# Natural examples of $(\infty,n)$-categories for large $n$

In Higher Topos Theory, Lurie argues that the coherence diagrams for fully weak $$n$$-categories with $$n>2$$ are 'so complicated as to be essentially unusable', before mentioning that several dramatic simplifications occur if we step up to $$\infty$$-categories but require all higher morphisms to be invertible.

What are some examples in nature (topological quantum field theory, string theory etc.) of notions that are best understood as a weak $$(\infty,n)$$-category, or simply a weak $$n$$-category, for 'large' $$n$$? (something which is naturally a 'fully weak' $$\infty$$-category would be interesting too)

Bordisms between $$n$$-dimensional manifolds have the former structure naturally and serve as the canonical example coming from TQFTs; are there any others?

(I believe the present question differs from this one, although the answers may overlap, as I'm looking for examples of $$\infty$$-categories that are 'motivated by nature' in the above sense and already believe higher categories are useful.)

I am also interested in natural occurrences of $$\infty$$-categories or weak $$n$$-categories for 'large' $$n$$ in areas of mathematics outside of category theory, so please share any good ones that occur to you. The use of $$2$$-category theory for the Galois theorem of Borceux and Janelidze wouldn't quite qualify, but if there is any area outside of category theory that seriously uses weak $$3$$-categories I would be interested to hear about it (invariants of $$3$$-manifolds?). This is why 'large' is in quotations; examples with $$n>>1$$ would be great, but I consider $$3$$ 'large' in certain contexts where $$2$$ isn't.

• I think this qualifies as a 'big list' question and should be CW accordingly, if someone could be so kind as to change it. – Alec Rhea Jul 6 '19 at 5:14
• I have notified mods so they can change it. – David Roberts Jul 6 '19 at 9:00
• @DavidRoberts Much appreciated. – Alec Rhea Jul 6 '19 at 9:03

For the second example (whose details I don't know), let $$\Vect(\cc)$$ denote the category of vector spaces, regarded as an $$(\infty,1)$$-category by taking its nerve. This admits a symmetric monoidal structure, and one can look at the $$(\infty,2)$$-category $$\mathrm{Mod}_{\Vect(\cc)}$$ of modules over it, taken in presentable $$(\infty,1)$$-categories. (This is related to the category of $$\cc$$-algebras, bimodules, and intertwiners.) This is again supposed to be a symmetric monoidal $$(\infty,2)$$-category, so one can look at "modules" over it to get an $$(\infty,3)$$-category, and so on. I think that the $$(\infty,n)$$-category you get this way is supposed to be the natural target of "physical" $$n$$-dimensional extended TQFTs, and that its Picard space is supposed/conjectured to be $$\Omega^\infty \Sigma^{n+1} I_{\cc^\times}$$, where $$I_{\cc^\times}$$ is the Brown-Comenetz dualizing spectrum (although I'm not sure of these statements, and would appreciate if someone confirmed/elaborated upon them!).
For the third example, let $$\mathcal{C}$$ denote a symmetric monoidal $$(\infty,1)$$-category; then you can inductively construct an $$(\infty,n+1)$$-category $$\mathrm{Mor}_n(\mathcal{C})$$ (the "higher Morita category") for any $$n$$ by defining its objects to be $$\mathbf{E}_n$$-algebras in $$\mathcal{C}$$, and such that the $$(\infty,n)$$-category of morphisms from $$R$$ and $$S$$ is the category $$\mathrm{Mor}_{n-1}(_R\mathrm{BMod}_S)$$, where $$_R\mathrm{BMod}_S$$ is the $$(\infty,1)$$-category of $$(A,B)$$-bimodules. If $$n=1$$, this is the category of associative algebras (in $$\mathcal{C}$$), bimodules, and bimodule homomorphisms (so the equivalences are Morita equivalences). You then get a TQFT $$Z_A:\mathrm{Bord}_{n+1}\to \mathrm{Mor}_n(\mathcal{C})$$ for every $$\mathbf{E}_n$$-algebra object $$A$$ of $$\mathcal{C}$$ which is fully dualizable in $$\mathrm{Mor}_n(\mathcal{C})$$ (this is a rather strong condition); this TQFT sends $$M^d$$ to the factorization/topological chiral homology $$\int_{\mathbf{R}^{n-d}\times M^d} A$$.
• Interesting, thanks for the reply -- where do $E_n$-spaces come up in mathematical practice? Are you talking about these guys? ncatlab.org/nlab/show/En-algebra – Alec Rhea Jul 6 '19 at 9:05
• @AlecRhea Technically yes, although that page does not mentions explicitly the examples of $E_n$-algebras in spaces, as far as I can tell. They are rather important: a group-like $E_n$-space is exactly a $n$-fold loop space, and $E_n$-structures on parameter spaces tend to produce interesting multiplicative structures on invariants (e.g. the Thom spectrum of an $E_n$-map is an $E_n$-ring spectrum). – Denis Nardin Jul 6 '19 at 10:25
• I also don't know about this stuff, but I thought the state of affairs was that we'd like to find a naturally occurring target with Picard space a shift of the Anderson dual of the sphere but that iterating "Mod" on $\mathbb{C}$ is definitely the wrong answer. For example, right away at level 2 you'll want super vector spaces to see that $\mathbb{Z}/2$ in $\pi_1S^0$. The next category is supposed to be the "Brauer-Wall category of super, simple algebras", and I don't think the next natural "delooped" category is known. Maybe the conformal net people have a guess about it? – Dylan Wilson Jul 6 '19 at 18:03
• @skd It seems misleading to call $\mathrm{Mor}_1(\mathcal C)$ the "category of associative algebras and Morita equivalences." It's the $(\infty,2)$-category of associative algebras, bimodules, and bimodule homomorphisms, in which it happens to be the case that the equivalences coincide with the Morita equivalences. – Kevin Arlin Jul 7 '19 at 19:09