Definition 12.18.3. Let $\mathcal{A}$ be an additive category. Let $A^{\bullet , \bullet }$ be a double complex. The associated simple complex, denoted $sA^\bullet$, also often called the associated total complex, denoted $\text{Tot}(A^{\bullet , \bullet })$, is given by

$sA^ n = \text{Tot}^ n(A^{\bullet , \bullet }) = \bigoplus \nolimits _{n = p + q} A^{p, q}$

(if it exists) with differential

$d_{sA^\bullet }^ n = d_{\text{Tot}(A^{\bullet , \bullet })}^ n = \sum \nolimits _{n = p + q} (d_1^{p, q} + (-1)^ p d_2^{p, q})$

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