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Add properties: cartesian filtered colimits + cocartesian cofiltered limits#84

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ScriptRaccoon wants to merge 9 commits intomainfrom
cartesian-filtered-colimits
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Add properties: cartesian filtered colimits + cocartesian cofiltered limits#84
ScriptRaccoon wants to merge 9 commits intomainfrom
cartesian-filtered-colimits

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@ScriptRaccoon ScriptRaccoon commented Apr 15, 2026

Several proofs that certain categories are not cartesian closed or do not have exact filtered colimits have used a different property: for every object $X$ the functor $X \times -$ preserves filtered colimits. This property has several connections to existing properties in the database, so it makes sense to add this here. My question on mathoverflow indicates that there is no established name yet, but cartesian filtered colimits has been suggested. So let's take this here.

Adding the property of having cartesian filtered colimits along with several implications indeed has clarified a lot of proofs in the database, and some could even be removed since they are now automatically generated (see the change in database/data/004_property-assignments/Meas.sql for example).

Another motivation to add this property was to decide if the partial order $[0,1]$ (regarded as a thin category) has exact filtered colimits; this was missing before this PR. This is now done automatically by combining several implications: cartesian closed + filtered colimits => cartesian filtered colimits, thin + cartesian filtered colimits => exact filtered colimits, and we already knew that it is cartesian closed because of thin + semi-strongly connected => locally cartesian closed.

Since the deduction system automatically dualizes implications, it makes sense to directly also add the dual property of having cocartesian cofiltered limits (which is even less common). This means that $X \sqcup -$ preserves cofiltered limits. Deciding this property for some categories has lead to interesting mathematics.

TODOS

  • decide cartesian filtered colimits for Ban, Met_c (then only LRS and Sch are left, which are weird anyway)
  • decide cocartesian cofiltered limits for Ban, Set_*, Top_*, Rng (then 0 are left)

@ScriptRaccoon ScriptRaccoon force-pushed the cartesian-filtered-colimits branch from 88db21e to 811ccc5 Compare April 15, 2026 11:31
@ScriptRaccoon ScriptRaccoon changed the title Add property: cartesian filtered colimits Add properties: cartesian filtered colimits + cocartesian cofiltered limits Apr 15, 2026
@ScriptRaccoon ScriptRaccoon force-pushed the cartesian-filtered-colimits branch from d328b31 to b0517d2 Compare April 15, 2026 20:09
@ScriptRaccoon ScriptRaccoon marked this pull request as draft April 15, 2026 20:09
@ScriptRaccoon ScriptRaccoon force-pushed the cartesian-filtered-colimits branch from c468cf6 to 44d0c90 Compare April 15, 2026 21:33
@ScriptRaccoon ScriptRaccoon force-pushed the cartesian-filtered-colimits branch from 44d0c90 to 5d18203 Compare April 15, 2026 21:34
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