More generally, are there any remarkable properties enjoyed by the $\infty$category of stable $\infty$categories?

$\begingroup$ Problably not, the problem is with the multiplicative structure (composition), e.g. the categories of DGAs or ring spectra are not stable. $\endgroup$ – Fernando Muro Mar 23 '11 at 11:47

$\begingroup$ @Fernando: Can you explain your comment? Are you saying that a DGA (or a ring spectrum) is somehow analogous to a stable $\infty$category? $\endgroup$ – Tom Goodwillie Mar 23 '11 at 12:44

1$\begingroup$ @Fernando: The categories of DGAs or ring spectra aren't stable, but these aren't the same as the categories of functors between their module categories. E.g. the category of RS bimodules (which is stable) maps to the category of functors Smod > Rmod. $\endgroup$ – Tyler Lawson Mar 23 '11 at 15:22

$\begingroup$ @Tom: I'm thinking of the $\infty$category of $\infty$categories as a nonadditive analog of the Morita model category of DGalgebras, or more generally DGcategories. This idea links up with Tyler's comment. @Tyler: you're right, but somehow bimodules represent morphisms, so the stable $\infty$categories or morphisms in the $\infty$category of $\infty$categories would be stable, but not necessarily the $\infty$category of $\infty$categories itself. (This looks like a tongue twister!). I'll try to think of how to properly explain what I mean. I might eventually produce a (counter)exm. $\endgroup$ – Fernando Muro Mar 23 '11 at 16:16
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''?)

$\begingroup$ As pretriangulated dgcategories are models for klinear stable (infinity,1)categories (for k being a field of characteristic zero), maybe it is of some relevance to compare what Tamarkin and Batanin had to say in their works on the question "What the dgcategories make ?" $\endgroup$ – Zoran Skoda Mar 23 '11 at 19:41

1$\begingroup$ To elaborate on Dustin's second paragraph, if $A\to B\to C$ is a Verdier localization sequence, then the square $$\matrix{A&\longrightarrow&B\cr \downarrow &&\downarrow\cr 0&\longrightarrow&C}$$ is both a pushout square and a pullback square, a suggestively stable property. I am curious which squares of stable categories should be considered exact. In a stable category, the question of exact squares reduces to exact sequences, but I don't see any symmetric way of making such a definition for squares of stable categories. $\endgroup$ – Ben Wieland Mar 24 '11 at 0:32