For a ring $R$, $\mathcal{D}(R)$ is the prototypical derived $\infty$-category. The mapping spaces encode Ext groups:

$\pi_n(\operatorname{Map}_{\mathcal{D}(R)}(M, N)) \cong \operatorname{Ext}_R^{-n}(M, N)$

for $R$-modules $M, N$ (viewed as complexes in degree $0$). In particular, $\pi_0 = \operatorname{Hom}_R(M, N)$ and $\pi_{-n} = \operatorname{Ext}_R^n(M, N)$ for $n > 0$. The homotopy category $\operatorname{h}\mathcal{D}(R)$ recovers the classical derived category $D(R)$.

For a topological space $X$ (or a site), the abelian category $\operatorname{Sh}(X, \mathbf{Ab})$ of sheaves of abelian groups has derived $\infty$-category $\mathcal{D}(\operatorname{Sh}(X))$. Sheaf cohomology is computed as:

$H^n(X, \mathcal{F}) \cong \pi_0 \operatorname{Map}_{\mathcal{D}(\operatorname{Sh}(X))}(\mathbb{Z}_X, \mathcal{F}[n])$

The standard t-structure has heart $\operatorname{Sh}(X, \mathbf{Ab})$, and the derived global sections functor $R\Gamma: \mathcal{D}(\operatorname{Sh}(X)) \to \mathcal{D}(\mathbb{Z})$ is an exact functor of stable $\infty$-categories.

The classical derived category $D^{\mathrm{cl}}(\mathcal{A}) = \operatorname{h}\mathcal{D}(\mathcal{A})$ records only $\pi_0$ of mapping spaces. The full $\infty$-category $\mathcal{D}(\mathcal{A})$ gains:

1. Functorial cones: the cofiber is a functor, not just defined up to non-canonical isomorphism.
2. Homotopy limits and colimits: these exist and are computed by the universal property.
3. Descent: sheaves of $\mathcal{D}(\mathcal{A})$-valued objects satisfy descent, while triangulated-valued sheaves do not.
4. Multiplicative structures: $E_\infty$-ring objects, module objects, and algebra objects make sense in $\mathcal{D}(\mathcal{A})$.

For a scheme $X$, the $\infty$-category $\operatorname{QCoh}(X)$ of quasi-coherent complexes is the derived $\infty$-category associated to the abelian category of quasi-coherent sheaves. For affine $X = \operatorname{Spec} R$:

$\operatorname{QCoh}(\operatorname{Spec} R) \simeq \mathcal{D}(R)$

For general $X$, $\operatorname{QCoh}(X)$ satisfies Zariski descent: for a Zariski cover $X = \bigcup U_i$,

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

This limit is taken in the $\infty$-category of stable $\infty$-categories.

The compact objects in $\mathcal{D}(R)$ are the perfect complexes: those quasi-isomorphic to bounded complexes of finitely generated projective $R$-modules. The full subcategory of perfect complexes $\operatorname{Perf}(R) \subset \mathcal{D}(R)$ is the smallest stable subcategory containing $R$ and closed under retracts.

$\mathcal{D}(R)$ is compactly generated: $\mathcal{D}(R) \simeq \operatorname{Ind}(\operatorname{Perf}(R))$. This characterization is central to noncommutative algebraic geometry (Kontsevich).

Two rings $R$ and $S$ are derived Morita equivalent if $\mathcal{D}(R) \simeq \mathcal{D}(S)$ as stable $\infty$-categories. This is equivalent to the existence of a tilting complex: a perfect complex $T \in \mathcal{D}(R)$ that generates $\mathcal{D}(R)$ and satisfies $\operatorname{End}_{\mathcal{D}(R)}(T) \cong S$.

For example, the path algebra of the Kronecker quiver $A = k(1 \rightrightarrows 2)$ is derived Morita equivalent to the category of coherent sheaves on $\mathbb{P}^1$, established through the tilting bundle $\mathcal{O} \oplus \mathcal{O}(1)$.

The derived $\infty$-category $\mathcal{D}(R)$ is enriched in spectra. The mapping spectrum satisfies:

$\pi_n(\operatorname{map}_{\mathcal{D}(R)}(C, D)) \cong H^{-n}(\mathbf{R}\operatorname{Hom}_R(C, D)) = \operatorname{Ext}_R^{-n}(C, D)$

This means the mapping spectrum is simply $\mathbf{R}\operatorname{Hom}_R(C, D)$ viewed as a spectrum. The underlying space is $\Omega^\infty \mathbf{R}\operatorname{Hom}_R(C, D)$.

