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

(if it exists) with differential

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

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