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Add property: ℵ₁-filtered colimits#179

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ScriptRaccoon merged 11 commits into
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aleph1-filtered-colimits
May 19, 2026
Merged

Add property: ℵ₁-filtered colimits#179
ScriptRaccoon merged 11 commits into
mainfrom
aleph1-filtered-colimits

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@ScriptRaccoon ScriptRaccoon commented May 14, 2026

This PR adds two properties:

  • has ℵ₁-filtered colimits
  • is ℵ₁-filtered

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:

Theorem. If a category is locally finite, essentially countable, and Cauchy complete, then it is ℵ₁-accessible.

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.

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@ykawase5048 Can you perhaps help here?

@ScriptRaccoon ScriptRaccoon force-pushed the aleph1-filtered-colimits branch 2 times, most recently from a60230d to db9ec99 Compare May 14, 2026 23:29
@ScriptRaccoon ScriptRaccoon force-pushed the aleph1-filtered-colimits branch 2 times, most recently from b60f325 to 89e1bdb Compare May 15, 2026 15:52
@ScriptRaccoon ScriptRaccoon changed the title Add property: ℵ₁-filtered colimits Add property: ℵ₁-filtered colimits (WIP) May 15, 2026
@ScriptRaccoon ScriptRaccoon force-pushed the aleph1-filtered-colimits branch from 89e1bdb to 4c98d04 Compare May 17, 2026 17:01
@ScriptRaccoon ScriptRaccoon force-pushed the aleph1-filtered-colimits branch 2 times, most recently from 9e8e4bd to b45f47a Compare May 19, 2026 08:50
@ScriptRaccoon ScriptRaccoon marked this pull request as ready for review May 19, 2026 08:51
@ScriptRaccoon ScriptRaccoon changed the title Add property: ℵ₁-filtered colimits (WIP) Add property: ℵ₁-filtered colimits May 19, 2026
the theorem states that every locally finite, essentially countable and cauchy complete category is ℵ₁-accessible
@ScriptRaccoon ScriptRaccoon force-pushed the aleph1-filtered-colimits branch from 0d1f0bd to 390b976 Compare May 19, 2026 22:18
@ScriptRaccoon ScriptRaccoon force-pushed the aleph1-filtered-colimits branch from 390b976 to 794b517 Compare May 19, 2026 22:36
@ScriptRaccoon ScriptRaccoon merged commit fa4e5c3 into main May 19, 2026
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@ScriptRaccoon ScriptRaccoon deleted the aleph1-filtered-colimits branch May 19, 2026 22:37
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