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:
- For every $0$-cell $y \in Y$, there exists a $0$-cell $x$ in $X$ such that $fx \sim y$
- 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?
