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Matteo Capucci (he/him) (Jul 11 2024 at 08:25):Do free cocompletions under a chosen class of weights preserve existing limits? My guess is yes (e.g. taking presheaves does) but I can't think of a formal justification.
Morgan Rogers (he/him) (Jul 11 2024 at 08:58):One way to see this in the presheaf case is to consider the Isbell adjunction between the free completion and cocompletion; both adjoints commute with the respective Yoneda embeddings. However, it's a non-trivial fact that the free cocompletion is complete and vice-versa, which leads me to suspect that the preservation of limits should not be taken for granted.
Ivan Di Liberti (Jul 11 2024 at 09:04):Matteo Capucci (he/him) said:
Do free cocompletions under a chosen class of weights preserve existing limits? My guess is yes (e.g. taking presheaves does) but I can't think of a formal justification.
Yes. They do. The justification can be given by "direct inspection", in the sense that they can all be realised as full subcategories of presheaves, and simply nothing changes with respect to the argument that works for presheaves, everything "happens" in presheaves.
I do not have a reference (@Nathanael Arkor ). Notice that it is (!) true for cocompletions preserving a certain class of existing colimits.
Ivan Di Liberti (Jul 11 2024 at 09:09):Morgan Rogers (he/him) said:
One way to see this in the presheaf case is to consider the Isbell adjunction between the free completion and cocompletion; both adjoints commute with the respective Yoneda embeddings. However, it's a non-trivial fact that the free cocompletion is complete and vice-versa, which leads me to suspect that the preservation of limits should not be taken for granted.
You "preserve the limits that exists", or if you want, the inclusion is as flat as it gets.
Morgan Rogers (he/him) (Jul 11 2024 at 09:09):So @Ivan Di Liberti you don't need to, say, be enriching/considering presheaves enriched over an accessible/lfp category for it to hold?
Ivan Di Liberti (Jul 11 2024 at 09:10):Morgan Rogers (he/him) said:
So Ivan Di Liberti you don't need to, say, be enriching/considering presheaves enriched over an accessible/lfp category for it to hold?
No, the point is really, at the end of the day, that Yoneda preserve all existing limits. This has no size constraints.
Morgan Rogers (he/him) (Jul 11 2024 at 09:17):Oh of course, it's just the fact that essentially by definition, ; this is how the universal property of (weighted) limits is usually given!
Morgan Rogers (he/him) (Jul 11 2024 at 09:20):Thinking about the Isbell adjunction (which really might be hard to show exists in full generality, I suppose) was way overkill.
Ivan Di Liberti (Jul 11 2024 at 09:21):Indeed for the Isbell adjunction to exist you do need that (small) presheaves are complete, and this is a constraint on both V (that needs to be bicomplete) and on the domain of the presheaves (accessibility will be sufficient when V locally presentable).
Matteo Capucci (he/him) (Jul 11 2024 at 09:23):Ivan Di Liberti said:
in the sense that they can all be realised as full subcategories of presheaves, and simply nothing changes with respect to the argument that works for presheaves,
ah, of course!
Matteo Capucci (he/him) (Jul 11 2024 at 09:23):thanks :D
Nathanael Arkor (Jul 11 2024 at 12:40):A particularly simple abstract proof of this fact is that every dense and fully faithful functor creates limits: this follows from the fact that fully faithful functors are right [[relative adjoints]], and right relative adjoints preserve limits if the root is dense.
Nathanael Arkor (Jul 11 2024 at 12:41):(And every free cocompletion (under any class of weights) is dense and fully faithful, essentially by definition.)