The derived tensor product $- \otimes_R^{\mathbf{L}} -: \mathcal{D}(R) \times \mathcal{D}(R) \to \mathcal{D}(R)$ gives $\mathcal{D}(R)$ a symmetric monoidal structure (when $R$ is commutative). One has:

$H_n(C \otimes_R^{\mathbf{L}} D) \cong \operatorname{Tor}_n^R(C, D)$

No explicit projective resolutions are needed: the $\infty$-categorical structure handles the derived corrections automatically. The unit is $R[0]$, and $\mathbf{R}\operatorname{Hom}_R(-, -)$ is the internal hom right adjoint to $\otimes_R^{\mathbf{L}}$.

For a morphism of schemes $f: X \to Y$, the pullback and pushforward are exact functors:

$f^*: \operatorname{QCoh}(Y) \rightleftarrows \operatorname{QCoh}(X) : f_*$

where $f^*$ is left adjoint to $f_*$. At the $\infty$-categorical level, these are simply functors between stable $\infty$-categories; no need to derive separately. When $f$ is not flat, $f^*$ is automatically the "derived pullback" $Lf^*$ in the classical sense.

For morphisms $f: X \to Y$ of schemes (with appropriate finiteness), the six functors $(f^*, f_*, f^!, f_!, \otimes, \underline{\operatorname{Hom}})$ are all exact functors between stable $\infty$-categories. The key adjunctions are:

• $f^* \dashv f_*$ (pullback--pushforward)
• $f_! \dashv f^!$ (proper pushforward--exceptional pullback)
• $(-) \otimes E \dashv \underline{\operatorname{Hom}}(E, -)$ (tensor--internal hom)

The $\infty$-categorical framework resolves the classical difficulties: all base-change formulas $f^* g_* \simeq g'_* f'^*$ hold as natural equivalences of functors, not just isomorphisms that must be checked for compatibility.

For a $k$-algebra $A$, Hochschild homology is computed as:

$\operatorname{HH}(A/k) \simeq A \otimes^{\mathbf{L}}_{A \otimes_k A^{\mathrm{op}}} A$

in $\mathcal{D}(k)$. The $\infty$-categorical framework gives $\operatorname{HH}(A/k)$ the structure of an $S^1$-object (a spectrum with circle action), leading naturally to cyclic homology $\operatorname{HC}(A/k) \simeq \operatorname{HH}(A/k)_{hS^1}$ as homotopy orbits. This $S^1$-structure is invisible at the level of chain complexes and requires the $\infty$-categorical enhancement.

The heart $\mathcal{D}(\mathcal{A})^\heartsuit$ of the standard t-structure is equivalent to $\mathcal{A}$ itself. This means the derived $\infty$-category $\mathcal{D}(\mathcal{A})$ "remembers" the abelian category it came from, via the t-structure.

However, different abelian categories can give rise to equivalent derived $\infty$-categories: if $\mathcal{D}(\mathcal{A}) \simeq \mathcal{D}(\mathcal{B})$, the derived equivalence may not respect the standard t-structures. In that case, $\mathcal{B}$ appears as the heart of a different t-structure on $\mathcal{D}(\mathcal{A})$.

A classical example: for the path algebra $A$ of the quiver $1 \to 2$ over a field $k$, one has $\mathcal{D}^b(\operatorname{mod}\text{-}A) \simeq \mathcal{D}^b(\operatorname{mod}\text{-}k^2)$. The standard t-structure on the left has heart $\operatorname{mod}\text{-}A$ (representations of $1 \to 2$), while the standard t-structure on the right has heart $\operatorname{mod}\text{-}k^2$ (pairs of vector spaces). These abelian categories are non-equivalent, but their derived categories coincide.

More generally, the Happel--Reiten--Smalo construction shows that bounded t-structures on $\mathcal{D}^b(\mathcal{A})$ with length heart correspond to torsion pairs on $\mathcal{A}$.

In derived algebraic geometry (Lurie, Toen--Vezzosi), a derived scheme is a locally ringed $\infty$-topos $(X, \mathcal{O}_X)$ where $\mathcal{O}_X$ is a sheaf of $E_\infty$-rings. The derived $\infty$-category $\operatorname{QCoh}(X)$ of quasi-coherent sheaves is a stable presentable $\infty$-category, serving as the primary invariant.

The cotangent complex $\mathbb{L}_{X/S} \in \operatorname{QCoh}(X)$ is naturally an object of the derived $\infty$-category, controlling deformation theory. Virtual fundamental classes in enumerative geometry arise from the obstruction theory encoded in $\mathbb{L}_{X/S}$.