For an abelian category $\mathcal{A}$ (with enough injectives or projectives), the derived category $D(\mathcal{A})$ has a canonical DG enhancement: the DG category $\mathbf{C}_{\mathrm{dg}}^{\mathrm{inj}}(\mathcal{A})$ of complexes of injective objects, with the standard Hom complex. Then:

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

Similarly, the DG category of h-projective complexes provides an enhancement using projective resolutions (when they exist).

For a quasi-compact quasi-separated scheme $X$, the derived category $D_{\mathrm{qc}}(X)$ of quasi-coherent sheaves has a DG enhancement given by the DG category of h-injective complexes of quasi-coherent sheaves. This enhancement is canonical and functorial in $X$.

For the bounded derived category $D^b(\operatorname{Coh}(X))$ of coherent sheaves on a Noetherian scheme, the DG enhancement is obtained by restricting to complexes with coherent cohomology.

For a DG algebra $A$, the derived category $D(A)$ is enhanced by the DG category $\operatorname{Mod}_A^{\mathrm{dg}}$ of DG modules (or more precisely, the full DG subcategory of h-projective or h-injective DG modules). The perfect derived category $D^{\mathrm{perf}}(A)$ is enhanced by the full DG subcategory of compact objects.

When $A$ is an ordinary ring (viewed as a DG algebra concentrated in degree $0$), this recovers the classical derived category enhancement.

For a Noetherian ring $R$ (or a scheme $X$), the singularity category $D_{\mathrm{sg}}(R) = D^b(\operatorname{mod}\text{-}R)/D^{\mathrm{perf}}(R)$ (the Verdier quotient) has a DG enhancement obtained by the DG quotient construction of Drinfeld. The DG quotient $\mathcal{A}/\mathcal{B}$ of a DG category $\mathcal{A}$ by a full DG subcategory $\mathcal{B}$ is constructed by formally adding null-homotopies for all objects of $\mathcal{B}$.

For a Gorenstein ring $R$, $D_{\mathrm{sg}}(R)$ is equivalent to the stable category of maximal Cohen--Macaulay modules $\underline{\operatorname{CM}}(R)$.

The homotopy category of matrix factorizations $\operatorname{HMF}(R, w)$ for a potential $w \in R$ has a natural DG enhancement: the DG category $\operatorname{MF}_{\mathrm{dg}}(R, w)$ of matrix factorizations with the standard Hom complex.

By Orlov's theorem, for a hypersurface $X_0 = \operatorname{Spec}(R/(w))$ in $\operatorname{Spec}(R)$:

$D_{\mathrm{sg}}(X_0) \simeq \operatorname{HMF}(R, w)$

Both sides have DG enhancements, and this equivalence lifts to a DG quasi-equivalence.

Lunts and Orlov (2010) proved that for a smooth projective variety $X$ over a field $k$, the DG enhancement of $D^b(\operatorname{Coh}(X))$ is unique up to Morita equivalence.

More precisely, any two DG enhancements of $D^b(\operatorname{Coh}(X))$ are connected by a zigzag of quasi-equivalences. This implies that any triangulated invariant that can be defined via DG enhancements (such as Hochschild cohomology, deformation theory, etc.) is well-defined for $D^b(\operatorname{Coh}(X))$.

If a triangulated category $\mathcal{T}$ has a compact generator $G$ (i.e., $\mathcal{T}$ is equivalent to $D^{\mathrm{perf}}(A)$ for $A = \operatorname{End}_{\mathcal{T}}(G)$), then any DG enhancement of $\mathcal{T}$ is Morita equivalent to the DG algebra $\tilde{A}$ (a DG lift of $A$), provided the enhancement exists.

The uniqueness question reduces to: is the DG algebra $\tilde{A}$ unique up to quasi-isomorphism? For smooth and proper DG categories, the answer is often yes (by deformation-theoretic arguments).

Muro, Schwede, and Strickland (2007) showed that DG enhancements are not always unique. Over $\mathbb{F}_p$, there exist "exotic" triangulated structures that arise from non-equivalent DG categories.

