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This title probably seems strange, so let me explain.

Out of the several different ways of modeling $(\infty, n)$-categories, complicial sets and comical sets 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 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 and this question by Giorgio Mossa are also relevant, asking similar questions for $n$ finite.

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    $\begingroup$ This is obviously not relevant to the main point but I would somewhat dispute the claim that "the $(\infty,\infty)$-category of all $(\infty,\infty)$-categories" is not a bad example. Some would even say that it is an excellent example of a bad example... $\endgroup$
    – Will Sawin
    Jan 29, 2021 at 0:38
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    $\begingroup$ So there are two non-equivalent definition of $(\infty,\infty)$-categories as explained here mathoverflow.net/a/134099/22131 I will refer to these as inductive and coinductive $(\infty,\infty)$-categories. If you are using the inductive definition, there is an $(\infty,\infty)$-category of cobordisms and an $(\infty,\infty)$-category of Higher spans. These becam trivial using the coinductive definition however. $\endgroup$
    – Simon Henry
    Jan 29, 2021 at 1:41
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    $\begingroup$ I appreciate the mononymy, but of course there are multiple "Theo"s who do mathematics :) In any case, @SimonHenry got to it before me, but spans are naturally an $(\infty,\infty)$-category. This is true also for spans-with-structure. An important example is the $(\infty,\infty)$-category of (shifted) symplectic manifolds and Lagrangian correspondences, which I believe is carefully defined in upcoming work by Calaque, Haugseng, and Scheimbauer. $\endgroup$
    – Theo Johnson-Freyd
    Jan 30, 2021 at 0:40
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    $\begingroup$ Actually, the category of Lagrangian correspondences can also be extended "down", where you form a sort of "loop spectrum" of $(\infty,\infty)$-categories — what Scheimbauer termed a tower in her PhD thesis. $\endgroup$
    – Theo Johnson-Freyd
    Jan 30, 2021 at 0:42
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    $\begingroup$ @TimCampion Not obviously, at least: an (inductive) $(\infty,\infty)$-category is a sequence $(X_n)$ where $X_n$ is an $(\infty,n)$-category with underlying $(\infty,n-1)$-category $X_{n-1}$. But for the Morita $(\infty,n)$-categories of $E_n$-algebras in a symmetric monoidal $\infty$-category $V$ then the objects are different for each $n$. But maybe there should be a Morita $(\infty,\infty)$-category of $V$-$(\infty,\infty)$-categories? (I don't think you can view $E_\infty$-algebras as special $(\infty,\infty)$-categories, as you can for $E_n$, since you are extending "down" not "up".) $\endgroup$
    – Rune Haugseng
    Feb 4, 2021 at 8:20

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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 (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.

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    $\begingroup$ I think Dominic Verity has given a description of this $(\infty,\infty)$-category as a complicial sets. (well, given that the precise connection between complicial sets and $(\infty,\infty)$-categories is still unclear one cannot prove that it really model this object, but it supposed too) I don't know if it is written out in his paper, I only heard gave him talk about this. $\endgroup$
    – Simon Henry
    Feb 3, 2021 at 22:58
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    $\begingroup$ As of a year ago my understanding from talking to Dom was that he had thought seriously about what would go into writing down such a definition, but had not written it up. But it's possible that's partly because he has high standards for what counts as writing and is very modest. $\endgroup$
    – Tim Campion ♦
    Feb 3, 2021 at 23:02
  • $\begingroup$ @TimCampion Thanks, this is a great answer! $\endgroup$
    – Arun Debray
    Feb 11, 2021 at 20:06

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