Add properties: cartesian filtered colimits + cocartesian cofiltered limits#84
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ScriptRaccoon wants to merge 9 commits intomainfrom
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Add properties: cartesian filtered colimits + cocartesian cofiltered limits#84ScriptRaccoon wants to merge 9 commits intomainfrom
ScriptRaccoon wants to merge 9 commits intomainfrom
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this settles in particular the category of sets (and many more)
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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.sqlfor 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 ofthin + 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
Ban,Met_c(then onlyLRSandSchare left, which are weird anyway)Ban,Set_*,Top_*,Rng(then 0 are left)