Definition 2.5.0.1. Let $\operatorname{\mathcal{A}}$ be an additive category (Definition 
). A chain complex with values in $\operatorname{\mathcal{A}}$ is a pair $( C_{\ast }, \partial )$, where $C_{\ast } = \{ C_ n \} _{n \in \operatorname{\mathbf{Z}}}$ is a collection of objects of $\operatorname{\mathcal{A}}$ and $\partial = \{ \partial _{n} \} _{n \in \operatorname{\mathbf{Z}}}$ is a collection of morphisms $\partial _{n}: C_ n \rightarrow C_{n-1}$ in $\operatorname{\mathcal{A}}$ with the property that each composition $\partial _{n} \circ \partial _{n+1}$ is the zero morphism from $C_{n+1}$ to $C_{n-1}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$