Definition 12.13.10. Let $\mathcal{A}$ be an abelian category.

1. A morphism of cochain complexes $f : A^\bullet \to B^\bullet$ of $\mathcal{A}$ is called a quasi-isomorphism if the induced maps $H^ i(f) : H^ i(A^\bullet ) \to H^ i(B^\bullet )$ is an isomorphism for all $i \in \mathbf{Z}$.

2. A cochain complex $A^\bullet$ is called acyclic if all of its cohomology objects $H^ i(A^\bullet )$ are zero.

Comment #1335 by jpg on

if the induced map ... is an isomorphism

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