$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Example Let $\operatorname{\mathcal{A}}$ be an additive category and suppose we are given a chain complex

\[ \cdots \rightarrow C_{2} \xrightarrow {\partial } C_{1} \xrightarrow {\partial } C_{0} \xrightarrow {\partial } C_{-1} \xrightarrow { \partial } C_{-2} \rightarrow \cdots \]

with values in $\operatorname{\mathcal{A}}$. A contracting homotopy for $( C_{\ast }, \partial )$ is a chain homotopy from the zero morphism $0: C_{\ast } \rightarrow C_{\ast }$ to the identity morphism $\operatorname{id}: C_{\ast } \rightarrow C_{\ast }$ (in the sense of Definition More concretely, a contracting homotopy is a system of morphisms $\{ h_{n}: C_{n} \rightarrow C_{n+1} \} _{n \in \operatorname{\mathbf{Z}}}$ which satisfy the identity $\operatorname{id}_{ C_{n} } = \partial \circ h_{n} + h_{n-1} \circ \partial $ for every integer $n$. We will say that the complex $(C_{\ast }, \partial )$ is contractible if it admits a contracting homotopy.