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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:

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

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

  3. 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.

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    $\begingroup$ I'm just shooting from the hip here, but are they the representably discrete objects in the $(\infty,1)$-category of $(\infty,n)$-categories? $\endgroup$ – Alexander Campbell Mar 7 at 22:15
  • $\begingroup$ @AlexanderCampbell That sounds promising. Now that you put it that way, it does seem that the characterization should be via mapping-in properties rather than mapping-out ones. This is a bit counterintuitive to me, since commonly one thinks of gaunt $n$-categories as ones whose mapping-out properties should be good. But it's staring us in the face from the types of lifting properties one gets! $\endgroup$ – Tim Campion Mar 7 at 22:26
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    $\begingroup$ I definitely agree with Alexander guess. Whatever model of $(\infty,n)$-category you are using, discreteness make them into strict infinity categories as every higher isomorphisms asserting coherence of operations is an equality, and "gauntness" is exactly the Rezk completness condition. (this is especially easy to see with the $\Theta_n$-Space model). $\endgroup$ – Simon Henry Mar 7 at 22:46
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    $\begingroup$ I think you can say (inductively) that an $(\infty,n)$-category is gaunt if its underlying $\infty$-groupoid is discrete and all its mapping $(\infty,n-1)$-categories are gaunt. $\endgroup$ – Rune Haugseng Mar 8 at 8:11
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Alexander Campbell's guess is correct.

Here is a reference. Lemma 10.2 of this paper

Clark Barwick, Christopher Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories, arXiv:1112.0040

shows that $Gaunt_n \simeq \tau_{\leq 0} Cat_{(\infty,n)}$. That is to say they are precisely the $(\infty,n)$-categories $G$ with the property that the space $Map(C,G)$ is homotopically discrete for all $C$.

They can also be described as the localization of $Cat_{(\infty,n)}$ at the single morphism $S^1 \times C_n \to C_n$, where $C_n$ is the free-walking $n$-cell, $S^1$ is the circle, and the map is projection. This description is model independent (for example the $n$-cell can be characterized model independently as in the proof of Lemma 4.8 in the same paper above), however it is easiest to check that this description is correct in a particular model such as Rezk's $\Theta_n$-spaces.

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