Definition 12.18.1. Let $\mathcal{A}$ be an additive category. A double complex in $\mathcal{A}$ is given by a system $(\{ A^{p, q}, d_1^{p, q}, d_2^{p, q}\} _{p, q\in \mathbf{Z}})$, where each $A^{p, q}$ is an object of $\mathcal{A}$ and $d_1^{p, q} : A^{p, q} \to A^{p + 1, q}$ and $d_2^{p, q} : A^{p, q} \to A^{p, q + 1}$ are morphisms of $\mathcal{A}$ such that the following rules hold:

1. $d_1^{p + 1, q} \circ d_1^{p, q} = 0$

2. $d_2^{p, q + 1} \circ d_2^{p, q} = 0$

3. $d_1^{p, q + 1} \circ d_2^{p, q} = d_2^{p + 1, q} \circ d_1^{p, q}$

for all $p, q \in \mathbf{Z}$.

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