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Add the categories of graded modules and cochain complexes - #327

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ScriptRaccoon merged 4 commits into
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Aug 12, 2026
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Add the categories of graded modules and cochain complexes#327
ScriptRaccoon merged 4 commits into
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graded-modules

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@ScriptRaccoon ScriptRaccoon commented Aug 12, 2026

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This PR adds the category $\mathbf{grMod}_G(R)$ of $G$-graded modules over a $G$-graded ring $R$, where $G$ is any monoid and $R$ is only assumed to be non-trivial. In this generality, two properties cannot be decided: (one-sorted) finitary algebraic and split abelian. However, they can be decided for three special cases, which are also added as separate entries to the database:

  • the category $\mathbf{Ab}^{(\mathbb{N},\leq)}$ of sequences of abelian groups,
  • the category $\mathbf{Ab}^{\mathbb{Z}}$ of $\mathbb{Z}$-graded abelian groups,
  • the category $\mathbf{Ch}(\mathbf{Ab})$ of cochain complexes of abelian groups.

Two of these have been mentioned in the issue #14.

All three have the category of graded modules as a parent (a feature only recently introduced in #323), so all properties except for the two mentioned above are inherited directly. Each of the three child categories is neither (one-sorted) finitary algebraic nor split abelian.

Currently, the three categories are indistinguishable. But this is just because some properties are missing from the database. For example, I think that $\mathbf{Ab}^{\mathbb{Z}}$ is hereditary, but $\mathbf{Ab}^{(\mathbb{N},\leq)}$ is not.

@ScriptRaccoon ScriptRaccoon changed the title Add the categories of graded modules and chain complexes Add the categories of graded modules and cochain complexes Aug 12, 2026
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ScriptRaccoon merged commit a2cc641 into main Aug 12, 2026
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ScriptRaccoon deleted the graded-modules branch August 12, 2026 07:50
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