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

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ScriptRaccoon merged 14 commits intomainfrom
cartesian-filtered-colimits
Apr 16, 2026
Merged

Add properties: cartesian filtered colimits + cocartesian cofiltered limits#84
ScriptRaccoon merged 14 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 (or its negation): 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 efficiently 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.

The property "cartesian filtered colimits" has been decided for almost all categories in the database. Exceptions: LRS, Sch (let's not talk about these...), and Met_c (I don't even know right now if filtered colimits exist).

The property "cocartesian cofiltered limits" has been deciced for all categories in the database. Some proofs are quite interesting, for example why Grp does not satisfy this property.

@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 3 times, most recently from 5d18203 to 0ee5b26 Compare April 16, 2026 10:04
@ScriptRaccoon ScriptRaccoon marked this pull request as ready for review April 16, 2026 10:05
@ScriptRaccoon ScriptRaccoon force-pushed the cartesian-filtered-colimits branch 2 times, most recently from b741286 to 86e749a Compare April 16, 2026 10:14
@ScriptRaccoon ScriptRaccoon force-pushed the cartesian-filtered-colimits branch from baa1e95 to c183ce7 Compare April 16, 2026 11:05
@ScriptRaccoon ScriptRaccoon merged commit 55423e0 into main Apr 16, 2026
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@ScriptRaccoon ScriptRaccoon deleted the cartesian-filtered-colimits branch April 16, 2026 11:14
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