(This is in a sense a follow-up to my [earlier question on a geometric definition of globular $\infty$-groupoids](https://mathoverflow.net/questions/435758/is-there-a-geometric-definition-of-globular-infty-groupoids-categories))




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We know by [Scholie 8.4.14 of Cisinski's thesis](http://www.numdam.org/item/?id=AST_2006__308__R1_0) that the globe category $\mathbb{G}$ is not a weak test category. Thus, there is no model structure on $\mathsf{Fun}(\mathbb{G}^\mathsf{op},\mathsf{Set})$ such that:


* This model structure models $\infty$-groupoids;
* The cofibrations in this model structure are precisely the monomorphisms;
* The $n$-globes $G\_n:=\mathrm{Hom}\_{\mathbb{G}}(-,[n])$ are contractible.


Nevertheless, there could still in theory be a model structure on $\mathsf{Fun}(\mathbb{G}^\mathsf{op},\mathsf{Set})$ modelling $\infty$-groupoids, even though it might be really bad behaved, and some of the constructions in it would be very ad hoc¹.


¹In particular we would need to somehow construct a geometric realisation functor without using the abstract procedure in [nLab, nerve and realization](https://ncatlab.org/nlab/show/nerve+and+realization), as that recovers only those homotopy types which are wedge sums of spheres (as Simon Henry [explained to me here](https://mathoverflow.net/a/435807)).


**Question.** Is the following statement true?


* There exists no model structure on $\mathsf{Fun}(\mathbb{G}^\mathsf{op},\mathsf{Set})$ that is Quillen equivalent to the Kan–Quillen model structure on simplicial sets.



Inexistence of a Kan–Quillen model structure on globular sets
Zhen Lin points out below that I've been way too cavalier with transferring model structures along a reflection. So the following answer is not clearly correct. I will leave this up as community wiki because I still think it addresses the spirit of the question, showing that spaces can be "modeled" in some sense by globular sets (or even just by graphs).




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Contrary to my guess in the comments, the answer is *no*: there does exist a model structure on globular sets (reflexive or otherwise) which is Quillen equivalent to the Kan-Quillen model structure on spaces.


To see this, note that if $\mathcal A$ is a reflective subcategory of $\mathcal B$, and if $\mathcal A$ has a model structure, then the model structure transfers to $\mathcal B$, and the resulting adjunction is a Quillen equivalence.


Now, as mentioned in the comments, the category $Gph$ of graphs (reflexive or otherwise) is a reflective subcategory of $Glob$. Moreover, the category $Pos$ of posets is a reflective subcategory of $Gph$. Thus $Pos$ is reflective subcategory of $Glob$. So it will suffice to find a model structure on $Pos$ which is Quillen equivalent to topological spaces. This is proven by [Raptis](https://dx.doi.org/10.4310/HHA.2010.v12.n2.a7), by transferring the [Thomason model structure](https://ncatlab.org/nlab/show/Thomason+model+structure) on $Cat$.


