Add property: ℵ₁-filtered colimits#179
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@ykawase5048 Can you perhaps help here? |
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the theorem states that every locally finite, essentially countable and cauchy complete category is ℵ₁-accessible
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This PR adds two properties:
As for being ℵ₁-filtered, all categories have been decided. For all categories except Sch, the existence of ℵ₁-filtered colimits has been decided.
For the categories Man, Grpc, and Setc, the database already contained proofs that they are not ℵ₁-accessible, but these proofs in fact showed that ℵ₁-filtered colimits do not exist. These were quick wins.
To decide the existence of ℵ₁-filtered colimits for various small categories, and to determine their ℵ₁-accessibility, the following theorem has been added together with its (long) proof:
In particular, ℵ₁-filtered colimits exist in this case. The theorem implies ℵ₁-accessibility of FinOrd,$\Delta$ , FI, FS, and B (which was previously open), as well as FinSet, FinGrp, and FinAb (for which proofs already existed, albeit imprecise ones).
This PR also adds a proof that Sp is ℵ₁-accessible, that BN is ℵ₁-accessible and that BOn has ℵ₁-filtered colimits (although it is not accessible since it is not well-powered). Questions on mathoverflow have shown that FreeAb and Metc do not have ℵ₁-filtered colimits. In particular, these categories are not ℵ₁-accessible, which has been open before. That being said, this PR already does some of the work for #91.
The only remaining category is Sch, the category of schemes. As of writing this, CatDat does not even record whether it has filtered colimits (presumably not, otherwise ind-schemes would not be a thing). Deciding whether it has ℵ₁-filtered colimits has therefore been postponed, as has already been done for several other properties of this category.