For the DG category $\mathbf{C}_{\mathrm{dg}}(R)$ of chain complexes of $R$-modules:

$N_{\mathrm{dg}}(\mathbf{C}_{\mathrm{dg}}^{\mathrm{inj}}(R)) \simeq D(R)$

where $\mathbf{C}_{\mathrm{dg}}^{\mathrm{inj}}(R)$ is the full DG subcategory of h-injective complexes, and $D(R)$ is the derived $\infty$-category. The mapping spaces satisfy:

$\pi_n(\operatorname{Map}_{D(R)}(C, D)) \cong H^{-n}(\operatorname{Hom}_{\mathrm{dg}}(C, D)) \cong \operatorname{Ext}^{-n}_R(C, D)$

For a DG algebra $A$ (viewed as a one-object DG category), the DG nerve $N_{\mathrm{dg}}(A)$ is a pointed $\infty$-category with one object $*$ and endomorphism spectrum:

$\operatorname{map}(*, *) \simeq |A|$

where $|A|$ denotes the spectrum associated to the chain complex underlying $A$. The DG nerve of the category of DG modules gives:

$N_{\mathrm{dg}}(\operatorname{Mod}_A^{\mathrm{dg, h-proj}}) \simeq \operatorname{Mod}_A^{\infty}$

the $\infty$-category of $A$-modules (in the derived sense).

For a smooth projective variety $X$ over a field $k$, the DG category of coherent sheaves $\operatorname{Coh}_{\mathrm{dg}}(X)$ maps under the DG nerve to:

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

the stable $\infty$-category enhancing $D^b(\operatorname{Coh}(X))$. This allows translation between DG-level results (Keller, Orlov) and $\infty$-categorical results (Lurie, Gaitsgory).

Given a small stable $\infty$-category $\mathcal{C}$ (linear over $k$), one can extract a DG category $\mathcal{C}^{\mathrm{dg}}$ by taking the mapping spectra and using the inverse Dold--Kan correspondence to produce chain complexes. This gives an inverse to $N_{\mathrm{dg}}$:

$\mathcal{C}^{\mathrm{dg}} \text{ with } \operatorname{Hom}_{\mathcal{C}^{\mathrm{dg}}}(X, Y) = \operatorname{DK}^{-1}(\operatorname{map}_{\mathcal{C}}(X, Y))$

The roundtrip $\mathcal{A} \mapsto N_{\mathrm{dg}}(\mathcal{A}) \mapsto N_{\mathrm{dg}}(\mathcal{A})^{\mathrm{dg}}$ recovers $\mathcal{A}$ up to quasi-equivalence.

For objects $X, Y$ in a DG category $\mathcal{A}$, the mapping space in $N_{\mathrm{dg}}(\mathcal{A})$ is:

$\operatorname{Map}_{N_{\mathrm{dg}}(\mathcal{A})}(X, Y) \simeq \Gamma(\tau_{\leq 0}\operatorname{Hom}_{\mathcal{A}}(X, Y))$

In particular:

• $\pi_0(\operatorname{Map}(X, Y)) = H^0(\operatorname{Hom}(X, Y))$ (morphisms in the homotopy category).
• $\pi_n(\operatorname{Map}(X, Y)) = H^{-n}(\operatorname{Hom}(X, Y))$ for $n > 0$ (higher homotopies).

The non-positive cohomology groups of the Hom complex become the homotopy groups of the mapping space.

A DG functor $F: \mathcal{A} \to \mathcal{B}$ induces an $\infty$-functor $N_{\mathrm{dg}}(F): N_{\mathrm{dg}}(\mathcal{A}) \to N_{\mathrm{dg}}(\mathcal{B})$. Key properties:

• If $F$ is a quasi-equivalence, then $N_{\mathrm{dg}}(F)$ is a categorical equivalence.
• If $F$ preserves cones (exact DG functor between pretriangulated categories), then $N_{\mathrm{dg}}(F)$ is an exact functor of stable $\infty$-categories.
• DG natural transformations map to $\infty$-natural transformations.

For a DG category $\mathcal{A}$, the Hochschild cohomology can be computed either at the DG level or the $\infty$-categorical level:

$HH^*(\mathcal{A}) = H^*(\operatorname{RHom}_{\mathcal{A}^{\mathrm{op}} \otimes \mathcal{A}}(\mathcal{A}, \mathcal{A})) \cong \pi_{-*}(\operatorname{map}_{\operatorname{Fun}(N_{\mathrm{dg}}(\mathcal{A}), N_{\mathrm{dg}}(\mathcal{A}))}(\operatorname{id}, \operatorname{id}))$

The DG nerve preserves the Hochschild complex, so computations can be performed in whichever framework is more convenient.

Over $\mathbb{Q}$ (or any field of characteristic $0$), the Dold--Kan correspondence gives an equivalence:

$\mathbf{Ch}_{\leq 0}(\mathbb{Q}) \simeq s\mathbf{Mod}_{\mathbb{Q}}$

between non-positively graded chain complexes and simplicial $\mathbb{Q}$-modules. This extends to an equivalence between DG categories and simplicial categories (over $\mathbb{Q}$), which then maps to $\infty$-categories via the homotopy coherent nerve.

In positive characteristic, the Dold--Kan correspondence still exists but the resulting simplicial objects carry less information, and the equivalence between DG categories and $\infty$-categories is more subtle.

Tabuada and Toen endowed $\mathbf{dgCat}_k$ with model structures:

1. Dwyer--Kan model structure: weak equivalences are quasi-equivalences. Fibrant objects are DG categories with "path objects" for Hom complexes.
2. Morita model structure: weak equivalences are Morita equivalences. Fibrant objects are idempotent-complete pretriangulated DG categories.

The DG nerve sends the Dwyer--Kan model structure to the Joyal model structure on simplicial sets. The Morita model structure corresponds to the $\infty$-category of idempotent-complete stable $\infty$-categories.

For DG categories $\mathcal{A}$ and $\mathcal{B}$, the DG category of DG functors $\operatorname{Fun}_{\mathrm{dg}}(\mathcal{A}, \mathcal{B})$ maps under the nerve to:

$N_{\mathrm{dg}}(\operatorname{Fun}_{\mathrm{dg}}(\mathcal{A}, \mathcal{B})) \simeq \operatorname{Fun}(N_{\mathrm{dg}}(\mathcal{A}), N_{\mathrm{dg}}(\mathcal{B}))$

the $\infty$-category of functors. This compatibility with internal Hom is essential for translating categorical constructions between the two frameworks.

Toen constructed the "derived category" (actually $\infty$-category) of DG categories:

$\operatorname{Hqe} = \mathbf{dgCat}_k[W_{\mathrm{qe}}^{-1}]$

This $\infty$-category has a symmetric monoidal structure (derived tensor product) and an internal Hom (given by DG bimodules). The DG nerve identifies $\operatorname{Hqe}$ with the $\infty$-category of $k$-linear $\infty$-categories.

For a morphism $f: X \to Y$ of schemes, the six-functor formalism $(f^*, f_*, f^!, f_!, \otimes, \underline{\operatorname{Hom}})$ can be constructed either:

1. At the DG level: using DG enhancements of derived categories and DG bimodules.
2. At the $\infty$-categorical level: using stable $\infty$-categories and $\infty$-functors.

The DG nerve theorem ensures these two approaches give equivalent results. In practice, DG methods are often more explicit (using resolutions), while $\infty$-categorical methods are more conceptual (using universal properties).