Definition 13.8.1. Let $\mathcal{A}$ be an additive category.

We set $\text{Comp}(\mathcal{A}) = \text{CoCh}(\mathcal{A})$ be the

*category of (cochain) complexes*.A complex $K^\bullet $ is said to be

*bounded below*if $K^ n = 0$ for all $n \ll 0$.A complex $K^\bullet $ is said to be

*bounded above*if $K^ n = 0$ for all $n \gg 0$.A complex $K^\bullet $ is said to be

*bounded*if $K^ n = 0$ for all $|n| \gg 0$.We let $\text{Comp}^{+}(\mathcal{A})$, $\text{Comp}^{-}(\mathcal{A})$, resp. $\text{Comp}^ b(\mathcal{A})$ be the full subcategory of $\text{Comp}(\mathcal{A})$ whose objects are the complexes which are bounded below, bounded above, resp. bounded.

We let $K(\mathcal{A})$ be the category with the same objects as $\text{Comp}(\mathcal{A})$ but as morphisms homotopy classes of maps of complexes (see Homology, Lemma 12.12.7).

We let $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, resp. $K^ b(\mathcal{A})$ be the full subcategory of $K(\mathcal{A})$ whose objects are bounded below, bounded above, resp. bounded complexes of $\mathcal{A}$.

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