The derived $\infty$-category $D(R)$ has the standard t-structure:

• $D(R)_{\geq 0}$: complexes $C$ with $H_n(C) = 0$ for $n < 0$.
• $D(R)_{\leq 0}$: complexes $C$ with $H_n(C) = 0$ for $n > 0$.

The heart is $D(R)^\heartsuit \simeq \operatorname{Mod}_R$ (the abelian category of $R$-modules). The truncation $\tau_{\leq 0}(C)$ is the good truncation of a chain complex.

The homotopy objects $\pi_n(C) = H_n(C)$ are the homology groups.

The $\infty$-category $\operatorname{Sp}$ has the standard t-structure:

• $\operatorname{Sp}_{\geq 0}$: connective spectra ($\pi_n = 0$ for $n < 0$).
• $\operatorname{Sp}_{\leq 0}$: coconnective spectra ($\pi_n = 0$ for $n > 0$).

The heart is $\operatorname{Sp}^\heartsuit \simeq \mathbf{Ab}$ (the category of abelian groups). The functor $\pi_0: \operatorname{Sp} \to \mathbf{Ab}$ is the "zeroth homotopy group."

On the derived $\infty$-category $D^b_c(X)$ of constructible sheaves on a complex algebraic variety $X$, the perverse t-structure has heart equal to the abelian category of perverse sheaves. This t-structure depends on a stratification of $X$ and is shifted relative to the standard t-structure.

Perverse sheaves are fundamental in geometric representation theory (Kazhdan--Lusztig theory, intersection cohomology).

In motivic stable homotopy theory, the $\infty$-category $\operatorname{SH}(k)$ of motivic spectra over a field $k$ has several t-structures, including the homotopy t-structure whose heart is related to Voevodsky's homotopy modules.

The truncation functors $\tau_{\geq n}$ and $\tau_{\leq n}$ fit into fiber sequences:

$\tau_{\geq n}(X) \to X \to \tau_{\leq n-1}(X)$

These give the Postnikov tower of $X$:

$\cdots \to \tau_{\leq 2} X \to \tau_{\leq 1} X \to \tau_{\leq 0} X$

with $X \simeq \lim_n \tau_{\leq n} X$ (when the t-structure is complete).

The homotopy objects are $\pi_n(X) = \tau_{\leq 0} \tau_{\geq 0} \Sigma^{-n} X \in \mathcal{C}^\heartsuit$, giving the "homology" of $X$ in the heart.

A fiber sequence $A \to B \to C$ in $\mathcal{C}$ gives a long exact sequence in the heart:

$\cdots \to \pi_{n+1}(C) \to \pi_n(A) \to \pi_n(B) \to \pi_n(C) \to \pi_{n-1}(A) \to \cdots$

This is the $\infty$-categorical generalization of the long exact sequence in homology for a short exact sequence of chain complexes.

A t-structure is left complete if $\mathcal{C} \simeq \lim_n \mathcal{C}_{\leq n}$ (every object is determined by its truncations). It is right complete if $\mathcal{C} \simeq \lim_n \mathcal{C}_{\geq -n}$.

The standard t-structure on $D(R)$ is both left and right complete. The standard t-structure on $\operatorname{Sp}$ is right complete but generally not left complete (there exist non-zero spectra with all homotopy groups zero in some exotic situations, though this requires careful set-theoretic considerations).

A t-structure on a presentable stable $\infty$-category $\mathcal{C}$ is accessible if $\mathcal{C}_{\geq 0}$ is presentable (equivalently, closed under filtered colimits). The standard t-structures on $D(R)$ and $\operatorname{Sp}$ are accessible.

Accessible t-structures are determined by their hearts: the functor $\mathcal{C} \to \operatorname{Fun}(\mathbb{Z}^{\mathrm{op}}, \mathcal{C}^\heartsuit)$ (sending $X$ to its Postnikov tower) is conservative for complete accessible t-structures.

An object $X \in \mathcal{C}$ is bounded (with respect to a t-structure) if $X \in \mathcal{C}_{\geq m} \cap \mathcal{C}_{\leq n}$ for some $m, n$. The full subcategory of bounded objects $\mathcal{C}^b$ is a stable $\infty$-category.

For $D(R)$: bounded objects are complexes with finitely many nonzero homology groups, giving $D^b(R)$.