The DG category $\mathbf{C}_{\mathrm{dg}}(R)$ of chain complexes over a ring $R$ is pretriangulated:

• Shifts: $(C[n])_p = C_{p+n}$ with differential $d_{C[n]} = (-1)^n d_C$.
• Cones: $\operatorname{Cone}(f: C \to D) = D \oplus C[1]$ with the standard cone differential.

The homotopy category $H^0(\mathbf{C}_{\mathrm{dg}}(R)) = K(R)$ (the homotopy category of chain complexes) is indeed triangulated.

The full DG subcategory $\mathbf{C}_{\mathrm{dg}}^{\mathrm{inj}}(\mathcal{A})$ of complexes of injective objects in an abelian category $\mathcal{A}$ (with enough injectives) is pretriangulated. Its homotopy category:

$H^0(\mathbf{C}_{\mathrm{dg}}^{\mathrm{inj}}(\mathcal{A})) \simeq D(\mathcal{A})$

Similarly, the DG category of h-projective complexes is pretriangulated with homotopy category $D(\mathcal{A})$.

For a DG algebra $A$, the full DG subcategory $\operatorname{Perf}_A^{\mathrm{dg}} \subset \operatorname{Mod}_A^{\mathrm{dg}}$ of perfect DG modules (compact objects) is pretriangulated. Its homotopy category is the perfect derived category:

$H^0(\operatorname{Perf}_A^{\mathrm{dg}}) = D^{\mathrm{perf}}(A)$

For an ordinary algebra $A$, perfect DG modules correspond to bounded complexes of finitely generated projective modules.

For a DG algebra $A$ viewed as a one-object DG category $\mathcal{A}$, the pretriangulated hull $\mathcal{A}^{\mathrm{pretr}}$ consists of:

• Shifts $A[n]$ for $n \in \mathbb{Z}$.
• Iterated cones (which are "twisted complexes" over $A$).

The homotopy category $H^0(\mathcal{A}^{\mathrm{pretr}})$ is the smallest triangulated subcategory of $D(A)$ containing $A$ and closed under direct summands. When $A$ is an ordinary ring, this is $D^{\mathrm{perf}}(A)$.

A twisted complex over a DG category $\mathcal{A}$ is a finite collection of objects $E_1, \ldots, E_n \in \mathcal{A}$ with shifts $n_1, \ldots, n_k \in \mathbb{Z}$ and a strictly upper-triangular matrix of morphisms $q_{ij} \in \operatorname{Hom}^{1+n_i-n_j}(E_i, E_j)$ for $i < j$, satisfying the Maurer--Cartan equation:

$dq + q \circ q = 0$

(where $q = (q_{ij})$ is viewed as an endomorphism of $\bigoplus E_i[n_i]$).

The DG category of twisted complexes $\operatorname{Tw}(\mathcal{A})$ is pretriangulated and provides a concrete model for $\mathcal{A}^{\mathrm{pretr}}$.

For a smooth projective variety $X$, the DG category $\operatorname{Coh}_{\mathrm{dg}}(X)$ obtained from injective resolutions of coherent sheaves is pretriangulated. It enhances $D^b(\operatorname{Coh}(X))$:

$H^0(\operatorname{Coh}_{\mathrm{dg}}(X)) \simeq D^b(\operatorname{Coh}(X))$

The pretriangulated structure ensures that cones of morphisms between complexes of coherent sheaves remain in the category.

For a pretriangulated DG category $\mathcal{A}$, the homotopy category $H^0(\mathcal{A})$ is triangulated with:

• Shift: $[1]$ induced by the DG shift $X \mapsto X[1]$.
• Distinguished triangles: $X \xrightarrow{f} Y \to \operatorname{Cone}(f) \to X[1]$ for closed degree-$0$ morphisms $f$.

The triangulated structure is automatic: all axioms (TR1)--(TR4) follow from the DG-level constructions. In particular, the octahedral axiom follows from the existence of iterated cones at the DG level.

