Let $A^\bullet = [A^0 \xrightarrow{d} A^1]$ (in degrees 0 and 1). Then $\tau_{\leq 0} A^\bullet = [\ker d]$ (concentrated in degree 0) and $\tau_{\geq 1} A^\bullet = [\mathrm{coker}\, d']$ where $d' : A^0/\ker d \to A^1$ is the induced map. The cohomology is $H^0 = \ker d$ and $H^1 = \mathrm{coker}\, d$.

Let $0 \to P_2 \to P_1 \to P_0 \to M \to 0$ be a projective resolution viewed as a complex $P^\bullet$ in degrees $-2, -1, 0$ with a quasi-isomorphism $P^\bullet \to M[0]$. Then $\tau_{\leq -1} P^\bullet = [P_2 \to \ker(P_1 \to P_0)]$ and $\tau_{\geq 0} P^\bullet \simeq M[0]$.

The standard t-structure gives $H^n = H^0 \circ [-n]$ where $H^0(X) = \tau_{\leq 0}\tau_{\geq 0} X$. This recovers the usual cohomology of complexes. For a distinguished triangle $X \to Y \to Z \to X[1]$, the long exact sequence $\cdots \to H^n(X) \to H^n(Y) \to H^n(Z) \to H^{n+1}(X) \to \cdots$ follows formally.

On $D(\mathbf{Vect}_k)$, every complex is formal: $A^\bullet \cong \bigoplus_n H^n(A)[-n]$. The truncation $\tau_{\leq 0}$ simply keeps the summands $H^n[-n]$ for $n \leq 0$. The t-structure is completely determined by the grading.

On $D(\mathbf{Ab})$, formality also holds (since $\mathbb{Z}$ is hereditary). The standard t-structure has heart $\mathbf{Ab}$. The truncation functors agree with the naive truncation on the level of cohomology groups.

For any integer $n$, the shifted t-structure $(D^{\leq n}, D^{\geq n})$ is also a valid t-structure on $D(\mathcal{A})$. Its heart consists of complexes with cohomology concentrated in degree $n$, which is again equivalent to $\mathcal{A}$ via $A \mapsto A[-n]$.

For a short exact sequence $0 \to A \to B \to C \to 0$ in $\mathcal{A}$, viewed as a distinguished triangle $A[0] \to B[0] \to C[0] \to A[1]$ in $D(\mathcal{A})$, all three terms lie in $D^{\leq 0} \cap D^{\geq 0}$. The truncation functors act as the identity on objects in the heart.

For $\mathcal{F}^\bullet \in D^b(\mathbf{Sh}(X))$, the truncation $\tau_{\leq n} \mathcal{F}^\bullet$ is a complex whose cohomology sheaves $\mathcal{H}^k$ vanish for $k > n$ and agree with those of $\mathcal{F}^\bullet$ for $k \leq n$. This is used extensively in the theory of perverse sheaves and intersection cohomology.

On $D^b_c(X)$ for a stratified space $X$, the standard t-structure has heart $\mathbf{Sh}_c(X)$ (constructible sheaves), while the perverse t-structure has heart $\mathrm{Perv}(X)$ (perverse sheaves). The perverse t-structure is a non-trivial deformation of the standard one, shifted differently on each stratum.

The inclusion $i : D^{\leq 0} \hookrightarrow D(\mathcal{A})$ has a right adjoint $\tau_{\leq 0}$, and $j : D^{\geq 0} \hookrightarrow D(\mathcal{A})$ has a left adjoint $\tau_{\geq 0}$. These adjunctions encode the universal property of truncation: $\mathrm{Hom}(X, \tau_{\leq 0} Y) \cong \mathrm{Hom}(X, Y)$ for $X \in D^{\leq 0}$.

For a left exact functor $F : \mathcal{A} \to \mathcal{B}$, the derived functor $RF$ maps $D^+(\mathcal{A}) \to D^+(\mathcal{B})$. The relationship $H^n(RF(X)) = R^nF(H^0(X))$ holds when $X \in D^{\geq 0} \cap D^{\leq 0}$. For general $X$, a spectral sequence $R^pF(H^q(X)) \Rightarrow H^{p+q}(RF(X))$ mediates.

The stupid (or brutal) truncation $\sigma_{\leq n}$ simply sets $X^k = 0$ for $k > n$ and keeps $X^k$ for $k \leq n$. Unlike the canonical truncation $\tau_{\leq n}$, the stupid truncation does not preserve quasi-isomorphism classes and does not define a functor on $D(\mathcal{A})$. However, $\sigma_{\leq n}$ is useful at the chain level.

The standard t-structure on $D^b(\mathcal{A})$ is non-degenerate: $\bigcap_n D^{\leq n} = 0$ and $\bigcap_n D^{\geq n} = 0$. This means every nonzero object has some nonzero cohomology, which is automatic for bounded complexes. For unbounded $D(\mathcal{A})$, non-degeneracy requires that no nonzero object has all cohomology groups zero (which holds by definition of the derived category).