Definition 12.13.4. Let \mathcal{A} be an abelian category.
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}.
A chain complex A_\bullet is called acyclic if all of its homology objects H_ i(A_\bullet ) are zero.
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