Lemma 12.14.9. Let $\mathcal{A}$ be an additive category. Suppose that $A^\bullet$ and $B^\bullet$ are cochain complexes. Given any morphism of cochain complexes $a : A^\bullet \to B^\bullet$ there is a bijection between the set of homotopies from $a$ to $a$ and $\mathop{\mathrm{Mor}}\nolimits _{\text{CoCh}(\mathcal{A})}(A^\bullet , B[-1]^\bullet )$. More generally, the set of homotopies between $a$ and $b$ is either empty or a principal homogeneous space under the group $\mathop{\mathrm{Mor}}\nolimits _{\text{CoCh}(\mathcal{A})}(A^\bullet , B[-1]^\bullet )$.

Proof. See above. $\square$

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