"Non-categorical" examples of $(\infty, \infty)$-categories

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.

• 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... Jan 29, 2021 at 0:38
• 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. Jan 29, 2021 at 1:41
• 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. Jan 30, 2021 at 0:40
• 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. Jan 30, 2021 at 0:42
• @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".) Feb 4, 2021 at 8:20

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.

• 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. Feb 3, 2021 at 22:58
• 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. Feb 3, 2021 at 23:02
• @TimCampion Thanks, this is a great answer! Feb 11, 2021 at 20:06