Stable ∞-category

Remarks

Equivalently, $\mathcal{C}$ is stable if it has a zero object and all finite limits and finite colimits, and its pullback squares are exactly its pushout squares.

Example: the ∞-category of spectra is stable. The zero spectrum is a zero object, and spectra have all finite limits and colimits, so clauses 1 and 2 hold. Clause 3 holds because the suspension functor on spectra is an equivalence. In a pointed ∞-category with finite limits and colimits, this forces fiber and cofiber sequences to agree.

Non-example: the ∞-category of pointed spaces has a zero object, the one-point space, and has fibers and cofibers, but clause 3 fails. The cofiber of $S^1 \to *$ is $S^2$, while the fiber of $* \to S^2$ is the loop space $\Omega S^2$. This is not equivalent to $S^1$, because $\pi_2(\Omega S^2) \cong \pi_3(S^2) \cong \Z$ while $\pi_2(S^1) = 0$.

The suspension $\Sigma X$ is the cofiber of $X \to 0$, and the loop object $\Omega X$ is the fiber of $0 \to X$. In a stable ∞-category, $\Sigma$ and $\Omega$ are inverse equivalences.