For a ring $R$, the DG category $\mathbf{C}_{\mathrm{dg}}(R)$ has:

• Objects: chain complexes of $R$-modules.
• $\operatorname{Hom}^n(C, D) = \prod_{p \in \mathbb{Z}} \operatorname{Hom}_R(C_p, D_{p+n})$ (degree $n$ graded maps).
• Differential: $(df)(x) = d_D(f(x)) - (-1)^{|f|} f(d_C(x))$.

A closed degree-$0$ element is a chain map. The homotopy category $H^0(\mathbf{C}_{\mathrm{dg}}(R))$ is the classical homotopy category $K(R)$ of chain complexes.

A DG algebra $(A, d)$ over $k$ can be viewed as a DG category with a single object $*$ and $\operatorname{Hom}(*, *) = A$. Composition is the multiplication of $A$.

Conversely, every DG category with a single object is a DG algebra. DG modules over $A$ correspond to DG functors $\mathcal{A} \to \mathbf{C}_{\mathrm{dg}}(k)$.

For a DG algebra $A$, the DG category $\operatorname{Mod}_A^{\mathrm{dg}}$ has:

• Objects: right DG $A$-modules (graded $A$-modules $M$ with differential satisfying $d(ma) = d(m)a + (-1)^{|m|}md(a)$).
• $\operatorname{Hom}^n(M, N) = \operatorname{Hom}_{A\text{-gr}}^n(M, N)$ (degree $n$ graded $A$-module maps).

The homotopy category $H^0(\operatorname{Mod}_A^{\mathrm{dg}})$ is the homotopy category of DG modules. The derived category $D(A)$ is obtained by further localizing at quasi-isomorphisms.

For an object $E$ in a DG category $\mathcal{A}$, the endomorphism DG algebra is $\operatorname{End}_{\mathcal{A}}(E) = \operatorname{Hom}_{\mathcal{A}}(E, E)$. The DG category is "generated" by $E$ if every object can be built from $E$ using shifts, cones, and direct summands.

For a scheme $X$ with a tilting generator $E \in D^b(\operatorname{Coh}(X))$, the endomorphism DG algebra $A = \operatorname{End}(E)$ determines the derived category: $D^b(\operatorname{Coh}(X)) \simeq D^{\mathrm{perf}}(A)$.

For a topological space $X$ and a ring $k$, the DG category $\operatorname{Shv}_{\mathrm{dg}}(X; k)$ of sheaves of $k$-modules on $X$ has:

• Objects: complexes of sheaves of $k$-modules.
• Hom complexes: $\operatorname{Hom}^n(\mathcal{F}, \mathcal{G}) = \Gamma(X, \underline{\operatorname{Hom}}^n(\mathcal{F}, \mathcal{G}))$.

The homotopy category $H^0(\operatorname{Shv}_{\mathrm{dg}}(X; k))$ is the homotopy category of complexes of sheaves, and localizing at quasi-isomorphisms gives $D(X; k)$.

For a regular ring $R$ and $w \in R$ (a "potential"), the DG category of matrix factorizations $\operatorname{MF}(R, w)$ has:

• Objects: pairs $(M_0 \xrightarrow{\phi_0} M_1 \xrightarrow{\phi_1} M_0)$ with $\phi_1 \phi_0 = w \cdot \operatorname{id}$ and $\phi_0 \phi_1 = w \cdot \operatorname{id}$.
• Hom complexes: $\mathbb{Z}/2$-graded morphisms with a differential induced by $\phi$.

Matrix factorizations appear in the Landau--Ginzburg model of mirror symmetry and Knorrer periodicity.

The wrapped Fukaya category of a symplectic manifold $(M, \omega)$ can be modeled as a DG category (or more naturally as an $A_\infty$-category). Objects are Lagrangian submanifolds with additional data (grading, spin structure), and the Hom complex is generated by intersection points with differential counting pseudo-holomorphic strips.

Homological mirror symmetry (Kontsevich) asserts a quasi-equivalence between the DG category of coherent sheaves on $X$ and the Fukaya DG category of the mirror $\check{X}$.

For a small DG category $\mathcal{A}$, the DG Yoneda embedding sends $X \in \mathcal{A}$ to the representable DG module $\hat{X} = \operatorname{Hom}_{\mathcal{A}}(-, X)$. This gives a fully faithful DG functor:

$\mathcal{A} \hookrightarrow \operatorname{Mod}_{\mathcal{A}}^{\mathrm{dg}}$

The image consists of the representable modules. The derived category $D(\mathcal{A})$ is obtained by localizing $\operatorname{Mod}_{\mathcal{A}}^{\mathrm{dg}}$ at quasi-isomorphisms, and $\mathcal{A}$ embeds into $D(\mathcal{A})$ via the Yoneda functor.

A DG bimodule over DG categories $\mathcal{A}$ and $\mathcal{B}$ is a DG functor $\mathcal{A}^{\mathrm{op}} \otimes \mathcal{B} \to \mathbf{C}_{\mathrm{dg}}(k)$. DG bimodules serve as morphisms in the "Morita $\infty$-category" of DG categories:

Two DG categories $\mathcal{A}$ and $\mathcal{B}$ are Morita equivalent if their derived categories of modules are equivalent: $D(\mathcal{A}) \simeq D(\mathcal{B})$. This is equivalent to the existence of a DG bimodule $M$ such that the tensor functor $- \otimes_{\mathcal{A}} M: D(\mathcal{A}) \to D(\mathcal{B})$ is an equivalence.

The tensor product $\mathcal{A} \otimes_k \mathcal{B}$ of DG categories has objects $\operatorname{Ob}(\mathcal{A}) \times \operatorname{Ob}(\mathcal{B})$ and Hom complexes:

$\operatorname{Hom}_{\mathcal{A} \otimes \mathcal{B}}((X_1, Y_1), (X_2, Y_2)) = \operatorname{Hom}_{\mathcal{A}}(X_1, X_2) \otimes_k \operatorname{Hom}_{\mathcal{B}}(Y_1, Y_2)$

This gives the category of DG categories a symmetric monoidal structure (over $k$). DG bimodules are equivalent to DG functors $\mathcal{A}^{\mathrm{op}} \otimes \mathcal{B} \to \mathbf{C}_{\mathrm{dg}}(k)$.

The DG nerve construction (Lurie) sends a DG category $\mathcal{A}$ over a field $k$ to a $k$-linear stable $\infty$-category $N_{\mathrm{dg}}(\mathcal{A})$. The homotopy category of $N_{\mathrm{dg}}(\mathcal{A})$ recovers $H^0(\mathcal{A})$.

For a DG category $\mathcal{A}$ over $\mathbb{Q}$, the DG nerve is equivalent to the simplicial nerve of the simplicial category obtained by the Dold--Kan correspondence: applying the Dold--Kan functor to each Hom complex gives a simplicial category, whose nerve is a quasi-category.

The $\infty$-category of DG categories (up to quasi-equivalence) is obtained by localizing the $1$-category $\mathbf{dgCat}_k$ at quasi-equivalences. Tabuada and Toen showed that $\mathbf{dgCat}_k$ admits a model structure where:

• Weak equivalences: quasi-equivalences.
• Fibrations: DG functors surjective on Hom complexes.

The underlying $\infty$-category $\mathbf{dgCat}_k[W^{-1}]$ is equivalent to the $\infty$-category of $k$-linear stable $\infty$-categories (when $k$ is a field of characteristic zero).