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Stream: learning: questions

Topic: Omega-Category Theory

Ruby Khondaker (she/her) (Jul 07 2025 at 21:28):

By this I mean $(\infty, \infty)$-categories, since $\infty$-categories are already taken :P

So I wanted to ask, what's the current "best technology" for working with $\omega$-categories? I've skimmed through Leinster's "Higher Operads, Higher Categories" so I'm familiar with this "algebraic" definition via an appropriate globular operad. But of course, in that book he doesn't expand on what weak $\omega$-functors should be.

I went to one of Riehl's talks recently and she recommended I look at complicial sets, which apparently give a notion of both $\omega$-category and $\omega$-functor? But it also seems like there's not a consensus on what $\omega$-category theory should be...

I guess a related question is - what would people want from a proposed notion of $\omega$-category theory? Presumably generalisations of things like universal properties, monads etc, but I'm not really familiar with the other kinds of things that naturally pop up in infinite-dimensional category theory...

One main reason I'm asking is that I've recently been toying around with my own definition of $\omega$-category (still very early stages and I really don't know if it works properly), so it'd be good to have a "checklist" of sensible things such objects (and the $\omega$-category of such objects) should satisfy!

John Baez (Jul 07 2025 at 22:06):

I don't have the energy for a systematic reply, and I barely follow research on $\omega$-categories anymore, since there are so many fun things to do that are easier, but you might check out the work on Grothendieck-Maltsiontis infinity-categories, which is a revival of Grothendieck's old dream. (And yes, these are actual $\omega$-categories, none of this $(\infty,1)$ stuff.)

Ruby Khondaker (she/her) (Jul 07 2025 at 22:08):

Ooh ok, I've never actually read Pursuing Stacks so perhaps now would be as good a time as any :)

John Baez (Jul 07 2025 at 22:09):

Sure, whenever you have time to read a 529-page book it's a good time to start!

Actually the new work may be easier to follow.

Also, even though it may be too lowbrow for you, you might be interested in

• Carlos Simpson, Some properties of the theory of n-categories.

since it's an attempt to characterize n-categories in a model-independent way.

Ruby Khondaker (she/her) (Jul 07 2025 at 22:10):

I think it'd be pretty difficult to find something too lowbrow for me! Also I did not realise it was that long, haha, I thought it was just a letter..

Madeleine Birchfield (Jul 07 2025 at 22:26):

There's also Zach Goldthorpe's Homotopy theories of $(\infty, \infty)$-categories as universal fixed points with respect to enrichment and a few other papers that define $(\infty, \infty)$-categories and $\omega$-categories in terms of $(\infty, 1)$-categories.

Ruby Khondaker (she/her) (Jul 07 2025 at 22:28):

Oh is that related to how you can do iterated enrichment in $(\infty, 1)$ categories to get $(\infty, n)$ categories…?

Madeleine Birchfield (Jul 07 2025 at 22:29):

Yeah.

Madeleine Birchfield (Jul 07 2025 at 22:30):

The embedding of $(\infty,n)\mathrm{Cat}$ into $(\infty,n + 1)\mathrm{Cat}$ has a left adjoint and a right adjoint, and taking the sequential limit of the left adjoints yields $\omega\mathrm{Cat}$ and taking the sequential limits of the right adjoints yields $(\infty,\infty)\mathrm{Cat}$.

Ruby Khondaker (she/her) (Jul 07 2025 at 22:31):

Wait what’s the difference between $\omega$ and $(\infty, \infty)$ here

Ruby Khondaker (she/her) (Jul 07 2025 at 22:31):

I was using them interchangeably

Madeleine Birchfield (Jul 07 2025 at 22:32):

Here the authors use $\omega$ for the coinductive $\infty$-catgories and $(\infty,\infty)$ for the inductive $\infty$-categories.

Ruby Khondaker (she/her) (Jul 07 2025 at 22:32):

Ohhh I see

Are these expected to be equivalent or actually different objects?

Madeleine Birchfield (Jul 07 2025 at 22:33):

They're different objects I think.

Ruby Khondaker (she/her) (Jul 07 2025 at 22:33):

Huh, weird..

Madeleine Birchfield (Jul 07 2025 at 22:34):

There's another article that is about inductive $\infty$-categories, where the author calls "$(\infty,\omega)$-categories", but I can't seem to find it at the moment.

