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 $\family {\operatorname{id}_{C_i} }_{i \mathop \in \Z}$.

Also see

• Category of Chain Complexes is Category
• Category of Chain Complexes is Abelian

Sources

• 1994: Charles Weibel: An Introduction to Homological Algebra: $\S 1.1$.