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Stream: learning: questions

Topic: ✔ Cubical-to-Globular translation

Ruby Khondaker (she/her) (Jul 09 2025 at 13:52):

Amar Hadzihasanovic said:

Read this paper.

thank you for this, i skimmed through it and it was very helpful!

Ruby Khondaker (she/her) (Jul 09 2025 at 13:53):

so "cubes with connections" seems to be a good "test category" for modelling higher category theory, in terms of being a good category of shapes?

Ruby Khondaker (she/her) (Jul 09 2025 at 13:53):

plus you can talk about things like "commutative shells"

Ruby Khondaker (she/her) (Jul 09 2025 at 13:53):

also seems like it makes internal homs and tensor products of (strict) cubical $\omega$-categories quite straightforward

Ruby Khondaker (she/her) (Jul 09 2025 at 13:54):

so does that mean you get a (strict) $\omega$-category of (strict) cubical $\omega$-categories?

Ruby Khondaker (she/her) (Jul 09 2025 at 14:57):

Ooh it seems like this paper might have some good stuff on cubical approach to weak $\omega$-categories - A cubical model for $(\infty, n)$-categories

Ruby Khondaker (she/her) (Jul 09 2025 at 14:58):

also comical set is a pretty funny name

Ruby Khondaker (she/her) (Jul 09 2025 at 15:27):

And we also have Equivalence of cubical and simplicial approaches to $(\infty, n)$-categories

Ruby Khondaker (she/her) (Jul 09 2025 at 15:27):

Amar Hadzihasanovic said:

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

if you don't mind me asking, would these papers count as cubical weak $\omega$-categories?

Amar Hadzihasanovic (Jul 09 2025 at 16:13):

I was following your instruction that by "weak $\omega$-categories" I should think of an algebraic model à la Batanin--Leinster.

Amar Hadzihasanovic (Jul 09 2025 at 16:13):

In which case no, they would not count.

Amar Hadzihasanovic (Jul 09 2025 at 16:13):

These are "geometric" models.

Ruby Khondaker (she/her) (Jul 09 2025 at 16:13):

Ah sorry for the confusion, I was looking for a geometric model

Amar Hadzihasanovic (Jul 09 2025 at 16:16):

I would also like to repeat that the category of "atoms" that @Clémence Chanavat and I use to define diagrammatic $(\infty, n)$-categories was specifically designed as "the best possible shape category" for constructions such as Gray tensor products, joins, and suspensions, so I would encourage you to actually check it out if that's what you are interested in.

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

Ooh ok, sure!

Amar Hadzihasanovic (Jul 09 2025 at 16:20):

The main difference is that categories such as the cube and simplex category are "naturally" suited to model undirected cells (as in higher groupoids) so to make them encode directed higher-categorical cells one needs to supplement some extra data in the form of a "marking" which picks what cells should be considered invertible and what cells should not be considered invertible.

Ruby Khondaker (she/her) (Jul 09 2025 at 16:21):

Whereas atoms don’t need such a marking…?

Amar Hadzihasanovic (Jul 09 2025 at 16:21):

Whereas the atom category has "naturally directed cells" and can support a model of $(\infty, \infty)$-categories on unmarked presheaves.

Ruby Khondaker (she/her) (Jul 09 2025 at 16:22):

Nice!!

Ruby Khondaker (she/her) (Jul 09 2025 at 16:22):

Are these “naturally directed” in the same way that n-globes are?

Amar Hadzihasanovic (Jul 09 2025 at 16:23):

(In one of the senses of $(\infty, \infty)$ discussed by Madeleine, the "coinductive" one, at least)

Amar Hadzihasanovic (Jul 09 2025 at 16:24):

Well atoms are best thought as something like "directed polytopal subdivisions" of $n$-globes, which are at once "nice" topological $n$-balls and composable pasting diagrams.

Ruby Khondaker (she/her) (Jul 09 2025 at 16:26):

Mhm mhm

Ruby Khondaker (she/her) (Jul 09 2025 at 16:26):

I skimmed through this paper if that’s what you meant

Ruby Khondaker (she/her) (Jul 09 2025 at 16:28):

Maybe this might be the end of my questions for now, since I now know what $\omega$-categories are supposed to be

Amar Hadzihasanovic (Jul 09 2025 at 16:28):

Yes, that's the paper written by Clémence specifically showing that the Gray product of diagrammatic sets is compatible with the model for $(\infty, n)$-categories.

Amar Hadzihasanovic (Jul 09 2025 at 16:28):

That and our joint referenced papers is all there is for now.

Ruby Khondaker (she/her) (Jul 09 2025 at 16:30):

So then, what’s next for $\omega$-category theory?

Amar Hadzihasanovic (Jul 09 2025 at 16:37):

I would tell you but then you might scoop me.

Amar Hadzihasanovic (Jul 09 2025 at 16:37):

(/s)

Ruby Khondaker (she/her) (Jul 09 2025 at 16:37):

pfffft, good one

Ruby Khondaker (she/her) (Jul 09 2025 at 16:38):

thank you so so much for your time and patience!!

Ruby Khondaker (she/her) (Jul 09 2025 at 16:38):

and all the links to awesome papers

Notification Bot (Jul 10 2025 at 08:31):

Ruby Khondaker (she/her) has marked this topic as resolved.