Ruby Khondaker (she/her) (Jul 07 2025 at 22:35):

I did not realise there was a whole zoo of definitions

Madeleine Birchfield (Jul 07 2025 at 22:35):

Never mind, found it. It's Félix Loubaton's Theory and models of $(\infty,\omega)$-categories.

Madeleine Birchfield (Jul 07 2025 at 22:37):

Ruby Khondaker (she/her) said:

I did not realise there was a whole zoo of definitions

Yeah, that's what happens when you have multiple dimensions of direction to deal with in higher category theory. Even for 2-category theory the combinatorial explosion of various notions of 2-categories is a sight to behold.

Madeleine Birchfield (Jul 07 2025 at 22:39):

Then when you get to the $\infty$-category case, you can decide whether you want the limit to $\infty$ to behave inductively like the natural numbers or coinductively like the extended natural numbers.

Ruby Khondaker (she/her) (Jul 07 2025 at 22:39):

Hmm, the main notion of 2-category I know of are bicategories

Amar Hadzihasanovic (Jul 08 2025 at 06:24):

Hi Ruby! For the introduction of my latest paper with Clémence Chanavat we tried to give a leisurely overview of the current zoo of definitions, so I would suggest that you read that instead of giving you a poorer summary :)

Ruby Khondaker (she/her) (Jul 08 2025 at 06:24):

Oh, thanks so much! I’ll take a look :)

Ruby Khondaker (she/her) (Jul 08 2025 at 06:25):

Woah it’s super recent

Amar Hadzihasanovic (Jul 08 2025 at 06:27):

Clémence and I have been very active precisely on the formulation of models that are better-suited to explicit computations and diagrammatics in the style of low-dim category theory (as opposed to homotopy theory), I am not enough of a physicist to know whether that suits your needs (and it is all pretty recent), you can also look at this.

For a more encyclopedic overview of what “working” mathematicians have been using you can look at this chapter written by Viktoriya Ozornova and Martina Rovelli for the Encyclopedia of mathematical physics.

Ruby Khondaker (she/her) (Jul 08 2025 at 06:35):

Ah, so the paper mentions the homotopy hypothesis - should I take this to mean that the homotopy type of any topological space may be realised as an $(\infty, 0)$ category in the model?

Nathanael Arkor (Jul 08 2025 at 06:43):

Madeleine Birchfield said:

The embedding of $(\infty,n)\mathrm{Cat}$ into $(\infty,n + 1)\mathrm{Cat}$ has a left adjoint and a right adjoint, and taking the sequential limit of the left adjoints yields $\omega\mathrm{Cat}$ and taking the sequential limits of the right adjoints yields $(\infty,\infty)\mathrm{Cat}$.

It should be noted that this terminology isn't standard.

Nathanael Arkor (Jul 08 2025 at 06:46):

There's a discussion of the difference between these approaches in §1.2 of Henry and Loubaton's An inductive model structure for strict $\infty$-categories.

Amar Hadzihasanovic (Jul 08 2025 at 06:47):

Ruby Khondaker (she/her) said:

Ah, so the paper mentions the homotopy hypothesis - should I take this to mean that the homotopy type of any topological space may be realised as an $(\infty, 0)$ category in the model?

Yes (well, the homotopy type of any CW-complex, technically)

Amar Hadzihasanovic (Jul 08 2025 at 06:49):

This is a theorem for all the "geometric" models and unproven for the "algebraic" models (except the one we define in our paper, which has not yet been compared to any of the pre-existing algebraic models).

John Baez (Jul 08 2025 at 07:06):

@Ruby Khondaker (she/her) - regarding Amar's technical remark, the point is simply that there are lots of nasty topological spaces, like the rational numbers, which we'd rather not think about in homotopy theory. So we invent the concept of [[weak homotopy equivalence]]. Any space is weakly homotopy equivalent to a nice one namely a [[CW complex]], and weakly homotopy equivalent CW complexes are homotopy equivalent. Thus, we say the homotopy type of any CW complex may be realized as an $\infty$-groupoid, but also that the 'weak homotopy type' of any topological space may be realized as an $\infty$-groupoid.