Under the DG nerve construction $N_{\mathrm{dg}}$, a pretriangulated DG category $\mathcal{A}$ maps to a stable $\infty$-category $N_{\mathrm{dg}}(\mathcal{A})$. The key correspondence:

| DG concept | Stable $\infty$-category concept | |---|---| | Pretriangulated DG category | Stable $\infty$-category | | Shift $X[1]$ | Suspension $\Sigma X$ | | Cone of $f: X \to Y$ | Cofiber of $f$ | | Closed degree-$0$ morphism | Morphism in $\mathcal{C}$ | | Homotopy ($d(h) = f - g$) | Path in $\operatorname{Map}(X, Y)$ |

The pretriangulated condition ensures that the DG nerve is stable (has a zero object, all finite limits and colimits, and $\Sigma$ is an equivalence).

A DG functor $F: \mathcal{A} \to \mathcal{B}$ between pretriangulated DG categories is exact if it preserves cones: $F(\operatorname{Cone}(f)) \cong \operatorname{Cone}(F(f))$ in $\mathcal{B}$ for all closed degree-$0$ morphisms $f$. The induced functor $H^0(F): H^0(\mathcal{A}) \to H^0(\mathcal{B})$ is then a triangulated functor.

Every DG functor between pretriangulated DG categories that preserves the zero object automatically preserves cones (since cones are constructed from shifts, direct sums, and the differential, all of which are preserved).

For a pretriangulated DG category $\mathcal{A}$ and a full pretriangulated DG subcategory $\mathcal{B} \subset \mathcal{A}$, the DG quotient $\mathcal{A}/\mathcal{B}$ (Drinfeld) is the DG category obtained by formally adjoining null-homotopies for all objects of $\mathcal{B}$: for each $B \in \mathcal{B}$, add a morphism $h_B \in \operatorname{Hom}^{-1}(B, B)$ with $d(h_B) = \operatorname{id}_B$.

The homotopy category satisfies $H^0(\mathcal{A}/\mathcal{B}) \simeq H^0(\mathcal{A})/H^0(\mathcal{B})$ (the Verdier quotient). The DG quotient lifts the Verdier quotient to the DG level.

The tensor product $\mathcal{A} \otimes_k \mathcal{B}$ of DG categories is generally not pretriangulated even if $\mathcal{A}$ and $\mathcal{B}$ are. One must take the pretriangulated hull $(\mathcal{A} \otimes_k \mathcal{B})^{\mathrm{pretr}}$.

Alternatively, one can work with the derived tensor product $\mathcal{A} \otimes_k^{\mathbf{L}} \mathcal{B}$ in the Morita model structure, which produces a DG category Morita equivalent to the expected result.

A semi-orthogonal decomposition $H^0(\mathcal{A}) = \langle \mathcal{T}_1, \mathcal{T}_2 \rangle$ lifts to the DG level: there exist full pretriangulated DG subcategories $\mathcal{A}_1, \mathcal{A}_2 \subset \mathcal{A}$ with $H^0(\mathcal{A}_i) = \mathcal{T}_i$ and a DG bimodule $M$ representing the "gluing data." The DG category $\mathcal{A}$ is recovered as the upper-triangular DG category:

$\mathcal{A} \simeq \left(\begin{smallmatrix} \mathcal{A}_1 & M \\ 0 & \mathcal{A}_2 \end{smallmatrix}\right)$

This lifting is possible because pretriangulated DG categories have well-defined orthogonal complements.

A pretriangulated DG category $\mathcal{A}$ (over $k$) is:

• Smooth (or homologically smooth) if the diagonal bimodule $\mathcal{A}$ is perfect as an $\mathcal{A}^{\mathrm{op}} \otimes_k \mathcal{A}$-module.
• Proper if $\bigoplus_n H^n(\operatorname{Hom}(X, Y))$ is finite-dimensional for all $X, Y$.

Smooth and proper DG categories are the noncommutative analogues of smooth projective varieties. They have well-defined Serre duality, finite-dimensional Hochschild cohomology, and satisfy the representability theorem for DG modules.