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Stream: learning: questions

Topic: Applied examples of braided monoidal categories

Nathaniel Virgo (Mar 26 2026 at 19:59):

I've had this question for ages and figure I might as well ask it:

Are there good examples of monoidal categories that are braided but not symmetric, that arise naturally in contexts other than quantum mechanics or knot theory? I'm most interested in examples that "turn out" to have a nice natural braiding, rather than being designed to study something that's already about strings moving around in space.

Nathaniel Virgo (Mar 26 2026 at 21:02):

Then my question is, where do $E_n$ algebras arise in applications? This is part of the motivation for my question actually - someone told me that $\infty$-operads are good for thinking about "degrees of braiding" in some sense, and then that led me to wonder about applications for braiding outside of physics.

I guess I'm hoping for places braiding might emerge in "ACT" type topics or computer science - places you might not necessarily expect to be dealing with topology.

Chad Nester (Mar 27 2026 at 01:12):

I'm also very interested in the answer to this question. I can't find any "computer science-y" examples that don't feel contrived.

Chad Nester (Mar 27 2026 at 01:15):

There's a sort of extensional braided combinatory algebra studied in https://arxiv.org/pdf/2405.10152, and from any one of these the authors recover a braided monoidal category. However, I don't know of any "naturally occurring" braided combinatory algebras (extensional or not).

Chad Nester (Mar 27 2026 at 01:42):

There is also this: https://www.kurims.kyoto-u.ac.jp/~hassei/papers/quantumdouble20111101.pdf

I'm not sure how 'applied' it is.

David Corfield (Mar 27 2026 at 08:53):

How about Melliès in Braided notions of dialogue categories. He does start off talking about representation theory of, e.g., quantum groups, but then writes:

Besides the connections to representation theory, this work is part of a research program in logic and computer science, whose general purpose is to recast the dialogical interpretation of proofs and programs in the language of contemporary algebra.

Bruno Gavranović (Mar 27 2026 at 14:34):

There was one example of a braided but not symmetric monoidal category in the context of profunctor optics, specifically affine traversals.

If I recall correctly, that was the monoidal structure we define in 5.2.10. in the actegories paper.

JR (Mar 27 2026 at 15:25):

Untitled.pdf

I asked a bot to build on an analogy I thought might work to discriminate between braided and symmetric. I don't think it's absolute trash.

David Wärn (Mar 27 2026 at 20:29):

I don't think it's absolute trash.

Sorry, but it is. And it doesn't answer the question.

John Baez (Mar 27 2026 at 22:01):

There's a weird connection between braidings and large cardinals, which I described here:

• Shelves and the infinite.

A [[shelf]] is a set with a binary operation that distributes over itself, usually written $\triangleright$. Any shelf $S$ gives a lax-braided monoidal category where the monoidal structure is the cartesian product of sets and the lax braiding is determined by

$B: S \times S \to S \times S$

$B: (x, y) \mapsto (x \triangleright y, x)$

I call this a 'lax braiding' because $B$ may not be invertible. If $B$ is invertible we call our shelf a [[quandle]].

Here's the weird part: there's a simple-sounding question about quotients of the free shelf on one generator whose answer is only known if we assume the existence of a certain extremely large cardinal. The Ackermann function is also relevant.

Nathaniel Virgo (Mar 28 2026 at 18:08):

Great - as a corollary we get a braided monoidal category from any group, which is exactly the kind of simple example I was looking for! I'll have fun thinking about this.

Todd Trimble (Mar 29 2026 at 13:16):

@John Baez wrote:

Here's the weird part: there's a simple-sounding question about quotients of the free shelf on one generator whose answer is only known if we assume the existence of a certain extremely large cardinal. The Ackermann function is also relevant.

You're referring to the Laver tables, I guess!

Besides this connection with self-distributivity, braids also crop up in connection with distributive laws between monads; this Cafe post gets into this.

IIRC, you get examples of braided monoidal structures by starting with $\mathbb{N}$-graded vector spaces with its standard signed symmetric monoidal structure, but replacing the signs $(-1)^{mn}$ by $q^{mn}$ for parameters $q$. Again IIRC, this example was given in the original Joyal-Street paper on braided monoidal categories, i.e., the Macquarie University Report version (before the journal version).

John Baez (Mar 29 2026 at 22:19):

Yes, my blog article explained the idea of Laver tables.