Remark 12.20.3 (Variant). It is often the case that the terms of a spectral sequence have additional structure, for example a grading or a bigrading. To accomodate this (and to get around certain technical issues) we introduce the following notion. Let $\mathcal{A}$ be an abelian category. Let $(T_ r)_{r \geq 1}$ be a sequence of *translation* or *shift* functors, i.e., $T_ r : \mathcal{A} \to \mathcal{A}$ is an isomorphism of categories. In this setting a *spectral sequence* is given by a system $(E_ r, d_ r)_{r \geq 1}$ where each $E_ r$ is an object of $\mathcal{A}$, each $d_ r : E_ r \to T_ rE_ r$ is a morphism such that $T_ rd_ r \circ d_ r = 0$ so that

is a complex and $E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(T_ r^{-1}d_ r)$ for $r \geq 1$. It is clear what a *morphism of spectral sequences* means in this setting. In this setting we can still define

and $Z_\infty $ and $B_\infty $ (if they exist) as above.

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