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Stream: learning: questions

Topic: Thomason model structure

James Deikun (Jan 21 2024 at 01:54):

The [[Thomason model structure]] is a model structure on $\mathsf{Cat}$ that presents the $(\infty,1)$-category of CW complexes, or $\infty$-groupoids. The weak equivalences of this model structure are the functors that induce a weak equivalence of the [[Kan-Quillen model structure]] on the standard simplicial nerve of the domain and codomain. There's a Quillen equivalence of the two model structures where the right adjoint is $\mathrm{Ex} \circ \mathrm{Ex} \circ \mathcal{N}$ and $\mathrm{Ex}$ is the right adjoint to [[barycentric subdivision]].

James Deikun (Jan 21 2024 at 01:58):

According to the theory of [[test categories]] there is also a Quillen adjunction $\int_\Delta \dashv \mathcal{N}_\Delta$ where the former is the [[category of simplices]] and the latter is $\mathsf{Cat}(\Delta/\!\! =,-)$. I'm rather confused about the relation between this adjunction and the former one. Does $\mathsf{Cat}$ in this other adjunction have the Thomason model structure as well, or is it a different one with the same weak equivalences? (And: is that actually a thing? I don't know of any examples offhand but it doesn't seem like it should be impossible based on the theorems I know...) And is this adjunction a Quillen equivalence? It seems like for a test category it should be. Does it somehow read out to be the same Quillen equivalence as above, or is it different?

Evan Cavallo (Jan 21 2024 at 15:36):

For the existence of model structures on Cat with the same weak equivalences as but different cofibrations than the Thomason model structure, there's Section 3 of

George Raptis, Homotopy theory of posets, Homology Homotopy Appl. 12 2 (2010) 211-230
https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/Homotopy-theory-of-posets/hha/1296223882.full

I (by no means an expert, just acquainted with some bits of Cisinski 06) am not even sure where to find a claim that $\int_\Delta \dashv \mathcal{N}_\Delta$ is a Quillen adjunction between anything.
Cisinski does bring up the Thomason model structure (Théorème 5.2.12 in Cisinski 06), but when he talks about Cat he is often just working with a [[basic localizer]] on it (so weak equivalences but not any notion of cofibration).

James Deikun (Jan 21 2024 at 15:48):

For a claim that $\int_\Delta \dashv \mathcal{N}_\Delta$ is a Quillen adjunction, see page 3 of https://www.math.uwo.ca/faculty/kapulkin/notes/test_categories.pdf toward the bottom (paragraph starting "Even though"). Now, it's not exactly specific as to what this is a Quillen adjunction between!

Evan Cavallo (Jan 22 2024 at 19:20):

I think it can't be the Thomason model structure.
If $\int_\Delta$ were a left Quillen functor, it would have to preserve cofibrant objects.
Every simplicial set is cofibrant, but any cofibrant object in the Thomason model structure is a poset (Proposition 5.7 in Thomason).
In particular the terminal object is cofibrant in simplicial sets, but $\int_\Delta 1 \cong \Delta$ is not a poset.

James Deikun (Jan 22 2024 at 20:21):

Aha, good catch!

James Deikun (Jan 22 2024 at 20:23):

Presumably whatever it is, if the claim isn't just an error, should match with all the model structures in Cisinski 06.