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There is a paper of Lafont, Metayer, and Worytkiewicz [1] that constructs a model structure on the category of strict $\omega$-categories that they call the folk model structure. This model structure has many nice properties, but it has a strange class of weak equivalences, which are maps $X\to Y$ that are coinductive equivalences.

The way that these equivalences are defined are as follows:

First, we say that two parallel $n$-cells of a strict $\omega$-category $X$ are equivalent, written $x \sim x^\prime$ if there exists a reversible $n+1$-cell $x\xrightarrow{\sim} x^\prime$.

We say that an $n$-cell $f:a\to b$ is reversible if there is an $n+1$-cell $g:b\to a$ such that $g\circ f \sim \operatorname{id}_a$ and $f\circ g \sim \operatorname{id}_b$.

This gives a coinductive notion of equivalence of $n$-cells, such that specifying an equivalence requires the specification of an infinite tower of reversible arrows witnessing the invertibility.

Then we say that a map $f:X\to Y$ of strict $\omega$-categories is a coinductive weak equivalence if the following two properties hold:

  1. For every $0$-cell $y \in Y$, there exists a $0$-cell $x$ in $X$ such that $fx \sim y$
  2. For any parallel pair of $n$-cells $x,x^\prime$ in $X$ and any $n+1$-cell $v:fx\to fx^\prime$ in $Y$, there exists an $n+1$-cell $u:x\to x^\prime$ such that $fu \sim v$.

Using this definition of weak equivalence, the authors give a set of generating cofibrations:

$$I=\{\partial O^n \hookrightarrow O^n | n\geq 0\}$$ where $O^n$ denotes the globular $n$-disk, and $\partial O^n = O^{n-1} \cup O^{n-1}$ is the union of two parallel $n-1$ disks along their boundaries.

The authors then verify the requirements of Jeff Smith's theorem and give a combinatorial model structure on the category of strict $\omega$-categories where the weak equivalences are as above and the cofibrations are given by $I-\operatorname{Cof}$.

The trouble with this model structure is that it models the projective limit of $n-\operatorname{Cat}$ along the cotruncation functors $n+1-\operatorname{Cat} \to n-\operatorname{Cat}$ given by collapsing all $n+1$-cells to identities.

In the homotopical models for weak $\omega$-categories, we can also exhibit an inductive model structure, where the equivalences are similar to the above ones, except that we also require that all towers of equivalence data terminate after a finite number of steps.

The problem with trying to find such a model on strict $\omega$-categories is that the trivial fibrations with respect to the set of generating cofibrations $I$ as above need not necessarily be inductive equivalences, as can be seen by taking, for example, the cofibrant replacement of the terminal strict $\omega$-category (which has no nontrivial isomorphisms of cells but is coinductively contractible). This means that to find an inductive model structure, we must change the generating cofibrations, and I am not aware of any obvious candidates.

The only possible idea I had was to adjoin to the generating cofibrations the set of maps $$I^\prime = \{\Sigma^n(C(G_2)) \to \Sigma^n(G_2)|n\geq 0\},$$ where $C(G_2)$ is a cofibrant replacement in the coinductive model structure of the contractible groupoid on two objects $G_2$ and $\Sigma^n$ denotes the $n$th power of the $2$-point suspension functor, where $\Sigma(X)$ is the strict $\omega$-category whose objects are $0,1$ and whose Hom-objects are given by $$\Sigma(X)(i,j)=\begin{cases}\ast, &\text{if}\qquad i=j\\ X, &\text{if}\qquad i<j\\ \emptyset, &\text{if} \qquad i>j\end{cases}.$$

But this seems somehow too strong a requirement, since it means that all cells in trivially fibrant objects with respect to this set of generating cofibrations should have completely strict inverses.

So the question: Is there any known model structure on strict $\omega$-categories where the weak equivalences are the inductive ones? Does the proposed set of generating cofibrations above seem plausible?

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    $\begingroup$ I don't know the answer to the question you ask (and I don't really have an intuition on whether the answer should be yes or no). But what should defintitely exists and somehow represents "inductive strict $\infty$-categories" is a model strucutre on "maked strict $\infty$-categories" where fibrant objects are those where the markings satisfies some stability property and where marked cell can be reversed up to higher marked cells, and equivalence between bifibrant objects are the things you can invert modulo a marked natural transformation. $\endgroup$ – Simon Henry Nov 5 '18 at 10:51
  • $\begingroup$ @SimonHenry Yes, indeed, this looks like the correct way to do it! The obvious adjunction to construct in this case comes from the functor $\Theta\times \Delta \to \operatorname{Cat}^+_\omega$ defined by the rule $([t],[n])\mapsto [t] \times \overline{\mathcal{O}^n}$ where $\overline{\mathcal{O}^n}$ is the nth oriental with its unique nontrivial globular n-cell marked. Interesting question: Does Ara's rigidity result still apply to this functor with markings introduced? Edit: Yes, probably. This functor still does not preserve cofibrations, so we end up back at the other question. $\endgroup$ – Harry Gindi Nov 5 '18 at 12:05
  • $\begingroup$ I don't really follow you last comment, I probably don't know the construction you are refering too, so maybe your idea is better. My idea was more to essentially follow the original strategy of Lafont, Metayer, Worytkiewicz appropriately corrected to take the marking into account. Probably the only place that require some works is in adapting the construction of cylinder/path objects in a way that treat the marking correctly. $\endgroup$ – Simon Henry Nov 5 '18 at 12:17
  • $\begingroup$ @SimonHenry No, no, I was agreeing! My comment was about constructing a nerve that is a right-Quillen functor to Rezk Θ-spaces. When markings are introduced, I was thinking maybe you could do what I said there, but I just realized that the construction I just defined fails to be functorial, so it's back to the drawing board. Anyway, yes, the idea of using markings is definitely the right idea! $\endgroup$ – Harry Gindi Nov 5 '18 at 12:23
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Denote by $tr_n$ the "intelligent truncation functor" which quotients an $\omega$-category by its $n$-cells. If I understand you correctly, the model structure you are looking for on strict $\omega$-categories would still have the property:

  • the maps $tr_{n+1} C (G_2) \to tr_n C(G_2)$ are trivial fibrations, and so in particular the maps $tr_n C(G_2) \to G_2$ too.

The issue is that the map $C(G_2) \to G_2$ is a trivial fibration for any model structure satisfying this property, which I believe contradicts what you are looking for (it is not an "inductive" trivial fibration).

This follows form the fact that $C(G_2)$ is the colimit of the sequence of projections $tr_{n+1} C(G_2) \to tr_n C(G_2)$

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  • $\begingroup$ Good point! It seemed like a longshot with bad properties, and indeed it was! $\endgroup$ – Harry Gindi Nov 5 '18 at 11:51

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