Is there a "geometric definition" of globular $\infty$-groupoids/categories?

Question

The nLab page on $\infty$-categories splits the known definitions of $\infty$-categories into two types:

• Algebraic $\infty$-categories, in which composition is expressed "externally", e.g. as a some kind of map "$X_1\times_{X_0}X_1\to X_1$";
• Geometric $\infty$-categories, in which composition is expressed "internally", being defined by means of existential assertions with unicity of composition holding only up to a contractible space of choices.

From what I understand (i.e. very little), the current known definitions of globular $\infty$-categories are all algebraic, including e.g. Batanin $\infty$-categories and Grothendieck–Maltsiniotis $\infty$-categories.

Part of the problem, as I understand it, is that the (reflexive or not) globe category $\mathbb{G}$ is not a test category. From what I gather, this means that we don't have a straightforward way to put a model structure on $\mathsf{Fun}(\mathbb{G}^\mathrm{op},\mathsf{Set})$ modelling the homotopy theory of $\infty$-groupoids, as described in the nLab page model structure on presheaves over a test category.

1. Does this rule out $\mathsf{Fun}(\mathbb{G}^\mathrm{op},\mathsf{Set})$ admitting any model structure modelling $\infty$-categories or $\infty$-groupoids?
2. Even if this is so, would it possible (at least in principle) to nevertheless develop a "geometric definition"¹ of globular $\infty$-categories without making use of model categories altogether², and then show that the resulting theory is equivalent to the usual homotopy theory of $\infty$-categories in some appropriate sense (other than "Quillen equivalent")?

¹Here I have in mind something like finding a family of globular sets that fulfills in the globular theory a similar role as to what the horns $\Lambda^n_k$ do for Kan complexes, quasicategories, and $(\infty,2)$-categories. A further requirement is that this family should be "convenient": it has to be small enough to be reasonably easy to work with, so e.g. globular versions of associahedra don't count.

²I've heard that we have a similar situation for Lurie's new model of $(\infty,2)$-categories, in the sense that there isn't a model structure on $\mathsf{sSet}$ recovering them as its fibrant objects. (It is this assertion that made me wonder about this question in the first place, but is it indeed true?)

Answer 1

In short there isn't: the problem is that if you just have globular sets - and if you want $k$-cells to model $k$-arrows following the globular structure - then globular sets have no way of expressing the idea that some cell $f$ is the composite of two cells $g$ and $h$. You can only express that two cells $g$ and $h$ are composable and $f$ is parallel to what their composite should be, but you have no way of saying that $f$ is equivalent to that composite, like you could do with simplicial or cubically shaped higher cells.

So this prevent you to define composition in a "geometric way" as a mere lifting property. Composition has to be an additional structure on the globular set.

This is closely related to the reason why Globular sets are not a test category: If you look at the type of spaces that are obtained as geometric realization of a globular set, you only get some very specific spaces (Exercice: they are all wedge sums of spheres) hence you can get all spaces this way.

Now there is I think a lot of comments one can add to this, and answer to your subquestions:

• The fact that globes are not a (weak) test category doesn't completely rule out the existence of a model structure on globular sets. What it rules out is the existence of a model structure where the cofibrations are monomorphisms and the globes are contractible.

• Lurie's model for $(\infty,2)$ has a model structure - it justs not on simplicial sets but on "simplicial sets with a class of marked 2-cells" (the "thin" cell) the problem is that his definition of $(\infty,2)$-category involved the notion of thin cell - it justs so happen that once a "simplicial sets with a notion of thin cell" is an $(\infty,2)$-category then the thin cell can be uniquely characterized, so that as long as we only look at the $(\infty,2)$-categories only then the functor forgetting these marked cells is fully faithful.

• There is a fairly canonical extension of the category of globular sets that still feel globular and is rich enough to encode compositions: This is Joyal category $\Theta$ and it has model structure (due to Ara following a conjecture of Joyal and Cisinski) that models $(\infty,n)$-category (in a "geometric way"). It is probably the simplest way to do this.

• The definition of "algebraic" vs " geometric" model isn't completely clear cut. I would be tempted to think about it in this way: A geometric model is one where you have a model category where all objects are cofibrants and only the vibrant objects should be thought of as $\infty$-something (the general objects are "presentations"). Morphisms between fibrant objects corresponds directly to "weak functor" already. An algebraic model structure is when you have a model structure where all objects are fibrants, are all considered as "$\infty$-something", but morphisms are some sort of "strict functor". Cofibrant objects corresponds to nice objects $X$ (typically, freely generated in some sense) which have the property that any "weak functor" $X \to Z$ is equivalent to a strict morphism $X \to Z$. In this point of view, Nikolaus construction is a general process that turn a geometric model into an algebraic model. Its dual version due to Ching and Riehl in the simplicial case, and Bourke and myself in the general case) take an algebraic model and turn it into a geometric model. But the problem is that if you apply to algebraic Globular $\infty$-groupoids (even if you assume their canonical model structure exists) then the model you get isn't globular anymore: It will have cells of fairly complex shapes that can encoded compoisiton.