$D(\mathcal{A}) = K(\mathcal{A}) / K_{\mathrm{ac}}(\mathcal{A})$, where $K_{\mathrm{ac}}(\mathcal{A})$ is the thick subcategory of acyclic complexes. The quasi-isomorphisms are exactly the morphisms whose cones are acyclic.

$D^b(\mathcal{A}) = K^b(\mathcal{A}) / K^b_{\mathrm{ac}}(\mathcal{A})$. Similarly for $D^+$ and $D^-$.

$D_{\mathrm{sg}}(R) = D^b(\mathrm{mod}\text{-}R) / \mathrm{Perf}(R)$: the singularity category is the Verdier quotient by perfect complexes. It is trivial iff $R$ has finite global dimension.

For a closed subvariety $Z \subseteq X$ with open complement $U$, there is a localization sequence $D^b_Z(X) \to D^b(X) \to D^b(U)$ where $D^b_Z(X)$ consists of complexes supported on $Z$. The Verdier quotient $D^b(X)/D^b_Z(X) \simeq D^b(U)$.

Morphisms in $\mathcal{T}/\mathcal{S}$ are represented by "roofs": $X \xleftarrow{s} X' \xrightarrow{f} Y$ where the cone of $s$ lies in $\mathcal{S}$. Two roofs are equivalent if they can be connected by a common refinement.

$D^b_c(X) / D^b_Z(X) \simeq D^b_c(U)$ for a closed subset $Z$ with complement $U$. This is the sheaf-theoretic localization sequence, crucial for excision arguments.

In a compactly generated triangulated category $\mathcal{T}$, the quotient $\mathcal{T}/\mathcal{T}^c$ (modding out compact objects) captures the "large-scale" behavior. This perspective is used in the theory of Brown representability.

For $R = \mathbb{Z}$, the quotient $D^b(\mathbb{Z}\text{-}\mathrm{mod}) / D^b(\mathrm{torsion})$ is equivalent to $D^b(\mathbb{Q}\text{-}\mathrm{Vect})$: modding out torsion modules gives rational vector spaces.

For an abelian category $\mathcal{A}$ and a Serre subcategory $\mathcal{S}$, $D^b(\mathcal{A})/D^b(\mathcal{S}) \simeq D^b(\mathcal{A}/\mathcal{S})$ under suitable hypotheses. This relates Verdier localization to Gabriel localization.

For a self-injective algebra $A$, the stable module category $\underline{\mathrm{mod}}\text{-}A = \mathrm{mod}\text{-}A / \mathrm{proj}\text{-}A$ can be realized as a Verdier quotient of the bounded derived category.

The localization sequence $\mathcal{S} \to \mathcal{T} \to \mathcal{T}/\mathcal{S}$ induces a long exact sequence in algebraic K-theory: $\cdots \to K_n(\mathcal{S}) \to K_n(\mathcal{T}) \to K_n(\mathcal{T}/\mathcal{S}) \to K_{n-1}(\mathcal{S}) \to \cdots$.

The Verdier quotient $\mathcal{T}/\mathcal{S}$ is universal among exact functors $\mathcal{T} \to \mathcal{T}'$ that send $\mathcal{S}$ to zero: any such functor factors uniquely through $Q : \mathcal{T} \to \mathcal{T}/\mathcal{S}$.