reference request – Product Smash and integers in a Grothendieck $ ( infty, 1) $ – topos

Let $ mathcal {H} $ to be a Grothendieck $ ( infty, 1) $-topos. According to this page in nlab, for all $ X in mathcal {H} $, the object suspension $ Sigma X $ is homotopy equivalent to the smash product $ B mathbb {Z} wedge X $, or $ B mathbb {Z} $ is "the discrete group's classification space of integers". In addition, for any sharp object $ X in mathcal {H} _ * $ and any group object $ G in Grp ( mathcal {H}) $, the article says we can "train the tensor product $ X otimes G in Grp ( mathcal {H}) $. "

My problem is this: none of these terminologies are explained, and the page does not provide any reference. More 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 as the map $ mathcal {H} to mathcal {H} _ * $ is monoidal symmetric (here $ mathcal {H} $ is given the Cartesian monoidal structure), but this is only a conjecture.

Is there a reference where all these notions are defined?