John Baez (Jul 08 2025 at 07:10):

Less technically, let's return to your original question "what would people want from a proposed notion of $\omega$-category theory?" Grothendieck proposed one thing we should want: the [[homotopy hypothesis]]. But this mainly concerns $\infty$-groupoids. For full-fledged $\omega$-categories James Dolan and I proposed a couple of other things we might want: the [[stabilization hypothesis]] and the [[tangle hypothesis]]. A certain special case of the tangle hypothesis, the 'stable' case, has received the most attention: this is called the [[cobordism hypothesis]]. Lurie formulated this as a hypothesis about $(\infty,n)$-categories, and outlined a proof that should work in any definition of $(\infty,n)$-category where a number of reasonable things can be shown.

Amar Hadzihasanovic (Jul 08 2025 at 07:11):

Yes, I guess perhaps now using “homotopy type” for “weak homotopy type” has become more standard, where I first learnt this stuff from there was still a distinction between the two.

John Baez (Jul 08 2025 at 07:12):

By the time people started doing "homotopy type theory", they were glad to not say they were doing "weak homotopy type theory". :upside_down:

Ruby Khondaker (she/her) (Jul 08 2025 at 07:48):

Yeah that is something that confused me a little with HoTT - definitions like contractibility seemed too “strong”? There contractibility is essentially defined as “homotopy equivalent to the point”, whereas really “weakly homotopy equivalent to the point” should suffice…

John Baez (Jul 08 2025 at 07:51):

They are working in a context where those are the same. By 'homotopy type', almost everybody now means 'weak homotopy type'. Essentially nobody in this branch of homotopy theory cares about weird topological spaces like $\mathbb{Q}$ or the Cantor set.

Ruby Khondaker (she/her) (Jul 08 2025 at 07:52):

Ah I see, I guess that’s just my unfamiliarity with it. When I see $\Sigma_{x : A} \Pi_{y : A} x =_A y$, my picture is “choose a point x, and contract A to x continuously”, and that feels like “strong” homotopy equivalence? Apologies for the misunderstanding

John Baez (Jul 08 2025 at 07:55):

There's no difference between weak homotopy equivalence and ordinary 'strong' homotopy equivalence of topological spaces until you start studying spaces that are locally nasty, and those are not relevant to homotopy type theory.

Amar Hadzihasanovic (Jul 08 2025 at 07:56):

A type in HoTT (can be/is) modelled by an object of the $(\infty, 1)$-category of spaces.
A model category can present this $(\infty, 1)$-category in such a way that the objects correspond to the bifibrant, i.e. fibrant-cofibrant objects of the model category.

Amar Hadzihasanovic (Jul 08 2025 at 07:57):

If you take the category of (nice enough) topological spaces with its classical model structure, all spaces are fibrant and only the (underlying spaces of) CW complexes are cofibrant.

Amar Hadzihasanovic (Jul 08 2025 at 07:58):

So the bifibrants are in fact the (underlying spaces of) CW-complexes, for which weak equivalence coincides with homotopy equivalence.

Ruby Khondaker (she/her) (Jul 08 2025 at 07:58):

I see! I recently read a little about model categories but still don’t have great intuition for them

Ruby Khondaker (she/her) (Jul 08 2025 at 07:59):

Maaaaybe because I haven’t actually taken a course in algebraic topology :sweat_smile:

Ruby Khondaker (she/her) (Jul 08 2025 at 08:24):

I also came across this paper - https://arxiv.org/pdf/1706.02866

Ruby Khondaker (she/her) (Jul 08 2025 at 08:24):

It’s some kind of type-theoretic model of $\omega$-categories? Does this allow you to get a notion of weak functor as well…?

Amar Hadzihasanovic (Jul 08 2025 at 08:30):

The type-theoretic model can be seen as an alternative presentation of the Grothendieck-Maltsiniotis model mentioned earlier by John, see this paper, in the sense that the category of models of the type theory (seen as a "generalised algebraic theory") is equivalent to the category of Grothendieck-Maltsiniotis weak $\omega$-categories and strict functors.

Ruby Khondaker (she/her) (Jul 08 2025 at 08:32):

Ah I see, so it’s still a fundamentally “algebraic” definition at heart, meaning you only get strict functors. Are they are any good “geometric” definitions floating around? Presumably it’d be significantly easier to define weak functors

Amar Hadzihasanovic (Jul 08 2025 at 08:32):

There is less work on weak functors between this structures, I believe I've been told that some work of Bourke (probably this) is applicable to giving such definitions, but I am not familiar enough with it.

