Definition:Chain Complex

Definition
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 $0 : C_i \to C_{i-2}$.

Visualization
A chain complex can be visualized as a diagram \begin{align*} \xymatrix{ \dots \ar[r] & C_{i+1} \ar[r]^{d_{i+1}} & C_{i} \ar[r]^{d_{i}} & C_{i-1} \ar[r]^{d_{i-1}} & C_{i-2} \ar[r] & \dots } \end{align*}

Also see

 * Definition:Differential Complex
 * Definition:Null Sequence (Homological Algebra)