Definition:Homology of Chain Complex

Definition
Let $\AA$ be an abelian category.

Let $C := \family {d_i : C_i \to C_{i - 1} }_{i \mathop \in \Z}$ be a chain complex in $\AA$.

The $j$-th homology (object) $\map {H_j} C$ of $C$ is defined as the cokernel of the canonical map $\Img {d_{i + 1} } \to \ker \paren {d_i}$.

The canonical map is induced by Homomorphisms Theorem for Categories with Zero Object since by definition $d_{j} \circ d_{j + 1} = 0$.

Also see

 * Chain Complex is Exact iff Zero Homology