Definition 12.13.8. Let $\mathcal{A}$ be an additive category. We say a morphism $a : A^\bullet \to B^\bullet $ is a *homotopy equivalence* if there exists a morphism $b : B^\bullet \to A^\bullet $ such that there exists a homotopy between $a \circ b$ and $\text{id}_ A$ and there exists a homotopy between $b \circ a$ and $\text{id}_ B$. If there exists such a morphism between $A^\bullet $ and $B^\bullet $, then we say that $A^\bullet $ and $B^\bullet $ are *homotopy equivalent*.

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