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(This is in a sense a follow-up to my earlier question on a geometric definition of globular $\infty$-groupoids)

We know by Scholie 8.4.14 of Cisinski's thesis 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, as that recovers only those homotopy types which are wedge sums of spheres (as Simon Henry explained to me here).

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.
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    $\begingroup$ I suspect the answer is no, but probably the tools do not exist to show this easily. Since the practical utility of having some model structure doing this would be questionable anyway, it's well worthwhile considering variations of the question. You might start by considering 1-globular sets, i.e. graphs (also, you should think about whether you want reflexive graphs/globular sets or not). On the one hand, one might try a brute-force enumeration of model structures as one can do in $Set$ ... $\endgroup$
    – Tim Campion
    Commented Dec 10, 2022 at 20:53
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    $\begingroup$ ... rather than $Glob$ or $Gph$, but I don't think this is feasible even for $Gph$. Note that already $Gph$ is a very rich category -- you can embed any accessible category into it. Consider, for example, that the category $Pos$ of posets embeds fully faithfully as the reflexive and transitive non-multi-graphs, and the realization functor on $Pos$ (which takes the realization of the 1-coskeletization) models the homotopy category via the Thomason model structure. So in some sense, even graphs can be used to model the homotopy category, even if it's not precisely as a model structure. $\endgroup$
    – Tim Campion
    Commented Dec 10, 2022 at 20:55
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    $\begingroup$ Oh, good point! I think a similar example in spirit to $\mathsf{Pos}$ is that of monoids: we can think of sets as one-object/vertex graphs, monoids as one-object categories, and while sets don't carry any higher homotopical information, there's that theorem that any connected CW-complex is homotopy equivalent to the classifying space of some monoid. $\endgroup$
    – Emily
    Commented Dec 10, 2022 at 21:38

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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).

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, by transferring the Thomason model structure on $Cat$.

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  • $\begingroup$ Amazing (and very surprising!), thank you =) $\endgroup$
    – Emily
    Commented Dec 10, 2022 at 21:38
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    $\begingroup$ You're welcome! I was surprised by this too! In fact, I think now that another answer of mine might be misleading. $\endgroup$
    – Tim Campion
    Commented Dec 10, 2022 at 21:44
  • $\begingroup$ And in turn that question and your answer was part of what led me to ask this one and my earlier one on the same subject! :) $\endgroup$
    – Emily
    Commented Dec 10, 2022 at 21:48
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    $\begingroup$ Note : You clearly get a right semi-model structure that way. $\endgroup$
    – Simon Henry
    Commented Dec 11, 2022 at 2:52
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    $\begingroup$ @Emily In short: Yes for the first question ( maybe through zig-zag though). I think the existence of a right semi-model category on $Psh(C)$ where cofibrations are the mono and representable are contractible is equivalent to $C$ being a weak test category but this was never written in details. The question with left or weak model structure does quite make sense because if cofibrations are the mono then every object is cofibrant and so weak => right-semi and Left-semi => Quillen. Premodel structure always exists and don't give you anything. $\endgroup$
    – Simon Henry
    Commented Dec 11, 2022 at 22:59

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