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David Michael Roberts (Nov 06 2024 at 12:17):For reference, I'm talking about notion introduced in the paper "Fibrations of bigroupoids" by Hardie, Kamps and Kieboom (essentially: you can lift n-arrows given a lift of their source, though in my case I'm starting with lifts of targets, for n=1,2)
I have a consstruction that uses the strict fibre of a surjective-on-objects fibration of bigroupoids, and I am claiming this is equivalent to the weak/homotopy fibre. This feels like something someone might have done, but I'm not entirely sure where.
In principle one could try something like restricting Lack's canonical model structure for bicategories (my 2-functor is in fact strict), and then checking that homotopy fibres correspond to weak fibres in the usual sense, and match up all the concepts, and so on. I don't have easy access to this paper.
Alternatively, one might imagine that someone has done some fairly nuts-and-bolts computation to check things work out, without resorting to model category tech. This feels unlikely.
Yet again, someone might have done enough work on (semi?-)model structures on Kan complexes/sSet and how the Duskin/geometric nerve of a bigroupoid is a certain kind of Kan complex, and then general model category stuff in the simplicial world takes over. This is rather far afield from what I am familiar with, and probably overkill.
Advice?
David Michael Roberts (Nov 06 2024 at 12:22):There is also this paper I just found, giving the model structure on bigroupoids and weak 2-functors: https://arxiv.org/abs/1809.00963
David Michael Roberts (Nov 06 2024 at 12:28):I guess I'm after a reference that I can point to that justifies _some_ approach, because I'm secretly wanting to show that the appropriately weak fibre of a map of 2-stacks of bigroupoids is computed using this much more tractable strict fibre that I can have a good handle on.
Oh, and I should say that this strict fibre is a legit 1-groupoid, rather than something like a bigroupoid merely equivalent to a 1-groupoid. That's what I'd rather work with!
Kevin Carlson (Nov 06 2024 at 16:59):Lack sounds like the right starting point to me. I’m not sure I know what difference you’re drawing between homotopy fibers and weak fibers so take this as only somewhat confident. Mike would probably know better if he’s around.
Kevin Carlson (Nov 06 2024 at 17:05):Here’s a copy of the paper. a-quillen-model-structure-for-bicategories.pdf
David Michael Roberts (Nov 06 2024 at 21:59):Well, either the homotopy fibres (defined using fibrant replacement etc) are equivalent to the weak fibres (defined using a general bicategorical construction), or not. This needs proving, and may well be the core of what needs to be done. I think the strict fibre in this case may well be the same as the homotopy pullback (since I have a fibration to start with and every object in Lack's model structure is fibrant), but this hasn't "solved the problem", since I still need to compare the weak and homotopy fibres.
David Michael Roberts (Nov 06 2024 at 22:20):Hmm. thinking about the path object in Lack's model structure, how every object is fibrant, means if we treat the model structure as a category of fibrant objects, then according to this proposition, the using path-object strict construction of the homotopy fibre product (which should be the weak fibre) this will give the homotopically correct limit.
But then this gives two constructions of the homotopy fibre, I think, and so they should be (bi)equivalent.
David Michael Roberts (Nov 07 2024 at 00:48):So to collect my thoughts:
David Michael Roberts (Nov 07 2024 at 00:51): David Michael Roberts (Nov 07 2024 at 01:02):OK, so this follows from this corollary, as it should.
Kevin Carlson (Nov 07 2024 at 01:31):Great!