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Stream: learning: questions

Topic: Cubical-to-Globular translation

Ruby Khondaker (she/her) (Jul 09 2025 at 07:13):

Recently I've been looking at ways to evaluate "cubical pasting diagrams" in terms of "globular pasting diagrams". Some motivation for this is the double category of squares, which is most naturally expressed using a cubical structure. I mean, it's in the name - square!

But of course, the usual definition of bicategories uses globular shapes. So you need to think a little about how to turn your ability to evaluate globular pasting diagrams to let you evaluate these "cubical" ones.

I presume there are analogs of this in higher dimensions too, but I quickly run out of visualisation power :sweat_smile: . Is there a general theory of how to evaluate cubical diagrams using globular ones? Or more generally, how to translate a cubical theory to a globular one?

Ruby Khondaker (she/her) (Jul 09 2025 at 07:16):

I guess maybe another way of phrasing this is - given a "globular" structure on a higher category (i.e. 0-cells, 1-cells, ... together with ways of composing them), is there a natural way to convert it to a "cubical" structure (i.e. 0-squares, 1-squares, ... together with ways of composing them)?

Amar Hadzihasanovic (Jul 09 2025 at 07:17):

Read this paper.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:18):

oh nice, that looks pretty much exactly like what i'm looking for! i'll take a look :)

Ruby Khondaker (she/her) (Jul 09 2025 at 07:18):

my main motivation for this is trying to understand the crans-gray tensor product

Ruby Khondaker (she/her) (Jul 09 2025 at 07:19):

i think this is probably easier to think about in the cubical setting because of the natural monoidal structure on the cube category?

Ruby Khondaker (she/her) (Jul 09 2025 at 07:19):

so you get day-convolution for free

Ruby Khondaker (she/her) (Jul 09 2025 at 07:20):

so my thought was - if i want to tensor together two globular $\omega$-categories, i can first translate them both into cubical $\omega$-categories, tensor them there, and then translate back!

Ruby Khondaker (she/her) (Jul 09 2025 at 07:20):

which hopefully recovers the usual tensor product of globular $\omega$-categories..

Amar Hadzihasanovic (Jul 09 2025 at 07:20):

Yes that's a way to define the tensor product of strict $\omega$-categories, but it's not the one that people have found most convenient.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:21):

hm, how about just weak $\omega$-categories?

Ruby Khondaker (she/her) (Jul 09 2025 at 07:21):

and is there a more convenient way?

Ruby Khondaker (she/her) (Jul 09 2025 at 07:22):

i guess for further context, my motivation for this is to try to understand functor categories for weak $\omega$-categories

Amar Hadzihasanovic (Jul 09 2025 at 07:22):

Personally I think that the diagrammatic model has the “best” definition of the Gray tensor product, but I am biased about that.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:22):

Ruby Khondaker (she/her) said:

i guess for further context, my motivation for this is to try to understand functor categories for weak $\omega$-categories

and because of tensor-hom, it would suffice to understand the tensor product of $\omega$-categories

Amar Hadzihasanovic (Jul 09 2025 at 07:22):

(It is a Day convolution but on a vastly more expressive category of shapes, which includes globes, cubes, and all sorts of other shapes.)

Ruby Khondaker (she/her) (Jul 09 2025 at 07:23):

ooh!

Ruby Khondaker (she/her) (Jul 09 2025 at 07:24):

but yeah the idea would be - the $n$-cells of a functor category $[\mathcal{X}, \mathcal{Y}]$ would just be maps $G_n \to [\mathcal{X}, \mathcal{Y}]$ where $G_n$ is the $n$-globe

Ruby Khondaker (she/her) (Jul 09 2025 at 07:24):

And then by tensor-hom those are equivalently described by maps $G_n \otimes \mathcal{X} \to \mathcal{Y}$ (or maybe the other order of tensoring, not quite sure)

Amar Hadzihasanovic (Jul 09 2025 at 07:25):

For strict $\omega$-categories, people have been using some machinery known as “Steiner theory”.
See the original article, this monograph by Ara and Maltsiniotis, and chapter 11 of my book.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:25):

because i have a good idea of what lax $\omega$-functors look like, but not natural transformations or higher transfors

Amar Hadzihasanovic (Jul 09 2025 at 07:26):

Yes, that is the idea.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:26):

is there not anything for weak $\omega$-categories yet?

