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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.

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