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Stream: learning: questions

Topic: flipping multicategories

Morgan Rogers (he/him) (Jan 22 2025 at 10:56):

You might know that multicategories are like categories in that they have a collection of objects and a collection of morphisms, but unlike categories the morphisms are allowed to have any (finite) list of objects as inputs. This means that we can no longer dualize by reversing the direction of the arrows. However, there is another duality that arises: we can get a new multicategory by reversing the order of the inputs. If you're more used to monoidal categories, think of the result of exchanging the arguments before tensoring.
Is there a standard name for this dualizing operation?

Morgan Rogers (he/him) (Jan 22 2025 at 10:58):

According to this nLab page stub, this is called the reverse monoidal category. A connection is made with the usual dualizing/reversing arrows operation via delooping, but that doesn't work so well for multicategories (does it?)

The first result when I searched for this was actually this set of notes apparently from a Prof. Pavel Etingof, who makes the wild choice in Definition 1.1.3 of denoting this operation as $\mathcal{C}^{\mathrm{op}}$ and using $\mathcal{C}^{\vee}$ for the usual opposite category!

Nathanael Arkor (Jan 22 2025 at 11:50):

I think "reverse" is the most common terminology, although I have also seen "twisted" (I think in the work of Power). The delooping perspective works for multicategories too: a multicategory is a one-object multibicategory, and its opposite is also a one-object multibicategory. (However, its "co" is not a multibicategory.)

Morgan Rogers (he/him) (Jan 22 2025 at 13:50):

Nathanael Arkor said:

a multicategory is a one-object multibicategory

I understand how a monoidal category is (the unlooping of) a 1-object bicategory: the objects of the monoidal category are morphisms in the bicategory, so the monoidal operation corresponds to composing them there. I don't immediately see what a "multibicategory" would need to be to perform an analogous construction. If the 1-morphisms of a bicategory are replaced with multimorphisms, we end up with "objects" graded by the natural numbers which can be composed in many ways - seems like too much composition. If we keep 1-morphisms, there is still a monoidal operation. If we don't allow morphisms to actually be composed (so we have a virtual composition operation only on 1-cells) then multibicategory is a funny name to choose, since there is nothing in the name suggesting the potential lack of composition.

Amar Hadzihasanovic (Jan 22 2025 at 14:47):

In a multibicategory, the objects and morphisms are just like in a bicategory, but the source of a 2-morphism is a composable sequence of 1-morphisms of any length, while the target is a single 1-morphism. For example, if you have 1-morphisms $f: x \to y$, $g: y \to z$, and $h: x \to z$, you can have a 2-morphism $\alpha: (f, g) \Rightarrow h$.

Amar Hadzihasanovic (Jan 22 2025 at 14:50):

The name is not great as it makes one think of a further “vertical” categorification of multicategories, rather than something that fits into the scheme
monoidal category : bicategory = multicategory : ?

Amar Hadzihasanovic (Jan 22 2025 at 14:51):

But that's the one that exists...

Amar Hadzihasanovic (Jan 22 2025 at 14:52):

It can also be seen as something like a “2-truncated opetopic set”.

Amar Hadzihasanovic (Jan 22 2025 at 14:53):

(I am 99.9% confident that the category of multibicategories & their morphisms is monadic over the category of “2-opetopic sets” i.e. presheaves on the category of opetopes of dimension 0, 1, 2)

Mike Shulman (Jan 22 2025 at 15:06):

I think the name "reverse" is a generalization from monoidal categories, where the reverse monoidal structure is defined by $A\otimes^{\rm rev} B = B\otimes A$.

Mike Shulman (Jan 22 2025 at 15:06):

A multibicategory is also the same as a [[virtual double category]] with only identity vertical arrows.

Mike Shulman (Jan 22 2025 at 15:21):

So another name for them is "virtual bicategory".

Morgan Rogers (he/him) (Jan 22 2025 at 15:54):

Amar Hadzihasanovic said:

In a multibicategory, the objects and morphisms are just like in a bicategory, but the source of a 2-morphism is a composable sequence of 1-morphisms of any length, while the target is a single 1-morphism.

You can actually compose 1-morphisms in a [[bicategory]], though. You cannot compose objects in a multicategory. I gather from your alternative description that it matches Mike's definition as a virtual double category with only identity vertical arrows makes sense and validates Nathanael's comment regarding taking the opposite. I certainly prefer the term "virtual bicategory" for these things.

Mike Shulman (Jan 22 2025 at 16:41):

This is implicit in the whole conversation, but just to have it written down clearly, the "delooping" operation $B$ shifts notions of "opposite", e.g. $B(C^{\rm rev}) = (BC)^{\rm op}$ and $B(C^{\rm op}) = (BC)^{\rm co}$.

Kevin Carlson (Jan 22 2025 at 17:41):

This looks to me to be an autoequivalence of the category of multicategories indexed by a sequence of elements of the symmetric groups on $n$ letters for every $n.$ (namely, the reversal permutations) Am I missing some reason why we couldn’t choose arbitrary permutations at every level? This would give a whole $\Pi_n \mathfrak S_n$ subgroup of the autoequivalence group of multicategories, which is pretty interesting in comparison to the mere $\mathbb Z/2$ autoequivalences of Cat.

Mike Shulman (Jan 22 2025 at 17:42):

If you choose arbitrary permutations at each level I don't expect it would be compatible with composition.

Kevin Carlson (Jan 22 2025 at 17:43):

I was trying to think about that but then what property of the reversal permutations are we relying on?

Kevin Carlson (Jan 22 2025 at 17:47):

I want to pick $\sigma=(\sigma_n)$ at each level and then define $M_\sigma((x),y)= M_((x)^\sigma_n,y)$ if $(x)$ has length $n.$ Then for the composition, a list of maps $((w))\to (x)$ is a list $(w)_i^{\sigma_{n_i}}\to x_i$…and OK, I think I see, the property is that the concatenation of several $\sigma_{n_i}$ must coincide with $\sigma_{\sum n_i}$.

Kevin Carlson (Jan 22 2025 at 17:48):

Mm…no, I don’t think that’s quite it yet.

Kevin Carlson (Jan 22 2025 at 17:52):

The thing that reversals do is that, if you decompose a string into substrings, reverse each substring, and then reverse the list of substrings, you've reversed the whole string. (For instance, in permutation notation, $(4321)=(12)(34)\cdot (13)(24).$) In general you'd want permutations $\sigma_n$ on $n$ letters such that, for any $n_1,\ldots,n_k,$ the permutation $\sigma_{\sum n_i}$ coincides with what you get when you apply the $\sigma_{n_i}$ in concatentation and then apply $\sigma_{k}$ in blocks. I'll forebear to try to create good notation for this.

Kevin Carlson (Jan 22 2025 at 17:52):

I wonder if there are any other lists of permutations other than the reversals that behave this way! It's like they admit a divide-and-conquer algorithm for computation, considering the context of reversing a string being a classic CS 101 algorithm.

Mike Shulman (Jan 22 2025 at 18:03):

I'm going to go out on a limb and conjecture that the identities and the reversals are the only lists of permutations with this property!

Kevin Carlson (Jan 22 2025 at 18:03):

Yeah, I think the divide-and-conquer axiom pretty quickly shows that all the permutations in the list are determined by what you do to (one and to) two elements and so you only get identity and reversal. Too bad. Or maybe it’s good!

Mike Shulman (Jan 22 2025 at 18:03):

Haha