t-structure

Remarks

The heart is always an abelian category, and the triangle in clause 3 is unique up to a unique isomorphism; this gives truncation functors $\tau_{\le 0}$ and $\tau_{\ge 1}$. Example: let $D(\mathcal{A})$ be the derived category of an abelian category, written with cochain complexes. Its standard t-structure has $\mathcal{D}^{\le 0}$ the complexes with $H^i = 0$ for $i > 0$ and $\mathcal{D}^{\ge 0}$ those with $H^i = 0$ for $i < 0$. The triangle of clause 3 is $\tau_{\le 0} X \to X \to \tau_{\ge 1} X \to$, built from the truncated complexes, and the heart is equivalent to $\mathcal{A}$. Non-example: if $\mathcal{D} \ne 0$, taking $\mathcal{D}^{\le 0} = \mathcal{D}^{\ge 0} = \mathcal{D}$ fails clause 2, because for $X \ne 0$ the identity of $X$ is a nonzero element of $\Hom(X, X)$, while $X$ lies in both $\mathcal{D}^{\le 0}$ and $\mathcal{D}^{\ge 1} = \mathcal{D}$.