The Stacks project

Remark 13.10.8. Let $\mathcal{A}$ be an additive category with countable direct sums. Let $\text{DoubleComp}(\mathcal{A})$ denote the category of double complexes in $\mathcal{A}$, see Homology, Section 12.18. We can use this category to construct two triangulated categories.

1. We can consider an object $A^{\bullet , \bullet }$ of $\text{DoubleComp}(\mathcal{A})$ as a complex of complexes as follows

   \[ \ldots \to A^{\bullet , -1} \to A^{\bullet , 0} \to A^{\bullet , 1} \to \ldots \]

   and take the homotopy category $K_{first}(\text{DoubleComp}(\mathcal{A}))$ with the corresponding triangulated structure given by Proposition 13.10.3. By Homology, Remark 12.18.6 the functor

   \[ \text{Tot} : K_{first}(\text{DoubleComp}(\mathcal{A})) \longrightarrow K(\mathcal{A}) \]

   is an exact functor of triangulated categories.

2. We can consider an object $A^{\bullet , \bullet }$ of $\text{DoubleComp}(\mathcal{A})$ as a complex of complexes as follows

   \[ \ldots \to A^{-1, \bullet } \to A^{0, \bullet } \to A^{1, \bullet } \to \ldots \]

   and take the homotopy category $K_{second}(\text{DoubleComp}(\mathcal{A}))$ with the corresponding triangulated structure given by Proposition 13.10.3. By Homology, Remark 12.18.7 the functor

   \[ \text{Tot} : K_{second}(\text{DoubleComp}(\mathcal{A})) \longrightarrow K(\mathcal{A}) \]

   is an exact functor of triangulated categories.