TITLE: What are _all_ of the exactness properties enjoyed by stable $\infty$-categories?
QUESTION [19 upvotes]: Alternate formulation of the question (I think): What's a precise version of the statement: "In a stable $\infty$-category, finite limits and finite colimits coincide"?
Recall that a stable $\infty$-category is a type of finitely complete and cocomplete $\infty$-category characterized by certain exactness conditions. Namely,

There is a zero object $0$, i.e. an object which is both initial and terminal.
Every pushout square is a pullback square and vice versa.

Item (2) takes advantage of a peculiar symmetry of the "square" category $S = \downarrow^\to_\to \downarrow$; namely $S$ can either be regarded as $S' \ast \mathrm{pt}$ where $S' = \cdot \leftarrow \cdot \rightarrow \cdot$ is the universal pushout diagram, or $S$ can be regarded as $S = \mathrm{pt} \ast S''$ where $S'' = \cdot \rightarrow \cdot \leftarrow \cdot$ is the universal pullback diagram. Hence it makes sense to ask, for a given $S$-diagram, whether it is a pullback, a pushout, or both. Item (1) similarly takes advantages of the identities $\mathrm{pt} = \emptyset \ast \mathrm{pt} = \mathrm{pt} \ast \emptyset$.
But I can't shake the feeling that notion of a stable infinity category somehow "transcends" this funny fact about the geometry of points and squares. For one thing, one can use a different "combinatorial basis" to characterize the exactness properties of a stable $\infty$-category, namely:
1.' The category is (pre)additive (i.e. finite products and coproducts coincide)
2.' The loops / suspension adjunction is an equivalence.
True, (2') may be regarded as a special case of (2) -- but it may also be regarded as a statement about the (co)tensoring of the category in finite spaces.
Both of these ways of defining stability say that certain limits and colimits "coincide", and my sense is that in a stable $\infty$-category, all finite limits and colimits coincide -- insofar as this makes sense.
Question: 
Is there a general notion of "a limit and colimit coinciding" which includes

zero objects
biproducts (= products which are also coproducts)
squares which are both pullbacks and pushouts
suspensions which are also deloopings

and if so, is it true that in in a stable $\infty$-category, finite limits and finite colimits coincide whenever this makes sense?
I would regard this as investigating a different sort of exactness to the exactness properties enjoyed by ($\infty$)-toposes. In the topos case, I think there are some good answers. For one, in a topos $C$, the functor $C^\mathrm{op} \to \mathsf{Cat}$, $X \mapsto C/X$ preserves limits. Foir another, a Grothendieck topos $C$ is what Street calls "lex total": there is a left exact left adjoint to the Yoneda embedding. It would be nice to have similar statements here which in some sense formulate a "maximal" list of exactness properties enjoyed by (presentable, perhaps) stable $\infty$-categories, rather than the "minimal" lists found in (1,2) and (1',2') above.

REPLY [2 votes]: The fundamental stability axiom (pushouts are pullbacks and vice-versa) can be seen geometrically as a particular instance of Verider--Lurie duality for the circle.  So, Lurie's formulation of Verdier duality (summarized below) for general spaces can be thought of as an answer to your question.
Let $X$ be a locally compact Hausdorff space.  Given a presheaf $F$ on $X$ valued in a pointed $\infty$-category $C$ which sends $\varnothing\subseteq X$ to a zero object, we can define a "dual cosheaf"
$$\mathbb DF(U)=\operatorname{cofib}(F(X)\to F(X\setminus U))$$
which is a precosheaf on $X$ valued in $C$ which sends $\varnothing$ to a zero object.  I will ignore the detail that $X\setminus U$ isn't usually an open subset of $X$ (but see Lurie HTT 7.3.4 for why this isn't too bad).  There is a natural map $F\to\mathbb D\mathbb DF$.
When $C$ is stable, $\mathbb D$ sends sheaves to cosheaves and the natural map $F\to\mathbb D\mathbb DF$ is an isomorphism (Lurie HA 5.5.5).  It's likely (just by considering very simple compact Hausdorff spaces and very simple (co)sheaves) that the converse is true (if $\mathbb D$ sends sheaves to cosheaves then $C$ is stable).
Remark: It would indeed be great to replace compact Hausdorff spaces in the above discussion with finite topological spaces.  Is there a duality result for finite topological spaces, or perhaps finite posets with their poset topology?
Remark: There should be a notion of "global stable $\infty$-category" and the initial such category (with marked object(s)) should be the global stable homotopy category (compatible families of $G$-spectra for all compact Lie groups $G$).  Then I expect there is a form of the above discussion where orbispaces or global spaces or local quotient stacks replace compact Hausdorff spaces.