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

$sA^ n = \bigoplus \nolimits _{n = p + q} A^{p, q}$

(if it exists) with differential

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

Alternatively, we often write $\text{Tot}(A^{\bullet , \bullet })$ to denote this complex.

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