Example 2.5.2.5 (Chain Complexes). Let $\operatorname{\mathcal{A}}$ be an additive category. We define a differential graded category $\operatorname{Ch}(\operatorname{\mathcal{A}})$ as follows:
The objects of $\operatorname{Ch}(\operatorname{\mathcal{A}})$ are chain complexes with values in $\operatorname{\mathcal{A}}$ (Definition 2.5.0.1).
If $C_{\ast }$ and $D_{\ast }$ are chain complexes with values in $\operatorname{\mathcal{A}}$, then $\operatorname{Hom}_{\operatorname{Ch}(\operatorname{\mathcal{A}})}( C_{\ast }, D_{\ast } )_{\ast }$ is the chain complex of abelian groups $[C,D]_{\ast }$ defined in Construction 2.5.0.10.
If $C_{\ast }$, $D_{\ast }$, and $E_{\ast }$ are chain complexes with values in $\operatorname{\mathcal{A}}$, then the composition law
\[ \circ : [D,E]_{e} \times [C,D]_{d} \rightarrow [C,E]_{d+e} \]is given by the formula $\{ g_{n} \} _{n \in \operatorname{\mathbf{Z}}} \circ \{ f_ n \} _{n \in \operatorname{\mathbf{Z}}} = \{ g_{n+d} \circ f_{n} \} _{n \in \operatorname{\mathbf{Z}}}$.
Note that if $C_{\ast }$ and $D_{\ast }$ are chain complexes with values in $\operatorname{\mathcal{A}}$, then a collection of maps $f = \{ f_ n: C_ n \rightarrow D_ n \} _{n \in \operatorname{\mathbf{Z}}}$ is a $0$-cycle of the chain complex $[C,D]_{\ast }$ if and only if it is a chain map from $C_{\ast }$ to $D_{\ast }$. Consequently, applying Construction 2.5.2.4 to the differential graded category $\operatorname{Ch}(\operatorname{\mathcal{A}})$ yields the ordinary category of chain complexes and chain maps. In other words, this construction supplies a $\operatorname{Ch}(\operatorname{\mathbf{Z}})$-enrichment of the category $\operatorname{Ch}(\operatorname{\mathcal{A}})$ introduced in Definition 2.5.0.3.