More generally, are there any remarkable properties enjoyed by the $\infty$category of stable $\infty$categories?
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No. It's pointed by the zero category, but then taking loops of a stable category C (in the sense of the pullback of 0 > C along itself) always gives the zero category, so loops is definitely not an equivalence. One important structural feature of the category of stable categories along these lines is that it has some nice cofiber sequences (Verdier localization sequences), but I'm not sure the categorical properties these satisfy have been axiomatized (into a possible definition of ``stable (infty,2)category''?) 