Amar Hadzihasanovic (Jul 09 2025 at 07:27):

What do you mean by weak $\omega$-categories?

Amar Hadzihasanovic (Jul 09 2025 at 07:27):

Clémence's paper that I have linked is about $(\infty, \infty)$-categories.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:28):

oh so the same thing i talked about in the #learning: questions > Omega-Category Theory thread - leinster's "geometric" definition of weak $(\infty, \infty)$ category

Amar Hadzihasanovic (Jul 09 2025 at 07:28):

No, I don't believe there's anything very good for those models.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:28):

oh no... do you have an idea of why that is? i guess that's exactly the sort of thing i'm trying to figure out now, haha

Amar Hadzihasanovic (Jul 09 2025 at 07:30):

Well, it's already very complicated to define the Gray tensor product for strict $\omega$-categories and these models are a more complicated version of those.

Amar Hadzihasanovic (Jul 09 2025 at 07:31):

That's not a technical answer, just a “from experience” answer.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:31):

mhm mhm

Ruby Khondaker (she/her) (Jul 09 2025 at 07:31):

Ruby Khondaker (she/her) said:

so my thought was - if i want to tensor together two globular $\omega$-categories, i can first translate them both into cubical $\omega$-categories, tensor them there, and then translate back!

would this sort of construction work?

Amar Hadzihasanovic (Jul 09 2025 at 07:32):

There is very little about cubical weak $\omega$-categories.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:33):

oh, interesting! how about cubical strict $\omega$-categories?

Ruby Khondaker (she/her) (Jul 09 2025 at 07:33):

i guess that's in the paper you linked

Ruby Khondaker (she/her) (Jul 09 2025 at 07:33):

hm i wonder if it would suffice to just weaken the strict definition

Ruby Khondaker (she/her) (Jul 09 2025 at 07:34):

so long as there's some "free strict cubical $\omega$-category monad" this should be doable?

Amar Hadzihasanovic (Jul 09 2025 at 07:36):

There is a series of papers by Camell Kachour on cubical weak $\omega$-categories, but I am not sure how solid this work is; the author writes stuff like

Sorry to say that, but my work must go beyond than those of Jacob Lurie, that I respect a lot, where applications of my approach much be much more easier and flexible for the future.

on their MathOverflow profile which does not exactly boost my confidence about them. The papers are also only self-cited.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:36):

ah... :sweat_smile:

Ruby Khondaker (she/her) (Jul 09 2025 at 07:37):

ok nice it does look like there's a notion of "free $\omega$-category" on cubical sets

Ruby Khondaker (she/her) (Jul 09 2025 at 07:37):

in the strict sense

Ruby Khondaker (she/her) (Jul 09 2025 at 07:38):

so then it should be pretty straightforward to define the weak version, if i just copy leinster's approach

Ruby Khondaker (she/her) (Jul 09 2025 at 07:38):

modulo figuring out how cubes work compared to globes

Amar Hadzihasanovic (Jul 09 2025 at 07:38):

Honestly I think that at the moment you do not display a very good sense of how difficult problems are.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:39):

yeah, i think that's fair, i am very new to all of this

Ruby Khondaker (she/her) (Jul 09 2025 at 07:39):

i guess i usually can't automatically see the reasons why a problem would be hard because of my lack of experience

Amar Hadzihasanovic (Jul 09 2025 at 07:39):

Yeah, that's what supervisors are for

Ruby Khondaker (she/her) (Jul 09 2025 at 07:39):

ah, my supervisor is also a physicist :sweat_smile:

Ruby Khondaker (she/her) (Jul 09 2025 at 07:41):

i guess i can just try out defining cubical weak $\omega$-categories and see why the "obvious" approach fails

Amar Hadzihasanovic (Jul 09 2025 at 07:42):

I would recommend reading a bit. Since we started this conversation yesterday, I have recommended multiple relevant and often very long papers.

Amar Hadzihasanovic (Jul 09 2025 at 07:42):

A good way to get a sense of the difficulties is to see what difficulties other authors have had to overcome.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:43):

mhm that’s true, though of course it takes quite a while to digest all of those!

