Kerodon

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Example 2.5.0.6. 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 2.5.0.5). 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.