For a ring $R$, the stable $\infty$-category $\mathcal{D}(R)$ has homotopy category $\operatorname{h}\mathcal{D}(R) = D(R)$, the classical derived category. The triangulated structure is:

• Shift: $C[1]$ (shift chain complex indices).
• Distinguished triangles: arise from short exact sequences $0 \to A \to B \to C \to 0$ of chain complexes, giving $A \to B \to C \to A[1]$.

The stable $\infty$-category $\mathcal{D}(R)$ resolves the well-known pathologies: non-functorial cones, non-existence of homotopy limits, failure of descent.

The homotopy category $\operatorname{h}\operatorname{Sp}$ of the $\infty$-category of spectra is the classical stable homotopy category. The triangulated structure has:

• Shift: $\Sigma E$ (suspension).
• Distinguished triangles: cofiber sequences of spectra.

The stable homotopy category was among the earliest triangulated categories studied, predating Verdier's axiomatization. The $\infty$-categorical enhancement $\operatorname{Sp}$ provides the correct framework for the smash product monoidal structure.

In a triangulated category $\mathcal{T}$, the cone of $f: X \to Y$ exists but is not functorial: given a morphism of triangles, the fill at the cone level exists but is not unique. In $\mathcal{C}$ the cofiber is a functor $\operatorname{cofib}: \operatorname{Fun}(\Delta^1, \mathcal{C}) \to \mathcal{C}$, fully functorial.

For example, in $D(\mathbb{Z})$, the exact triangle $\mathbb{Z} \xrightarrow{2} \mathbb{Z} \to \mathbb{Z}/2 \to \mathbb{Z}[1]$ has cone $\mathbb{Z}/2$. A morphism of this triangle to itself by $(\operatorname{id}, \operatorname{id})$ admits any element of $\operatorname{End}(\mathbb{Z}/2) = \mathbb{Z}/2$ as the fill, demonstrating non-uniqueness.

The homotopy limit of a tower $\cdots \to X_2 \to X_1 \to X_0$ does not generally exist in a triangulated category. The Milnor exact sequence

$0 \to \lim{}^1 [-, X_n] \to [-, \operatorname{holim} X_n] \to \lim [-, X_n] \to 0$

requires $\operatorname{holim}$ to exist, which triangulated axioms alone do not guarantee. In a stable $\infty$-category with appropriate completeness, homotopy limits are well-defined by universal properties.

In a stable $\infty$-category, given composable morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$, taking iterated cofibers produces a $3 \times 3$ diagram where all rows and columns are cofiber sequences. The octahedral axiom is simply the assertion that this diagram exists.

The proof is completely formal: no choices are involved, unlike the triangulated setting where the octahedral axiom must be verified separately and often with significant effort.

For a Zariski covering of a scheme $X$, descent requires:

$\operatorname{QCoh}(X) \simeq \lim\left(\prod_i \operatorname{QCoh}(U_i) \rightrightarrows \prod_{i,j} \operatorname{QCoh}(U_{ij}) \cdots\right)$

This limit makes sense in the $\infty$-category of stable $\infty$-categories but not at the triangulated level. Triangulated categories form only a $2$-category, without good limit properties.

Algebraic K-theory $K(\mathcal{C})$ of a stable $\infty$-category $\mathcal{C}$ is defined via the $S_\bullet$-construction (Waldhausen). K-theory is not a triangulated invariant: there exist triangulated equivalences $\operatorname{h}\mathcal{C} \simeq \operatorname{h}\mathcal{D}$ with $K(\mathcal{C}) \not\simeq K(\mathcal{D})$.

Schlichting showed that certain triangulated categories of the form $D^b(\mathcal{A})$ and $D^b(\mathcal{B})$ can be equivalent as triangulated categories while having different $K_1$-groups.

Muro, Schwede, and Strickland (2007) constructed triangulated categories admitting no stable $\infty$-categorical enhancement at all. Specifically, certain "exotic" triangulated structures on categories of $\mathbb{Z}/p$-modules violate necessary conditions for liftability.

Conversely, Canonaco and Stellari proved uniqueness of DG enhancements for the bounded derived category $D^b(\operatorname{Coh}(X))$ of a smooth projective variety $X$, showing that in algebro-geometric situations, the stable enhancement is often unique.