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Joe Moeller (Jun 09 2020 at 22:44):I was reading about Gabriel Ulmer duality: https://ncatlab.org/nlab/show/Gabriel-Ulmer+duality
It says that the 2-categories [finitely complete cats, finite limit preserving functors, natural transformations] and [locally finitely presentable cats, finitary right adjoint functors, natural transformations] are equivalent.
Joe Moeller (Jun 09 2020 at 22:45):I was wondering if there's a generalization for locally -presentable categories, for some cardinal , or locally presentable categories?
John Baez (Jun 09 2020 at 23:06):Good question! Nitpick: there's an "op" in the equivalence you mentioned, since it comes from the map sending a finitely complete category to its category of models , which is contravariant.
Joe Moeller (Jun 09 2020 at 23:08):Fixed.
Nathanael Arkor (Jun 09 2020 at 23:44):Yes, this follows from the general duality theorem of Centazzo–Vitale's A duality relative to a limit doctrine, where the limit doctrine is chosen to be the doctrine of -small limits (see Adámek–Borceux–Lack–Rosický's A classification of accessible categories).
Joe Moeller (Jun 09 2020 at 23:45):Thanks!
John Baez (Jun 09 2020 at 23:58):Great!
Jens Hemelaer (Jun 10 2020 at 06:00):Joe Moeller said:
I was wondering if there's a generalization for locally -presentable categories, for some cardinal , or locally presentable categories?
The statement (based on Centazzo-Vitale) is written down in detail in Gabriel-Ulmer duality for topoi and its relation with site presentations by Ivan Di Liberti and Julia Ramos González.