Which free strict $\omega$-categories are also free as weak $(\infty,\infty)$-categories?

Question

There are a number of formalisms available for presenting free strict $\omega$-categories -- Street's parity complexes, Steiner's directed complexes, computads, polygraphs,... Typically one has a certain category $\mathcal C$ of combinatorial data, a straightforward notion of "map" from $C \in \mathcal C$ to a strict $\omega$-category $X$, and a free functor $F: \mathcal C \to \omega Cat$ with an explicit combinatorial description of $F$. The free functor will have the universal property that strict $\omega$-functors $F(C) \to X$ are in bijection with "maps" $C \to X$ whenever $X$ is a strict $\omega$-category.

Depending on one's choice of model, there may still be a clear notion of "map" from $C \in \mathcal C$ to a weak $(\infty,\infty)$-category $Y$. I'm interested to know conditions on $C \in \mathcal C$ guaranteeing that "maps" $C \to Y$ are in bijection with morphisms $i(F(C)) \to Y$, where $i: \omega Cat \to (\infty,\infty)Cat$ is the inclusion from strict $\omega$-categories to weak $(\infty,\infty)$-categories. Of course, this may be model-dependent. I find myself needing a result of this form in a particular setting, but I'd be interested seeing results of this kind for any choice of $\mathcal C$ and any model of weak $(\infty,\infty)$-categories -- I'd be particularly happy if the model of weak $(\infty,\infty)$-categories is nonalgebraic in nature.

Question: What is an example of

• a category $\mathcal C$ of "presentations of certain strict $\omega$-categories",

• a free functor $F: \mathcal C \to \omega Cat$ from $\mathcal C$ to strict $\omega$-categories with an explicit combinatorial description,

• a 1-category $(\infty,\infty)Cat$ which "models" the $\infty$-category of weak $(\infty,\infty)$-categories (e.g. via a model structure or whatever) with inclusion functor $i: \omega Cat \to (\infty,\infty)Cat$,

• a straightforward notion of "map" from objects $C \in \mathcal C$ to objects of $(\infty,\infty) Cat$,

• and a (not completely vacuous) condition $\Phi$ on the objects of $\mathcal C$

such that

• Objects $C \in \mathcal C$ satisfying $\Phi$ have the property that $Hom(iF(C),Y)$ is naturally isomorphic to the set of maps from $C$ to $Y$, for all (suitably fibrant, perhaps) $Y \in (\infty,\infty)Cat$?

I'm happy to see quite restrictive conditions $\Phi$; in fact in my case I don't need to understand much more than Gray tensor powers of the arrow category $\bullet \to \bullet$. I suspect that something along the lines of "$F(C)$ is gaunt" or "loop-free" or something may often do the trick, but I'd be happy with something more or less restrictive.

I suppose I'd also be happy to see examples with "$n$" replacing "$\omega$" and "$(\infty,n)$" replacing "$(\infty,\infty)$".

And let me stress that I'm not looking for some kind of fancy $\infty$-categorical freeness -- when I say $Hom(i(F(C)),Y)$ above, I mean $Hom$ in whatever 1-category is being used to model $(\infty,\infty)Cat$. Although if there are results showing something fancier, that would be interesting to hear about too.

Answer 1

If you don't mind, I'll talk about strinct $\infty$-categories, but weak $(\infty,n)$-category to avoid discussing the 'problem' regarding the non uniqueness of the meaning of $(\infty,\infty)$-categories mentioned Here.

Also I don't think what follows completely answer the question, as Harry said in the comment, this is a quite open problem.

So, I think, a very simple, and 'essentially' unique, way to obtain such situation is to construct model categories for $(\infty,n)$-category, which are categories of either presheaf of set or presheaf of spaces over a small category $C$ of 'diagrams'. (You can generally jump from a description as presheaf of sets to a description as presheaf of spaces, using some variant of the simplicial completion techniques as illustrated for example in the case of the category $\Theta$ Here.)

Obviously, if you have such a model then the objects of your category $C$ have all the properties you expect. Also notes that generally presheaf over $C$, or at least some presheaves over $C$, still corresponds to some kind of diagrams and so you get this sort of construction not justs for objects of $C$ but also for more general diagrams build out of objects of $C$. (basically, the cofibrant object of your model structure).

(And of course model that are not exactly presheaves can also give some partial answer to your question as long as they contain some subcategories of cofibrant objects that can be thought of as diagrams.)

While producing a rigorous argument for the converse will be difficult, it definitely feels also true: as soon as you have a rich enough class of diagrams $C$ with the sort of properties you are requiring one should be able to prove some sort of Nerve theorem (an $\infty$-categorical version of monads with arities or Nervous monads) to show that $(\infty,n)$-categories can be represented as presehaves of spaces on $C$ satisfying some segal type conditions, for which you'll be able to build a projective/injective model structure on $[C^{op},sSet]$. This model structure will also often (especially if $C$ is rich enough) have a "simplicial decompletion" on the category of presheaf of sets of $C$.

So it remains to look at examples of such model for $(\infty,n)$-categories... But here much work is left to be done (and to be completely honest this is something that I'm very interested in and actively working on) One reason for this is that people have generally tried to come up with small and simple models, while here we ask about very big models.

The first that comes to mind is obviously the category $\Theta_n$ (see for example Dimitri's paper mentioned above) which obviously fits into this picture, and Verity complicial set models which does the job for Street Orientals, with the only problem that this model hasn't been compared to others ones.

But maybe the most interesting example already worked out regarding your question is the so-called "Colossal model" constructed in Barwick and Schommer-Pries' paper On the unicity of the theory of higher categories, which does this for a large class of gaunt categories. But I'll have to read again this paper before I can say something more precise here.

Finally my own work on polygraphs and the Simpson conjecture (here and enter link description here) is basicaly an attempt to prove this result for the very large class of all "non-unital polygraphs".

Here the hope is that $(\infty,n)$-category can be represented by a model structures on presehaves of sets and or space over the category of "plex" (the representable in the category of polygraphs) probably with some "stratification" in the spirit of the complicial model. So far I have been focusing mostly on modeling $\infty$-groupoids for simplicity, but I expect the extention to $\infty$-category will not be the hardest part. Even in the groupoid case I'm stuck for technical reason for general polygraphs, and I can only make the theory works for "regular polygraphs", but this is still a fairly large class (containing $\Theta, \Delta$ and many other things and closed under the Gray tensor products) and I've shown that regular polygraphs form a presheaf category and carries a natural model structure that models all $\infty$-groupoids. I believe extending this to a model structure on "stratified regular polygraphs" modeling weak $(\infty,n)$-categories should be possible with a bit of work (I mean by that: it is probably a lot of work, but very feasible if someone want to spend some time on it) and this would constitute a very good answer to your question.

Also notes that in everything I have discussed above the functor "$F$" of your question is not really expecte to be fully faithful. I don't know if this is a requirement you have or if you are happy staying with "polygraphic morphisms".

Answer 2

Important progress on this question has been made by Yuki Maehara, who shows in Orientals as free $\omega$-categories that the complicial nerve of the $n$th oriental is a complicial anodyne extension of the $n$-simplex (providing a fibrant replacement thereof in Verity's model structure on complicial sets). In other words, at least in the complicial set model, the orientals -- which a priori have a strict universal property -- have the expected weak universal property too.