Amar Hadzihasanovic (Jul 08 2025 at 08:33):

The "geometric" definitions are what everyone who is actually "doing things with $(\infty, n)$-categories” (as opposed to doing “meta-level” work) currently uses. And yes their natural functors are weak functors because in a certain sense there is no other way that they could be.

Ruby Khondaker (she/her) (Jul 08 2025 at 08:34):

I guess I’m just thinking about how quasicategories neatly describe $(\infty, 1)$ categories as “presheaves with properties”, so that the obvious maps between them give weak functors.

Amar Hadzihasanovic (Jul 08 2025 at 08:34):

Geometric definitions typically are not equipped with a choice of composites (rather, just a contractible space of equally good choices) so there is no fixed choice that a functor should strictly preserve.

Ruby Khondaker (she/her) (Jul 08 2025 at 08:36):

Part of the reason I’m asking is that I’ve been toying around with my own definition of $\omega$-category, and I’m trying to see whether it’s either:

• Wrong for a reason I haven’t figured out
• Already present somewhere in the literature

Hence why I want to know more about the “geometric” definitions of $\omega$-categories

Amar Hadzihasanovic (Jul 08 2025 at 08:37):

You should probably give some details about what your proposed model looks like.

Ruby Khondaker (she/her) (Jul 08 2025 at 08:38):

Hmmm I would but that would be spoilers? Or I guess, I don’t want to get scooped… what sorts of details would you need?

Amar Hadzihasanovic (Jul 08 2025 at 08:40):

I think for a young researcher in a “niche” field, it is a VASTLY greater problem to get others to even care about your work, than to avoid being “scooped”...

Ruby Khondaker (she/her) (Jul 08 2025 at 08:42):

Right, that’s true… well, I think some details I can give is that it’s a “geometric” def where the shapes are basically globular. I was inspired by Leinster’s “algebraic” def using his contractible globular operad, and I think I figured out a way to make it “geometric”? As far as I can tell, categories, bicategories, quasicategories and current defs of weak omega-categories embed into the model

Amar Hadzihasanovic (Jul 08 2025 at 08:44):

What does "basically globular" mean?

Ruby Khondaker (she/her) (Jul 08 2025 at 08:44):

Shapes being used are globular sets (and pasting diagrams of those)

Amar Hadzihasanovic (Jul 08 2025 at 08:45):

Are you using presheaves on the Theta category?

Ruby Khondaker (she/her) (Jul 08 2025 at 08:45):

Hmm I don’t think so

Amar Hadzihasanovic (Jul 08 2025 at 08:47):

Pasting diagrams of globes are the objects of Theta. One of the mainstream geometric definitions uses simplicial presheaves on Theta. This is the Rezk model. See [[Theta space]]

Ruby Khondaker (she/her) (Jul 08 2025 at 08:47):

Ooh ok, I’ll take a look! Perhaps this is what I was rediscovering

Amar Hadzihasanovic (Jul 08 2025 at 08:49):

If that's what you were rediscovering you are lucky because it is one of the best-developed models!

Ruby Khondaker (she/her) (Jul 08 2025 at 08:49):

I guess I’m not sure what to do in the incredibly unlikely scenario that what I’m discovering is genuinely new

Ruby Khondaker (she/her) (Jul 08 2025 at 08:49):

The issue is that my approach seems so blindingly obvious that it feels like it must either be wrong or just already be known

Amar Hadzihasanovic (Jul 08 2025 at 08:53):

My advice would be that at an early stage in your career it is important to think carefully about what you are trying to achieve.
Since there is already this "zoo" of definitions, you should only try to contribute “yet another definition” if you believe it can achieve something that others cannot. In the case of myself and Clémence, it was to have a model that supports diagrammatic reasoning & proves a strictification theorem, which was an open problem.

Amar Hadzihasanovic (Jul 08 2025 at 08:55):

And even with that, I cannot say for sure whether this will lead do interest/adoption for these models, since in the end most people study higher category theory for some reason, whether it is to prove something about quantum field theories or about algebraic geometry.

Ruby Khondaker (she/her) (Jul 08 2025 at 08:55):

I suppose I started this all out for selfish reasons, I was dissatisfied with how higher category theory was presented and wanted to figure out if there was a simpler way to go about things? Now it feels like I’ve figured out a simpler way, which makes it easier for me to think about higher category theory.

Unfortunately I don’t really know enough about the field to know what sorts of open problems there are… I guess I just wanted to “reorganise” things that make the concepts simpler in my head, and maybe that’ll work for others too!

