For simplify, let's just consider the following category first:

  • object: "spectral sequences", that is a sequence of pages $(E_r)_{r\ge0}$, where each page $E_r$ is a differential object in an abelian category $\mathcal{A}$, equipped with a sequences of isomorphisms $H(E_r)\cong E_{r+1}$.
  • morphisms: a sequences of morphisms of differential objects $u_r\colon E_r\to E'_r$ such that the following diagrams commute. \begin{array} HH(E_r) & \stackrel{H(u_r)}{\longrightarrow} & H(E'_r) \\ \downarrow{\cong} & & \downarrow{\cong} \\ E_{r+1} & \stackrel{u_{r+1}}{\longrightarrow} & E'_{r+1} \end{array}

My question is: what can we say about this category? Is it an abelian category? If $\mathcal{A}$ satisfies more properties such as AB3,AB4,AB5, will the above category satisfy? If $\mathcal{A}$ is an abelian tensor category, or furthermore cocomplete tensor category, rigid tensor category etc., what about the above category?

If answer to aboves are positive, how about the following category?

  • object: convergent spectral sequences equipped with target object. Here the bigraded version of spectral sequences may be needed to state the conditional convergences.
  • morphisms: a morphism between spectral sequences together with a morphism between their target objects satisfying the compatible conditions.

Update I feel that in general these cannot be true. Unless certain taking limit/colimit functor is exact, it shouldn't expect the limit/colimit of spectral sequences exists or have good properties. For example, consider kernels and cokernels.

  • $\begingroup$ The category of spectral sequences is not abelian. This is a remark in Grothendieck's Tohoku paper. It is naturally $\mathsf{Ab}$-enriched. I believe it has biproducts and a zero object, since homology behaves well with biproducts. It is probably not possible to define kernels and cokernels pointwise, because they are not exact and behave not well with homology. If someone has an answer to this question after so many years, I would like to hear it. $\endgroup$
    – Merle
    May 19 '21 at 9:15

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