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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.

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    $\begingroup$ This is easy to see using Rezk's $\Theta_n$-space model for $(\infty,n)$-categories. The category of gaunt n-categories is equivalent (via the cellular nerve functor) to the full (simplicially enriched) subcategory of Rezk $\Theta_n$-spaces spanned by the "discrete-valued" ones. $\endgroup$ Mar 27 '20 at 22:05
  • $\begingroup$ @AlexanderCampbell Thanks. I think what confuses me now, as discussed with Simon below, is the fact that the cellular nerve doesn't generally produce a functor $Cat_n \to Cat_{(\infty,n)}$ unless Rezk completion is applied, which makes the situation a bit murky. Do you have any insight on how to produce the "correct" functor $Cat_n \to Cat_{(\infty,n)}$ and check that it agrees with the cellular nerve on $Gaunt_n$? $\endgroup$
    – Tim Campion
    Mar 28 '20 at 15:06
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    $\begingroup$ A reasonable guess (which we know works for $n=2$) of a right Quillen functor from $n$-categories to $n$-quasi-categories is the nerve/singular functor induced by a Reedy cofibrant replacement of the full inclusion $\Theta_n \to \mathbf{Cat}_n$. Compose this with Ara's right Quillen functor from $n$-quasi-categories to Rezk $\Theta_n$-spaces to get a right Quillen "classifying diagram" functor. Since gaunt $n$-categories "see" weak equivalences as isomorphisms, this agrees on gaunt $n$-categories with the full inclusion I mentioned above. $\endgroup$ Mar 29 '20 at 21:17
  • $\begingroup$ @AlexanderCampbell That does sound like a reasonable guess. I wonder how it compares in general to taking the Rezk completion of the cellular nerve as Simon suggests below... I also wonder if there's a model-independently characterization of the "correct" functor... $\endgroup$
    – Tim Campion
    Mar 29 '20 at 22:07
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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.

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  • $\begingroup$ Thanks! I'm still a bit confused. (1) This gives fully-faithfulness at the 1-categorical level, but how do I know that I still have fully-faithfulness after deriving these functors? (2) You seem to be identifying a strict $n$-category with its cellular nerve as Alex does above -- but in general isn't this the wrong functor? It carries non-gaunt $n$-categories to non-complete $\Theta_n$-spaces. This shouldn't be a problem because we're restricting to gaunt $n$-categories in the end, but it seems to me that we need to set up a model structure on $Gaunt_n$ or something to derive this functor... $\endgroup$
    – Tim Campion
    Mar 27 '20 at 22:16
  • $\begingroup$ @TimCampion : For (1), when I talk about $\infty$-categories as functor $\Theta^{op} \to$ Space, I mean by "Space" the $(\infty,1)$-category of spaces. So there is no $1$-categorical level in what I say. $\endgroup$ Mar 27 '20 at 22:28
  • $\begingroup$ Regarding (2) I have to admit I didn't think about it. In this point of view, isn't the "right functor" taking the cellular nerve of a strict $\infty$-category (to get a Cellular set seen as a discrete cellular space) and then taking the Rezk completion ? if so then Gaunt object are already Rezk complete, so Rezk completion does nothing. $\endgroup$ Mar 27 '20 at 22:30
  • $\begingroup$ I'm not 100% sure this is the "right functor", since it's the composite of a right adjoint and a left adjoint, so it's not clear to me that the resulting functor is a right adjoint. I'd be most comfortable constructing the "right functor" as something like Rezk's classifying diagram. Probably this is still equivalent to the cellular nerve for gaunt $n$-categories, but there's something to check. $\endgroup$
    – Tim Campion
    Mar 28 '20 at 15:01

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