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.


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$\newcommand{\Vect}{\mathrm{Vect}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\cc}{\mathbf{C}}$Here are three (related) examples. The first one is simple (although not really related to physics): an $\mathbf{E}_n$-space is "just" an $(\infty,n)$-category with one object, one $1$-morphism, ..., and one $(n-1)$-morphism. This can be generalized: an $\mathbf{E}_k$-monoidal $(\infty,n)$-category can be thought of as an $(\infty,n+k)$-category with one object, ..., and one $(k-1)$-morphism.

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 $\int_{\mathbf{R}^{n-d}\times M^d} A$.

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    $\begingroup$ 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 $\endgroup$
    – Alec Rhea
    Commented Jul 6, 2019 at 9:05
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    $\begingroup$ @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). $\endgroup$ Commented Jul 6, 2019 at 10:25
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    $\begingroup$ 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? $\endgroup$ Commented Jul 6, 2019 at 18:03
  • $\begingroup$ Ah, yep. Thanks. $\endgroup$
    – skd
    Commented Jul 6, 2019 at 20:42
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    $\begingroup$ @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. $\endgroup$ Commented Jul 7, 2019 at 19:09

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