This title probably seems strange, so let me explain.


Out of the several different ways of modeling $(\infty, n)$-categories, [complicial
sets](https://arxiv.org/abs/1610.06801) and [comical sets](https://arxiv.org/abs/2005.07603) allow $n = \infty$,
providing mathematical definitions of $(\infty, \infty)$-categories. I've asked people a few times for interesting
examples of $(\infty, \infty)$-categories that could fit into these definitions, and I've always gotten the answer:
the $(\infty, \infty)$-category of (small) $(\infty, \infty)$-categories.


This is not a bad example, and I think it's cool, but I would like to know what kinds of examples are out there
other than just categories of categories. For example, for $(\infty, n)$-categories with $n$ finite,
"non-categorical" examples include $(\infty, n)$-categories of bordisms as well as the Morita $(\infty,
n)$-category of $E\_{n-1}$-algebras in an $(\infty, 1)$-category: people care about bordisms and $E\_{n-1}$-algebras
before learning that they have this higher-categorical structure.


I'm interested in hearing about examples like these for $(\infty, \infty)$-categories. It doesn't matter a lot to
me whether something's been rigorously shown to be an example of one of these models or not; and maybe your favorite
example is a different kind of $(\infty, \infty)$-category, such as the ones discussed in [Theo's
question](https://mathoverflow.net/questions/134057) from several years ago; that's also welcome.


What would be really neat is an example of a new phenomenon at the $n = \infty$ level, so an example of an
$(\infty, \infty)$-category that's not similar to an $(\infty, n)$-category example for any $n$, but that seems
like a lot to ask for.


In addition to Theo's question that I linked above, [this question by Alec
Rhea](https://mathoverflow.net/questions/335550) and [this question by Giorgio
Mossa](https://mathoverflow.net/questions/73772) are also relevant, asking similar questions for $n$ finite.



"Non-categorical" examples of $(\infty, \infty)$-categories
As mentioned by Simon Henry: **The $(\infty,\infty)$-category of cobordisms.**


(Not constructed, but if you did it you could presumably have any of the usual bells and whistles you might want.)


To clarify Simon Henry's comment: The statement is that that $(\infty,\infty)$-category of cobordisms in the coinductive setting is an $\infty$-groupoid by [Cheng's theorem](https://link.springer.com/article/10.1007/s10485-007-9081-8) (so it's whatever Thom spectrum you expect by GMTW). In the inductive setting, Cheng's theorem doesn't hold. So non-invertible $(\infty,\infty)$-TFT's should be a thing. I think nobody's formally written down this $(\infty,\infty)$-category -- I assume because $(\infty,n)$-TFTs are hard enough so there's not much demand for it. Please challenge that assumption!


One nice thing about complicial sets (and I guess also comical sets) is that they (ought to) naturally put you in the (more general) inductive setting, and you might hope they'd be a good place to construct these (∞,∞)-categories.


Anyway, this ticks a few boxes:


1. The inductive / coinductive distinction is arguably a "new phenomenon" (though maybe it's just a "new complication"), and this example already illustrates how it works.
2. It's a super-canonical example, and should be super-interesting for all the reasons its lower brethren are.
3. It's not a category of categories.


