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Definition Let $( C_{\ast }, \partial _ C)$ and $( D_{\ast }, \partial _{D} )$ be chain complexes with values in an additive category $\operatorname{\mathcal{A}}$. A chain map from $( C_{\ast }, \partial _ C)$ and $( D_{\ast }, \partial _{D} )$ is a collection $f = \{ f_ n \} _{n \in \operatorname{\mathbf{Z}}}$, where each $f_ n$ is a morphism from $C_ n$ to $D_{n}$ in the category $\operatorname{\mathcal{A}}$, for which each of the diagrams

\[ \xymatrix@R =50pt@C=50pt{ C_{n} \ar [r]^-{ \partial _{C} } \ar [d]^{ f_ n } & C_{n-1} \ar [d]^{ f_{n-1} } \\ D_{n} \ar [r]^-{ \partial _{D} } & D_{n-1} } \]

is commutative.

If $\operatorname{\mathcal{A}}$ is an additive category, we let $\operatorname{Ch}(\operatorname{\mathcal{A}})$ denote the category whose objects are chain complexes with values in $\operatorname{\mathcal{A}}$ and whose morphisms are chain maps.