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Definition (Chain Homotopy). Let $\operatorname{\mathcal{A}}$ be an additive category and let $(C_{\ast }, \partial _ C)$ and $(D_{\ast }, \partial _ D)$ be chain complexes with values in $\operatorname{\mathcal{A}}$. Let $f = \{ f_ n \} _{n \in \operatorname{\mathbf{Z}}}$ and $f' = \{ f'_ n \} _{n \in \operatorname{\mathbf{Z}}}$ be chain maps from $C_{\ast }$ to $D_{\ast }$. A chain homotopy from $f$ to $f'$ is a collection of maps $h = \{ h_ n: C_ n \rightarrow D_{n+1} \} $ which satisfy the identity

\[ f'_ n - f_{n} = \partial _{D} \circ h_{n} + h_{n-1} \circ \partial _ C \]

for every integer $n$.

We say that $f$ and $f'$ are chain homotopic if there exists a chain homotopy from $f$ to $f'$. We will say that $f$ is a chain homotopy equivalence if there exists a chain map $g: D_{\ast } \rightarrow C_{\ast }$ such that $g \circ f$ and $f \circ g$ are chain homotopic to the identity morphisms $\operatorname{id}_{ C_{\ast } }$ and $\operatorname{id}_{ D_{\ast } }$, respectively.