For the standard t-structure on $D(\mathcal{A})$, the heart $D^{\leq 0} \cap D^{\geq 0}$ consists of complexes with cohomology concentrated in degree 0. The equivalence $\mathcal{A} \xrightarrow{\sim} D^{\leq 0} \cap D^{\geq 0}$ sends $A \mapsto A[0]$. This is indeed abelian, confirming the theorem in its simplest case.

The heart of the perverse t-structure on $D^b_c(X)$ is $\mathrm{Perv}(X)$. By the theorem, perverse sheaves form an abelian category. This is non-obvious because perverse sheaves are complexes of sheaves, and the notion of kernel/cokernel for perverse sheaves differs from that for ordinary sheaves. A short exact sequence in $\mathrm{Perv}(X)$ is a distinguished triangle all of whose vertices are perverse.

For a map $f : A \to B$ in the heart of $D^b(\mathcal{A})$, the cone $C_f$ is a complex with $H^{-1}(C_f) = \ker f$ and $H^0(C_f) = \mathrm{coker}\, f$. The kernel in the heart is $H^{-1}(C_f)[1] = (\ker f)[0]$ and the cokernel is $H^0(C_f)[0] = (\mathrm{coker}\, f)[0]$. This matches the usual kernel and cokernel in $\mathcal{A}$.

For a torsion pair $(\mathcal{T}, \mathcal{F})$ in $\mathcal{A}$, the tilted t-structure on $D^b(\mathcal{A})$ has heart $\mathcal{A}^\sharp$ consisting of complexes with $H^0 \in \mathcal{F}$, $H^{-1} \in \mathcal{T}$, and $H^n = 0$ otherwise. The theorem guarantees $\mathcal{A}^\sharp$ is abelian, even though it is not a subcategory of $\mathcal{A}$.

Weight structures (co-t-structures) also have hearts, but the theorem does not apply since the axioms differ. The heart of a weight structure is additive and idempotent complete but typically not abelian. For example, the heart of the Chow weight structure on $DM^{\mathrm{eff}}$ is the category of Chow motives, which is additive but not abelian.

In the heart $\mathcal{A}$, a short exact sequence $0 \to K \to A \to Q \to 0$ corresponds to a distinguished triangle $K \to A \to Q \to K[1]$ where $K, A, Q \in \mathcal{A}$. The connecting morphism $Q \to K[1]$ classifies the extension: it lies in $\mathrm{Hom}_{\mathcal{T}}(Q, K[1]) = \mathrm{Ext}^1_{\mathcal{A}}(Q, K)$.

For objects $A, B$ in the heart $\mathcal{A}$, $\mathrm{Ext}^n_{\mathcal{A}}(A, B) = \mathrm{Hom}_{\mathcal{T}}(A, B[n])$. This identifies the Ext groups computed in $\mathcal{A}$ with Hom spaces in the ambient triangulated category. For the standard t-structure this recovers the classical identification $\mathrm{Ext}^n_{\mathcal{A}}(A, B) = \mathrm{Hom}_{D(\mathcal{A})}(A, B[n])$.

The functor $H^0 : \mathcal{T} \to \mathcal{A}$ is cohomological: it maps distinguished triangles $X \to Y \to Z \to X[1]$ to exact sequences $H^0(X) \to H^0(Y) \to H^0(Z)$ in $\mathcal{A}$. Note that $H^0$ is NOT exact in the sense of triangulated functors; it is exact in the abelian sense when restricted to short exact sequences in the heart.

An object $I \in \mathcal{A}$ is injective in the heart iff $\mathrm{Ext}^1_{\mathcal{A}}(-, I) = 0$, equivalently $\mathrm{Hom}_{\mathcal{T}}(-, I[1]) = 0$ on $\mathcal{A}$. The heart need not have enough injectives, even if $\mathcal{A}$ does. For the standard t-structure on $D^b(\mathcal{A})$, the injectives in the heart are exactly the injective objects of $\mathcal{A}$.

The category $D^b(\mathrm{Coh}(\mathbb{P}^1))$ admits many t-structures with non-equivalent hearts. The standard heart is $\mathrm{Coh}(\mathbb{P}^1)$. Tilting gives a heart equivalent to representations of the Kronecker quiver $\bullet \rightrightarrows \bullet$. Both are abelian by the theorem, but they are very different abelian categories.

Given the heart $\mathcal{A}$ of a bounded t-structure on $\mathcal{T}$, the realization functor $\mathrm{real} : D^b(\mathcal{A}) \to \mathcal{T}$ is the exact functor extending the inclusion $\mathcal{A} \hookrightarrow \mathcal{T}$. The theorem guarantees $\mathcal{A}$ is abelian, so $D^b(\mathcal{A})$ exists. The realization functor is always exact but not always an equivalence.

A Bridgeland stability condition on $D^b(\mathrm{Coh}(X))$ consists of a bounded t-structure with heart $\mathcal{A}$ together with a central charge $Z : K_0(\mathcal{A}) \to \mathbb{C}$. The theorem ensures $\mathcal{A}$ is abelian, which is needed for the Harder-Narasimhan property. The space of stability conditions is a complex manifold parametrizing different abelian hearts.

The heart $\mathcal{A}$ of a bounded t-structure uniquely determines the t-structure via $\mathcal{T}^{\leq 0} = \langle \mathcal{A}, \mathcal{A}[1], \mathcal{A}[2], \ldots \rangle$ (extension closure). Conversely, not every abelian subcategory of $\mathcal{T}$ arises as the heart of a t-structure. The heart must satisfy specific generation and orthogonality conditions.