Amar Hadzihasanovic (Jul 09 2025 at 07:43):

I'm all for “rediscovering” but you cannot reinvent the totality of higher category theory alone.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:44):

oh I’m not trying anything as ambitious as that! I just wanna understand functor categories for weak $(\infty, \infty)$ categories is all

Ruby Khondaker (she/her) (Jul 09 2025 at 07:44):

so that I can understand the weak $(\infty, \infty)$ category of weak $(\infty, \infty)$ categories

Amar Hadzihasanovic (Jul 09 2025 at 07:46):

Ruby Khondaker (she/her) said:

ah, my supervisor is also a physicist :sweat_smile:

If you're in Oxford, there's plenty of other people that you can talk to!

Ruby Khondaker (she/her) (Jul 09 2025 at 07:46):

to be clear i totally get where you're coming from

Ruby Khondaker (she/her) (Jul 09 2025 at 07:46):

i can understand how frustrating it must be from your end for me to be asking a bunch of silly questions that already have answers in the literature

Ruby Khondaker (she/her) (Jul 09 2025 at 07:46):

when the obvious thing to do would just be "read the literature lol"

Ruby Khondaker (she/her) (Jul 09 2025 at 07:47):

i guess i get a different sort of appreciation for the difficulty of a problem if i try it myself and then run into the stumbling blocks naturally

Ruby Khondaker (she/her) (Jul 09 2025 at 07:47):

not necessarily better or worse, just different

Amar Hadzihasanovic (Jul 09 2025 at 07:47):

Oh don't worry about me, I'm not duty-bound to answer!

Ruby Khondaker (she/her) (Jul 09 2025 at 07:47):

ah no no i wasn't saying you were!

Ruby Khondaker (she/her) (Jul 09 2025 at 07:48):

just that i can totally get how it's annoying on your end

Amar Hadzihasanovic (Jul 09 2025 at 07:48):

Yeah but it's not, I would just switch off if I was annoyed.

Ruby Khondaker (she/her) (Jul 09 2025 at 07:49):

ah, apologies, tone is harder to read over text

Ruby Khondaker (she/her) (Jul 09 2025 at 07:49):

Amar Hadzihasanovic said:

I would recommend reading a bit. Since we started this conversation yesterday, I have recommended multiple relevant and often very long papers.

when you said this i thought this was in some kind of exasperated tone :sweat_smile:

Ruby Khondaker (she/her) (Jul 09 2025 at 07:50):

Amar Hadzihasanovic said:

Ruby Khondaker (she/her) said:

ah, my supervisor is also a physicist :sweat_smile:

If you're in Oxford, there's plenty of other people that you can talk to!

also any you'd recommend? i'd be pretty interested!

Amar Hadzihasanovic (Jul 09 2025 at 07:53):

Christopher Douglas, Andre Henriques and Ulrike Tillmann would be people with whom you can certainly discuss higher category theory.

Amar Hadzihasanovic (Jul 09 2025 at 07:53):

Just the first that come to my mind.

Amar Hadzihasanovic (Jul 09 2025 at 07:57):

And no, I meant the reading as "honest" advice, in the sense that I think it may help you gain some appreciation of what difficulties are involved with e.g. Gray products or algebraic weak higher categories, since often these would be discussed in the papers, and at the moment it seems that you underestimate them (which is not a criticism).

Amar Hadzihasanovic (Jul 09 2025 at 07:58):

It is useful to have a better assessment of the difficulties since during a PhD you don't necessarily want to start a project that takes you into a rabbit hole (note that this is advice that I myself have not followed in my PhD, so take it with a pinch of hypocrisy).

Ruby Khondaker (she/her) (Jul 09 2025 at 08:06):

ah thanks for the advice! i'll see if i can talk to those people sometime

Ruby Khondaker (she/her) (Jul 09 2025 at 08:06):

though i guess i haven't really thought of this as a "project" per se

Ruby Khondaker (she/her) (Jul 09 2025 at 08:06):

this is more just, like, a hobby for me

Ruby Khondaker (she/her) (Jul 09 2025 at 11:36):

Hm if i managed to come up with a definition of weak cubical $n$-category would that be good

Ruby Khondaker (she/her) (Jul 09 2025 at 11:37):

I don't exactly know how I'd be able to check such a definition

Ruby Khondaker (she/her) (Jul 09 2025 at 11:40):

Or maybe, what would a good "target" to aim for be?