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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.
Patrick Nicodemus (Jun 29 2025 at 19:11):@Nathanael Arkor the original paper by Grandis and Pare proves that a 2-category is complete iff its double category of quintets is complete. I was hoping for a more precise theorem in the form of a kind of double Grothendieck construction: given a diagram 2-category P with Cat valued presheaf of weights J, and F : P -> A, construct a double category and a functor in quintets of A whose limit is the weighted limit of F with weights in J.
Does such a theorem exist or should I prove this
Nathanael Arkor (Jun 29 2025 at 22:00):Yes, this does exist, though for some reason it's not recorded in the Grandis–Paré paper. It was first presented by Paré in a talk, and is written up as Proposition 1.4 of Grandis–Paré's Persistent double limits and flexible weighted limits. I believe it's also covered in Chapter 2 of Verity's thesis Enriched Categories, Internal Categories and Change of Base.
Nathanael Arkor (Jun 29 2025 at 22:01):Note that, for these purposes, it is more appropriate to view a 2-category as a loosely-discrete double category, rather than via the quintet construction.
Patrick Nicodemus (Jun 30 2025 at 00:48):Nathanael Arkor said:
Note that, for these purposes, it is more appropriate to view a 2-category as a loosely-discrete double category, rather than via the quintet construction.
Ah, I was worried you might say something like that! This paper gives a definition of fibrations between double categories and proves that a functor between 2-categories is a 2-fibration iff it's a fibration as a functor between the associated double categories of quintets. https://arxiv.orgabs/2205.15240. I had hoped to use this theorem here, but based on what you are saying, I will still have to do a little work in order to prove this limit lifting theorem.
Nathanael Arkor (Jun 30 2025 at 08:45):I think the discussion in CLPS misses an important point, which is that it is not really reasonable to compare fibrations of double categories and 2-fibrations directly in the way they do (although it is elucidating that they do so, to demonstrate this point). Suppose we have a discrete 2-fibration. This corresponds to a 2-functor . We can then take the double category of elements of this 2-functor to get a fibration of double categories.
Nathanael Arkor (Jun 30 2025 at 08:48):However, if you care about general 2-fibrations, which correspond to 2-functors into , then I suspect fibrations of double categories are insufficient, since these take values in spans of categories, and that one really needs a three-dimensional notion.
Patrick Nicodemus (Jun 30 2025 at 16:50):Well, it seems okay to restrict to discrete 2-fibrations/2-presheaves of categories here if I am interested in weighted limits, right?
Nathanael Arkor (Jun 30 2025 at 17:06):Fibrations appear in two different roles in your setting: the 2-fibration along which you want to lift limits; and the discrete 2-fibrations associated with the weights for the limits.
Nathanael Arkor (Jun 30 2025 at 17:07):Presumably you don't just want to lift limits along discrete 2-fibrations?
Patrick Nicodemus (Jul 06 2025 at 13:06):Nathanael Arkor said:
Presumably you don't just want to lift limits along discrete 2-fibrations?
Yes, this is true. I will work this out and see if there are problems. Thank you