Recall that a strict $n$-category $C$ is called gaunt if every $k$-morphism in $C$ with a weak inverse is an identity, for all $k$; let $Gaunt_n$ denote the strict 1-category of gaunt $n$-categories. Another way of saying this is that $C \in Gaunt_n$ iff $C$ has the unique right lifting property with respect to the canonical $n$-functor $E_k \to C_{k-1}$, where $E_k$ is the free $k$-equivalence and $C_{k-1}$ is the free $(k-1)$-morphism. Thus we have a characterization of the essential image of the fully faithful functor $Gaunt_n \to Cat_n^{str}$ into the strict 1-category of strict $n$-categories.

Now consider the composite inclusion $Gaunt_n \to Cat_n^{str} \to Cat_{(\infty,n)}$ into the $(\infty,1)$-category of weak $(\infty,n)$-categories. I believe this inclusion is also fully faithful; can we characterize its essential image? We can't repeat the same characterization as before, because the image of the canonical map $E_k \to C_{k-1}$ is already an equivalence in $Cat_{(\infty,n)}$.

**Question:** What is a (model-independent) characterization of the essential image $Gaunt_n \to Cat_{(\infty,n)}$?

I have some hope that there is a nice answer, from the following model-dependent considerations. In every model of $Cat_{(\infty,n)}$ I've thought about, the objects of $Cat_{(\infty,n)}$ are the fibrant objects of a model structure on some 1-category $\mathcal K$, defined by a (non-unique) right lifting property against the acyclic cofibrations of $\mathcal K$. It seems to me that in all cases, the following facts hold:

The functor $Cat_n^{str} \to Cat_{(\infty,n)}$ lifts to a canonical functor $Cat_n^{str} \to \mathcal K$;

The composite functor $Gaunt_n \to Cat_n^{str} \to \mathcal K$ is fully faithful;

The essential image of $Gaunt_n \subseteq \mathcal K$ can be characterized as the objects which satisfy the stronger

*unique*right lifting property against the acyclic cofibrations of $\mathcal K$.

To flesh out (3) a bit, the lifting properties for fibrant objects in these model structures are generally characterized by "horn fillers" and "univalence maps" (a.k.a. "completeness" or "Rezk" or "2-out-of-6" maps). Unique lifting against the horn fillers seems to generally pick out objects which can be thought of as strict $n$-categories presented via a sort of "naive nerve" which doesn't quite handle equivalences appropriately; if in addition an object lifts against the univalence maps, it is forced to be gaunt (and in this case the "naive nerve" coincides with the "genuine nerve"). For example, if the Duskin nerve of a 2-category is univalent, then the 2-category is gaunt.

I find it striking that the model-dependent description of the essential image $Gaunt_n \subseteq \mathcal K$ seems to always take the same form across models $\mathcal K$, and I'm wondering if these parallel characterizations are really model-dependent avatars of something which can be said model-independently.

outproperties should be good. But it's staring us in the face from the types of lifting properties one gets! $\endgroup$ – Tim Campion Mar 7 '20 at 22:26