We write the subcategory generated by all the $\mathcal{P}(\phi)$ within the interval $\phi\in(\phi_{0},\phi_{1}]$ as $\mathcal{P}(\phi_{0}<\phi\leq\phi_{1})$. The important property is that $\mathcal{P}(\phi_{0}<\phi\leq\phi_{0}+1)$ is an abelian category, known as the heart of a bounded $t$-structure. The advantage is that while in general triangulated categories only distinguished triangles make sense, for abelian categories we can talk about exact sequences, subobjects and quotients.