed object $X$ in $\mathcal{E}$ and $S^1\_\mathcal{E}$ is defined as the (homotopy) coequaliser of the two canonical maps
$$
X \vee S^1\_\mathcal{E} \mathrel{\mathop{\rightrightarrows}^{\mathrm{pt}}\_{\mathrm{incl.}}} X\times S^1\_\mathcal{E}
$$
where the wedge sum $X \vee S^1\_\mathcal{E}$ is the (homotopy) pushout of $X\leftarrow \mathrm{pt} \to S^1\_\mathcal{E}$.


