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Patrick Nicodemus (Jun 11 2025 at 04:42):If is a Grothendieck fibration, and has limits, and the fibers of have limits that are preserved by reindexing functors, then has all limits.
I've been trying to generalize this to 2-categories but I am realizing now that of course many interesting 2-limits are weighted rather than conical.
Is there a variant of this theorem for weighted limits, with a proof that doesn't just involve reducing it to the conical case?
Are all 2-limits reducible to conical limits via the Grothendieck construction? I assume so but I have not checked.
Nathanael Arkor (Jun 11 2025 at 07:09):Are all 2-limits reducible to conical limits via the Grothendieck construction?
No, it is necessary to enhance the domain of a 2-functor to something more expressive. This is the motivation for [[double limits]] and [[marked 2-limits]].
Patrick Nicodemus (Jun 11 2025 at 21:29):Ah, alright, cool. I've never heard of this theorem that they're equivalent, I'll check it out.