The $\infty$-category $\operatorname{Sp}$ of spectra is the prototypical stable $\infty$-category. It is the stabilization of pointed spaces: $\operatorname{Sp} = \lim(\cdots \xrightarrow{\Omega} \mathcal{S}_* \xrightarrow{\Omega} \mathcal{S}_* \xrightarrow{\Omega} \mathcal{S}_*)$.

In $\operatorname{Sp}$, the suspension $\Sigma$ is an equivalence by design. The homotopy groups of a spectrum $E$ are $\pi_n(E) = \operatorname{colim}_k \pi_{n+k}(E_k)$, which can be negative.

For a ring $R$, the derived $\infty$-category $D(R)$ (the $\infty$-categorical enhancement of the derived category) is a stable $\infty$-category. The suspension is the shift functor $\Sigma = [1]$, and the loop functor is $\Omega = [-1]$.

Pushout-pullback squares in $D(R)$ correspond to exact triangles: $A \to B \to C \to A[1]$ is an exact triangle iff the square with $A \to B$, $0 \to C$, and the connecting map is both a pushout and a pullback.

For a ring spectrum $R$ (an $E_1$-ring in the $\infty$-categorical sense), the $\infty$-category $\operatorname{Mod}_R$ of $R$-module spectra is stable. When $R$ is an Eilenberg--MacLane spectrum $HR$ for an ordinary ring $R$, $\operatorname{Mod}_{HR} \simeq D(R)$.

For a scheme $X$, the $\infty$-category $\operatorname{QCoh}(X)$ of quasi-coherent complexes is stable. The suspension is the shift $[1]$, and exact triangles correspond to short exact sequences of complexes.

If $\mathcal{C}$ is stable and $K$ is any simplicial set, then $\operatorname{Fun}(K, \mathcal{C})$ is stable. In particular, the $\infty$-category $\operatorname{Fun}(\mathcal{A}, \operatorname{Sp})$ of "spectral presheaves" is stable.

The zero object is the constant functor at $0$, and suspension/loop are computed pointwise.

In $D(R)$, a fiber sequence $A \to B \to C$ gives the familiar long exact sequence in homology:

$\cdots \to H_{n+1}(C) \to H_n(A) \to H_n(B) \to H_n(C) \to H_{n-1}(A) \to \cdots$

The connecting homomorphism $H_{n+1}(C) \to H_n(A)$ comes from the boundary map in the exact triangle.

In $\operatorname{Sp}$, the cofiber sequence $S^0 \xrightarrow{2} S^0 \to M(2)$ (where $M(2)$ is the mod-$2$ Moore spectrum) gives the long exact sequence:

$\cdots \to \pi_{n+1}(M(2)) \to \pi_n(S^0) \xrightarrow{2} \pi_n(S^0) \to \pi_n(M(2)) \to \cdots$

relating the homotopy groups of the sphere spectrum to those of the Moore spectrum.

In a stable $\infty$-category, the homotopy category $\operatorname{h}\mathcal{C}$ is automatically additive: for any objects $X, Y$, the map $X \sqcup Y \to X \times Y$ (from coproduct to product) is an equivalence. The common value is the biproduct $X \oplus Y$.

The hom-sets $[X, Y] = \pi_0 \operatorname{Map}(X, Y)$ acquire an abelian group structure (from the loop space structure on $Y$), making $\operatorname{h}\mathcal{C}$ a preadditive category.

A stable $\infty$-category $\mathcal{C}$ is naturally enriched in spectra: for objects $X, Y$, there is a mapping spectrum

$\operatorname{map}_{\mathcal{C}}(X, Y) \in \operatorname{Sp}$

with $\Omega^\infty \operatorname{map}_{\mathcal{C}}(X, Y) \simeq \operatorname{Map}_{\mathcal{C}}(X, Y)$. The homotopy groups are:

$\pi_n(\operatorname{map}_{\mathcal{C}}(X, Y)) \cong [X, \Sigma^{-n} Y] \cong [\Sigma^n X, Y]$

For $\mathcal{C} = D(R)$: $\pi_n(\operatorname{map}(C, D)) = \operatorname{Ext}^{-n}_R(C, D)$.

A functor $F: \mathcal{C} \to \mathcal{D}$ between stable $\infty$-categories is exact if it preserves finite limits (equivalently, finite colimits, equivalently, zero objects and pushouts, equivalently, zero objects and pullbacks). Exact functors preserve fiber and cofiber sequences.

The $\infty$-category $\operatorname{Fun}^{\mathrm{ex}}(\mathcal{C}, \mathcal{D})$ of exact functors is itself stable.