Additionally, for non-smooth or non-proper categories, uniqueness can fail. Dugger and Shipley constructed examples of distinct DG algebras $A$ and $B$ (over $\mathbb{F}_p$) with equivalent derived categories $D(A) \simeq D(B)$ as triangulated categories but non-quasi-isomorphic DG algebras.

Not every triangulated category admits a DG enhancement. Muro, Schwede, and Strickland constructed a triangulated category over $\mathbb{F}_2$ that admits no DG enhancement (or even $A_\infty$-enhancement). The obstruction lies in the failure of certain Toda bracket relations.

However, all triangulated categories arising in "nature" (derived categories, stable homotopy categories, etc.) do admit DG enhancements, so non-enhanceable examples are somewhat exotic.

In a DG enhanced triangulated category, cones become functorial: the cofiber of a closed degree-$0$ morphism $f: X \to Y$ in $\mathcal{A}$ is $\operatorname{Cone}(f) = Y \oplus X[1]$ with differential $d_{\operatorname{Cone}} = \left(\begin{smallmatrix} d_Y & f \\ 0 & -d_{X[1]} \end{smallmatrix}\right)$, and this construction is functorial in $f$ at the DG level.

At the homotopy category level ($H^0$), one can only say that the cone is unique up to non-canonical isomorphism. The DG enhancement restores the canonical choice.

The Hochschild cohomology $HH^*(\mathcal{T})$ of a triangulated category is not well-defined intrinsically (it depends on the enhancement). Given a DG enhancement $\mathcal{A}$ of $\mathcal{T}$:

$HH^*(\mathcal{T}) := HH^*(\mathcal{A}) = H^*(\operatorname{RHom}_{\mathcal{A}^{\mathrm{op}} \otimes \mathcal{A}}(\mathcal{A}, \mathcal{A}))$

This is invariant under quasi-equivalence of DG categories. When the enhancement is unique (e.g., for smooth projective varieties), $HH^*(\mathcal{T})$ is a well-defined invariant of $\mathcal{T}$.

The DG enhancement governs deformations: deformations of a DG category $\mathcal{A}$ over $k$ are controlled by the Hochschild cohomology $HH^2(\mathcal{A})$ (first-order deformations) and $HH^3(\mathcal{A})$ (obstructions).

Without a DG enhancement, one cannot define deformations of a triangulated category. The enhancement is essential for noncommutative algebraic geometry (Kontsevich, Keller).

For smooth projective varieties $X, Y$ over a field $k$, every exact functor $D^b(\operatorname{Coh}(X)) \to D^b(\operatorname{Coh}(Y))$ that is a DG functor (at the enhanced level) is a Fourier--Mukai functor: it is isomorphic to $\Phi_{\mathcal{P}} = \mathbf{R}p_{2*}(p_1^* (-) \otimes^{\mathbf{L}} \mathcal{P})$ for a unique (up to isomorphism) kernel $\mathcal{P} \in D^b(\operatorname{Coh}(X \times Y))$.

The proof (Toen, Lunts--Orlov) uses the DG enhancement in an essential way: the result is not accessible from the triangulated category alone.

DG enhancements allow gluing constructions that are impossible at the triangulated level. For a semi-orthogonal decomposition $\mathcal{T} = \langle \mathcal{T}_1, \mathcal{T}_2 \rangle$, the DG enhanced version provides a gluing DG bimodule $M$ such that $\mathcal{A}$ is the upper-triangular DG category:

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

The reconstruction of $\mathcal{A}$ from $\mathcal{A}_1$, $\mathcal{A}_2$, and $M$ works at the DG level but fails at the triangulated level.

A DG enhancement (over a field $k$ of characteristic $0$) is essentially the same as a $k$-linear stable $\infty$-category enhancement, via the DG nerve:

$N_{\mathrm{dg}}: \mathbf{dgCat}_k \to \operatorname{Cat}_\infty^{k\text{-lin, st}}$

is an equivalence between the $\infty$-category of DG categories (localized at quasi-equivalences) and that of $k$-linear stable $\infty$-categories.

In positive characteristic, DG categories provide a strictly algebraic model, while stable $\infty$-categories work universally.