Is the inclusion functor from gaunt strict $n$-categories to weak $(\infty,n)$-categories fully faithful?

Question

I'm now second-guessing an assertion I made here so let me ask it as a question.

• Let $Cat_n$ be the 1-category of strict $n$-categories;

• Let $\widetilde{Cat_n}$ be the $(\infty,1)$-category obtained from $Cat_n$ by localizing at the weak equivalences (as presented by the folk model structure);

• Let $Gaunt_n \subseteq Cat_n$ be the fully faithful inclusion of the gaunt $n$-categories, i.e. those strict $n$-categories where every equivalence is an identity.

• Let $\widetilde{Cat_n} \to Cat_{(\infty,n)}$ be the "inclusion" functor (which is probably not fully faithful for $n \geq 3$).

Questions:

1. Is the composite functor $Gaunt_n \to Cat_n \to \widetilde{Cat_n}$ a fully faithful functor of $(\infty,1)$-categories?

2. Is the composite "inclusion" functor $Gaunt_n \to Cat_n \to \widetilde{Cat_n} \to Cat_{(\infty,n)}$ a fully faithful functor of $(\infty,1)$-categories?

I'm pretty sure the answer to (1) is yes, and intuitively the answer to (2) should also be yes, but I'm not quite sure.

Barwick and Schommer-Pries do show that if we further restrict to the inclusion $\Theta_n \to Cat_{(\infty,n)}$, or even the slightly larger $\Upsilon_n \to Cat_{(\infty,n)}$, then we get a fully faithful $(\infty,1)$-functor. But I'm not sure about all gaunt $n$-categories.

Answer 1

I will see an $(\infty,\infty)$-category as a functor $\Theta^{op} \to \text{Space}$ that satisfies the usual Segal condition, i.e., i.e. preserve the pushouts encoding the various type of compositions (the globular sum), and the Rezk completeness condition at all level (the map from the space of $n$-cell to the space of "invertible $n+1$-cells" defined in the appropriate way is an equivalence). These naturally form an $(\infty,1)$-category.

This corresponds to an "inductive" (by opposition to coinductive style) definition of weak $(\infty,\infty)$-category, but if you add the assumption that every cell of dimension $>n$ is invertible, then you recover something equivalent to other classical definition of weak $(\infty,n)$-categories. You can also do everything with $\Theta_n$ directly to avoid this.

I claim that $\text{gaunt}$, as a $1$-category identifies with the full subcategory of these functor $\Theta^{op} \to \text{Space}$ as above that takes values in discrete spaces.

Indeed, a functor $\Theta^{op} \to \text{Set}$ satisfiying the Segal conditions is the same as a strict $\infty$-category by the classical $\Theta$-nerve theorem.

If you unfold what the Rezk completeness condition means in this special case it exactly means that every isomorphism in the category (in the strict sense) is an identity, hence that your category is Gaunt.

$\text{gaunt}_n$ corresponds to these that are further more $n$-categories: if your category is Gaunt and all cell of dimension $>n$ are invertible, then it only has identity cell of dimenion $>n$, so it is a $n$-category.