The [nLab page on $\infty$-categories](https://ncatlab.org/nlab/show/infinity-category) splits the known definitions of $\infty$-categories into two types:


* [Algebraic $\infty$-categories](https://ncatlab.org/nlab/show/algebraic+definition+of+higher+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](https://ncatlab.org/nlab/show/geometric+definition+of+higher+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*](https://ncatlab.org/nlab/show/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](https://kerodon.net/tag/01W4), 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?)



Is there a "geometric definition" of globular $\infty$-groupoids/categories?
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 $\Theta$](https://ncatlab.org/nlab/show/Theta+category) and it has model structure (due to [Ara](https://arxiv.org/abs/1206.4354) 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](https://arxiv.org/abs/1003.1342) is a general process that turn a geometric model into an algebraic model. Its dual version due to [Ching and Riehl](https://arxiv.org/abs/1403.5303) in the simplicial case, and [Bourke and myself](https://arxiv.org/abs/2005.05384) 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.


