Are there any stable $(\infty,1)$-topoi?

Question

Can a stable $(\infty,1)$-category be an $(\infty,1)$-topos?

Answer 1

No, for essentially the same reason that there are no interesting abelian categories that are also toposes: an abelian category (resp. stable $(\infty, 1)$-category) has a zero object, i.e. an initial object that is also terminal, but initial objects in toposes (resp. $(\infty, 1)$-toposes) are strict, so the only topos (resp. $(\infty, 1)$-topos) that has a zero object is the trivial one.

Answer 2

This is basically Zhen Lin's answer for nontrivial cartesian closed categories: Assume that a nontrivial cartesian closed category $\mathcal{C}$ has a zero object. Then, let $C\in\mathcal{C}$. We would have that $0=0\times C=C$ (the functor $-\times C:\mathcal{C}\to\mathcal{C}$ commutes with empty colimits). Thus $\mathcal{C}$ would be a trivial cartesian closed category; hence any nontrivial cartesian closed category cannot have a zero object.