The Stacks project

Definition 12.13.4. Let $\mathcal{A}$ be an abelian category.

1. A morphism of chain complexes $f : A_\bullet \to B_\bullet $ is called a quasi-isomorphism if the induced map $H_ i(f) : H_ i(A_\bullet ) \to H_ i(B_\bullet )$ is an isomorphism for all $i \in \mathbf{Z}$.

2. A chain complex $A_\bullet $ is called acyclic if all of its homology objects $H_ i(A_\bullet )$ are zero.