\begin{definition}[Definition:Chain Complex]
Let $\AA$ be an abelian category.
A '''chain complex''' in $\AA$ is a family of objects $\family {C_i}_{i \mathop \in \Z}$ of $\AA$ and a family of morphisms $\family {d_i : C_i \to C_{i - 1} }_{i \mathop \in \Z}$, such that for all $i \in \Z$, the composition $d_{i - 1} \circ d_i$ is the zero morphism $0 : C_i \to C_{i - 2}$.
\end{definition}