Amar Hadzihasanovic (Jul 08 2025 at 08:57):

So if you want to publish something that uses higher categories,

• either you publish in the “meta-theory” of higher categories and then there must be some new insight or some open problem in the theory that your model is solving,
• or you publish in “applied” higher category theory and then using a non-standard model just means more pain in getting your papers read and accepted.

Ruby Khondaker (she/her) (Jul 08 2025 at 08:59):

Yeah then maybe it shouldn’t be a paper but an article or something? I don’t know, it’d just be cool to share my idea and see if others like it… I’m not even a researcher in this field really, being a physicist and all

John Baez (Jul 08 2025 at 09:00):

You are free to share your idea whenever you want.

Ruby Khondaker (she/her) (Jul 08 2025 at 09:01):

Ok! Then I’ll keep working on it until I feel like it’s substantial enough to share

Amar Hadzihasanovic (Jul 08 2025 at 09:01):

From our conversation, it seems to me like you do not have such a clear picture of the state of higher category theory; the Rezk “Theta-space” model is perhaps the most used model (I guess the Tamsamani--Simpson model is a contender) of $(\infty, n)$-categories around, and there are good reasons for it, besides the fact that plenty of machinery has been developed around it. So perhaps it is best to familiarise yourself with the “mainstream” before saying you are “dissatisfied” with it.

Ruby Khondaker (she/her) (Jul 08 2025 at 09:02):

Yeah apologies I know this makes me seem like a crank and all - when I say “dissatisfied” I don’t mean that I’m upset at the way the field is going, I just mean I was annoyed I couldn’t understand it lol

Amar Hadzihasanovic (Jul 08 2025 at 09:02):

This is not to say that there are no reasons to be dissatisfied! If you read the introduction to the paper I shared, we basically start by saying how the current state is great for some sub-fields, but not so great for others :)

Ruby Khondaker (she/her) (Jul 08 2025 at 09:03):

It sounds like I should familiarise myself with the Theta-space model! Maybe I’ll like it enough that I won’t even need to worry about my own toy model

Ruby Khondaker (she/her) (Jul 08 2025 at 09:03):

And yeah apologies again, I am very new to all of this

Amar Hadzihasanovic (Jul 08 2025 at 09:03):

No need to apologise! We're all learning together, there was nothing offensive about what you said.

Ruby Khondaker (she/her) (Jul 08 2025 at 09:04):

I just am very self-conscious of coming across as a crank

Ruby Khondaker (she/her) (Jul 08 2025 at 09:04):

Because it feels like I am doing all the behaviours a crank would typically do

Ruby Khondaker (she/her) (Jul 08 2025 at 09:04):

Being “dissatisfied” with the mainstream and being “secretive” about my own pet model

Ruby Khondaker (she/her) (Jul 08 2025 at 09:05):

I am very much trying not to be a crank

Ruby Khondaker (she/her) (Jul 08 2025 at 09:05):

But I don’t know if I’m being one anyway and I just can’t tell :(

Amar Hadzihasanovic (Jul 08 2025 at 09:06):

Cranks are not typically self-conscious about being such so I think you can interpret your own self-doubts as evidence against those doubts ;)

John Baez (Jul 08 2025 at 09:06):

I often invent ideas when I'm annoyed at how I can't understand what other people are saying. Most of these ideas turn out to be already known. It's still fun to reinvent things. Now and then these ideas turn out to be new, or at least have some new aspect.

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

Yeah that’s the whole reason I’m doing this

Ruby Khondaker (she/her) (Jul 08 2025 at 09:07):

It’s a skill issue on my part of finding higher cat theory hard

Amar Hadzihasanovic (Jul 08 2025 at 09:07):

Oh but it is hard, nobody says it isn't.

Ruby Khondaker (she/her) (Jul 08 2025 at 09:07):

But if I’ve figured out a way of rephrasing it that makes it easier, maybe others will find it easier to think about higher cat theory too

John Baez (Jul 08 2025 at 09:07):

If.

I think everyone finds higher category theory to be hard.

