This question is an extension of my previous question last year (see [2020]) in which I asked about the (consensus of a) definition of a weak $n$-category.


Here are some background: while strict $n$-categories are easily defined, they are not sufficient for $n>2$. Therefore weak $n$-categories need to be defined. What a definition of a weak $n$-category should satisfy was proposed in [BD1995]. However, many proposals have since been given (see [Lei2001] or [2020]). And as David White pointed out in [2020], we had not reached to a consensus yet.


This question focuses on a smaller part of the problem.


**Question:** In order to prove that different models of $n$-categories to be equivalent, there must be a well-defined notion of a $(n+1)$-category to start with. So how is it possible to really prove the equivalence?


I guess this relates to a philosophical problem that in order to justify (anything) one needs to justify the setting in which we justify. Before entering the realm of formalized arguments, we can postulate some desired results (as in [BD1995], or called "specification" in compsci's term), but nothing can stop people from building different settings (or called "implementation" in compsci's term). How could such problem be resolute without brute-force [translating](https://math.stackexchange.com/questions/3869884/translating-developments-over-different-foundations) results from different foundations?


* [2020]: [Definition of a $n$-category](https://mathoverflow.net/questions/374783/definition-of-an-n-category/374795)
* [BD1995]: Higher-dimensional Algebra and Topological Quantum Field Theory-[John C. Baez and James Dolan]-[arXiv:q-alg--9503002]
* [Lei2001]: [A Survey of Definitions of n-Category](http://www.tac.mta.ca/tac/volumes/10/1/10-01abs.html)-[Tom Leinster]-[Theory and Applications of Categories, Vol. 10, 2002, No. 1, pp 1-70]



Equivalences of $n$-categories
As Marc Hoyois indicates in the comments, historically this *was* a major obstruction, past tense "was". My feeling is that these days, there is a nice perspective that whatever weak $n$-categories are, they are the objects of some $(\infty,1)$-category $n\mathrm{Cat}$. (Of course, weak $n$-categories are the objects of more than an $(\infty,1)$-category. But the extra $n$ dimensions in $n\mathrm{Cat}$ should be recoverable from looking at exponential objects.) It was a substantial feat to develop a theory of $(\infty,1)$-categories, but it has been more or less done.


Moreover, surely $n\mathrm{Cat}$ will be not just some weird $(\infty,1)$-category, but in fact a *presentable* $(\infty,1)$-category, and these can be "presented" by model categories. So the question is "just" one of finding the correct Quillen equivalence class of model categories, where "correct" means that it should match your intuition about weak $n$-categories.


Note that I am not saying that there is complete consensus about which presentable $(\infty,1)$-category deserves the name $n\mathrm{Cat}$. I certainly have opinions on the matter (namely: set $0\mathrm{Cat}:= \mathrm{Set}$, and define inductively $n\mathrm{Cat}$ to be the $(\infty,1)$-category of $(\infty,1)$-categories enriched in $(n{-}1)\mathrm{Cat}$), but I don't have the sociological data to conclude that my opinions are shared by the majority, let alone that there exists a consensus.


I should also emphasize that, although I do believe there to be a single correct answer to the question of which presentable $(\infty,1)$-category should be called "$n\mathrm{Cat}$", this answer doesn't satisfy, or at least doesn't obviously satisfy, all natural desiderata. Most notably, it is not very algebraic. I mean, it isn't very *non*-algebraic — it is about as algebraic as is the notion of "Kan simplicial set" — but it isn't very algebraic either. However, I counter that the search for a highly algebraic theory of $n$-categories is most likely a fool's errand. Any such theory will in particular include a highly algebraic description of homotopy $n$-types. Postnikov and Whitehead provide a "lowly algebraic" description of homotopy $n$-types, and I am pessimistic about there being anything better.


