Add _make_named_class_key for several categories, miscellaneous changes

[user202729]

Fixes #39154

Let C = Category.join([NumberFields(), QuotientFields(), MetricSpaces()]), then there are 4 categories being created

• Modules(C)
• VectorSpaces(C)
• Modules(C).Filtered()
• VectorSpaces(C).Filtered()

The last 2 have type FilteredModulesCategory, which inherit from RegressiveCovariantConstructionCategory, Category_over_base_ring which in turn inherits from CategoryWithParameters. Subclasses of CategoryWithParameters need to override _make_named_class_key correctly so that category A and B can only return same key if their parent class, element class, subcategory class etc. are identical. But FilteredModulesCategory takes _make_named_class_key from Category_over_base_ring, which returns the base category (C) for the key.

With this change, the returned key is e.g. Modules(C).parent_class and VectorSpaces(C).parent_class respectively, which is different.

Side note: by how things are implemented in SageMath, Modules(C) actually returns the same thing as VectorSpaces(C). The above diagram is only for illustration purpose.

Side note 2: Modules(C).Filtered() is currently a subcategory of VectorSpaces(C)… so the above is not a diamond, rather it's a linear chain.

Also make Modules(Fields())).Filtered() ≤ VectorSpaces(Fields()), and a few other changes e.g. make _test_category_graph works for hom categories.

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• I have created tests covering the changes.
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[tobiasdiez]

Thanks a lot! I can confirm that the linked bug is indeed fixed by these changes. Tests pass as well (modulo the usual random issues).

I don't really feel comfortable reviewing such fundamental changes in the category framework. So it would be awesome if one of the experts have a look as well.

[user202729]

Actually while working on this I realize the bug runs deeper than it looked. Just see how many tests #39170 breaks.

The current implementation is suboptimal as well (but yes, at least it works slightly better than the previous one).

I think @simon-king-jena is the best one tocomment on this (as author of #11943 and #11935 ) butthe PR was 13 years ago so I've no idea if the author is still contactable.

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There's a better way to implement this. Will copy from #39170 once it works.

[user202729]

I implement the cleaner solution (just getattr(category, base)), as well as modifying the documentation to add what I found out during the hours of debugging. Hopefully it saves whoever comes across this function some time.

The debug option to check the MRO is not in though (#39170). That would take more time.