# Definition:Category of Chain Complexes

## Definition

Let $\AA$ be an abelian category.

The category of chain complexes of $\AA$, denoted $\map {\mathbf{Ch}} \AA$, is the metacategory with:

 Objects: chain complexes in $\AA$ Morphisms: morphisms of chain complexes in $\AA$ Composition: If $\family {C_i}_{i \mathop \in \Z}$, $\family {D_i}_{i \mathop \in \Z}$ and $\family {E_i}_{i \mathop \in \Z}$ are chain complexes in $\AA$ and $\family {f_i : C_i \to D_i}_{i \mathop \in \Z}$ and $\family {g_i : D_i \to E_i}_{i \mathop \in \Z}$ are morphisms of chain complexes, then the composition of $\family {f_i : C_i \to D_i}_{i \mathop \in \Z}$ and $\family {g_i: D_i \to E_i}_{i \mathop \in \Z}$ is defined as $\family {g_i \circ f_i: C_i \to E_i}_{i \mathop \in \Z}$. Identity morphisms: If $\family {C_i}_{i \mathop \in \Z}$ is a chain complex in $\AA$, the identity morphism of $\family {C_i}_{i \mathop \in \Z}$ is defined as $\family {\operatorname{id}_{C_i} }_{i \mathop \in \Z}$.