chain homotopy equivalence

1. Definition

Let \(\mathcal{A}\) be an abelian category and \(\mathrm{Ch_*}(\mathcal{A})\) be the category of chain complexes. Suppose \(M_n,M_n' \in \mathrm{Ob}(\mathrm{Ch_*}(\mathcal{A}))\) are chain complexes.

Then a chain homotopy equivalence between \(M_n,M_n'\) consists of chain maps

\begin{align*} f: M_n \rightarrow M_n' \\ g: M_n' \rightarrow M_n \end{align*}

such that

\begin{align*} f \circ g \cong& \mathrm{id}_{M_n} \\ g \circ f \cong& \mathrm{id}_{M_n'} \\ \end{align*}