Definition:Morphism of Chain Complexes

Definition

Let $\AA$ be an abelian category.

Let $\family {C_i, d_i}_{i \mathop \in \Z}$ and $\family {C_i', d_i'}_{i \mathop \in \Z}$ be chain complexes in $\AA$.

A morphism of chain complexes from $\family {C_i}_{i \mathop \in \Z}$ to $\family {D_i}_{i \mathop \in \Z}$ is a family of morphisms $\family {f_i: C_i \to C_i'}_{i \mathop \in \Z}$, such that:

$\forall i \in \Z: f_{i-1} \circ d_i = d_i' \circ f_i$

Visualization

A morphism of chain complexes can be visualized by a commutative diagram:

$\begin{xy} \xymatrix{ \dots \ar[r] & C_i \ar[r]^{d_i} \ar[d]^{f_i} & C_{i-1} \ar[d]^{f_{i-1}} \ar[r] & \dots \\ \dots \ar[r] & C_i' \ar[r]^{d_i'} & C_{i-1}' \ar[r] & \dots } \end{xy}$

Also see

• Definition:Homomorphism of Complexes

Sources

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