Definition 2.5.1.1. Let $\operatorname{\mathcal{A}}$ be an additive category, let $C_{\ast }$ be a chain complex with values in $\operatorname{\mathcal{A}}$, and let $n$ be an integer. We will say that $C_{\ast }$ is *concentrated in degrees $\geq n$* if objects $C_{m} \in \operatorname{\mathcal{A}}$ are zero for $m < n$. Similarly, we say that $C_{\ast }$ is *concentrated in degrees $\leq n$* if the objects $C_ m$ are zero for $m > n$. We let $\operatorname{Ch}(\operatorname{\mathcal{A}})_{\geq n}$ denote the full subcategory of $\operatorname{Ch}(\operatorname{\mathcal{A}})$ spanned by those chain complexes which are concentrated in degrees $\geq n$, and $\operatorname{Ch}(\operatorname{\mathcal{A}})_{\leq n}$ the full subcategory spanned by those chain complexes which are concentrated in degrees $\leq n$.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$