Kerodon

Example 2.5.4.2. Let $\operatorname{\mathcal{A}}$ be an additive category, let $C_{\ast }$ and $D_{\ast }$ be chain complexes with values in $\operatorname{\mathcal{A}}$, and let $f,f': C_{\ast } \rightarrow D_{\ast }$ be chain maps, which we regard as $0$-cycles in the mapping complex $\operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathcal{A}})}( C_{\ast }, D_{\ast } )_{\ast }$ in the differential graded category $\operatorname{Ch}(\operatorname{\mathcal{A}})$ of Example 2.5.2.5. Let $h = \{ h_ n: C_{n} \rightarrow D_{n+1} \} _{n \in \operatorname{\mathbf{Z}}}$ be a collection of morphisms, which we regard as a $1$-chain of $\operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathcal{A}})}( C_{\ast }, D_{\ast } )_{\ast }$. Then $h$ is a homotopy from $f$ to $f'$ (in the sense of Definition 2.5.4.1) if and only if it is a chain homotopy from $f$ to $f'$ (in the sense of Definition 2.5.0.5). In particular, $f$ and $f'$ are homotopic morphisms of the differential graded category $\operatorname{Ch}(\operatorname{\mathcal{A}})$ (in the sense of Definition 2.5.4.1) if and only if they are chain homotopic (in the sense of Definition 2.5.0.5).