Definition 13.13.2. Let $\mathcal{A}$ be an abelian category.

Let $\alpha : K^\bullet \to L^\bullet $ be a morphism of $K(\text{Fil}^ f(\mathcal{A}))$. We say that $\alpha $ is a

*filtered quasi-isomorphism*if the morphism $\text{gr}(\alpha )$ is a quasi-isomorphism.Let $K^\bullet $ be an object of $K(\text{Fil}^ f(\mathcal{A}))$. We say that $K^\bullet $ is

*filtered acyclic*if the complex $\text{gr}(K^\bullet )$ is acyclic.

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