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

All I’d want is for people to say “oh hey have you seen Ruby’s way of thinking about this? It makes it conceptually clearer”

John Baez (Jul 08 2025 at 09:09):

Ah yes, the young scientist's quest for glory and fame. :wink:

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

Well a girl’s gotta dream a little right :P

Amar Hadzihasanovic (Jul 08 2025 at 09:09):

I think one thing to be wary of is the monad tutorial fallacy:

After struggling to understand them for a week, looking at examples, writing code, reading things other people have written, he finally has an “aha!” moment: everything is suddenly clear, and Joe Understands Monads! What has really happened, of course, is that Joe’s brain has fit all the details together into a higher-level abstraction, a metaphor which Joe can use to get an intuitive grasp of monads; let us suppose that Joe’s metaphor is that Monads are Like Burritos. Here is where Joe badly misinterprets his own thought process: “Of course!” Joe thinks. “It’s all so simple now. The key to understanding monads is that they are Like Burritos. If only I had thought of this before!” The problem, of course, is that if Joe HAD thought of this before, it wouldn’t have helped: the week of struggling through details was a necessary and integral part of forming Joe’s Burrito intuition, not a sad consequence of his failure to hit upon the idea sooner.

(I am actually quoting John quoting this here)

Ruby Khondaker (she/her) (Jul 08 2025 at 09:10):

Ooh yes I have heard of this phenomenon, I should look into that more carefully to make sure I’m not falling into that trap

Ruby Khondaker (she/her) (Jul 08 2025 at 09:11):

The only reason I’m being secretive is that if I share an incomplete version of the idea, and then some actual researcher tightens it up, I can’t really say I deserve any credit for it

Ruby Khondaker (she/her) (Jul 08 2025 at 09:11):

Hence why I want to make it more substantial before sharing

John Baez (Jul 08 2025 at 09:14):

That makes sense. I just advise not being so talkative about your secrecy.

Ruby Khondaker (she/her) (Jul 08 2025 at 09:14):

oh? Why’s that?

John Baez (Jul 08 2025 at 09:19):

I guess it's just a social norm in science, not to announce that you have a secret. It just sounds funny: "I have an idea but I'm not going to tell you what it is".

Ruby Khondaker (she/her) (Jul 08 2025 at 09:19):

Ahhh I see, apologies - I’m still very new to academia so I haven’t quite picked up all the social norms

Ruby Khondaker (she/her) (Jul 08 2025 at 09:19):

I’ll bear this in mind in the future :)

John Baez (Jul 08 2025 at 09:19):

Centuries ago it was different: scientists would post anagrams to announce that they had discovered something without revealing what it was.

Ruby Khondaker (she/her) (Jul 08 2025 at 09:20):

pfft, I had no idea :joy:

Ruby Khondaker (she/her) (Jul 08 2025 at 09:23):

I guess the three main outcomes I can foresee are either:

• I understand theta spaces well enough that I don’t need to worry about my pet “higher cat theory tutorial”
• I still don’t get theta spaces, try to develop my “higher cat theory tutorial” and realise it’s wrong (which hopefully helps me understand higher cat theory better)
• I try to develop my “higher cat theory” tutorial and figure out it’s right, in which case I can try sharing it with others to see what they think of it

Amar Hadzihasanovic (Jul 08 2025 at 09:23):

Also, to repeat myself, you will find that, perhaps with the exception of a small number of extremely competitive fields (the sort of stuff that may get you a Fields medal) you will find that the idea that anyone is going around looking for good ideas to “scoop” is very far from reality...

Ruby Khondaker (she/her) (Jul 08 2025 at 09:24):

Yeah clearly I don’t have a good idea of how actual academia works :P

Amar Hadzihasanovic (Jul 08 2025 at 09:24):

And that if you try to tell anyone about your new model, the reaction is likely to be “good for you, come back when you have done something that nobody has done with any other model AND that I care about, then I'll take a real interest”

Ruby Khondaker (she/her) (Jul 08 2025 at 09:41):

I see!

Ruby Khondaker (she/her) (Jul 08 2025 at 09:41):

Then, what sorts of things has nobody done with any model, and that people care about?

Ruby Khondaker (she/her) (Jul 08 2025 at 09:43):

That would probably be a good “test” to see if what I’m doing is “just another monad tutorial” or whether it’s actually useful in practice :)

John Baez (Jul 08 2025 at 09:47):

There are tons of things nobody has done yet, and since your model is top secret we can't easily guess which of those things your model might be useful for.

Ruby Khondaker (she/her) (Jul 08 2025 at 09:48):

Right… maybe learning more about theta spaces might give me a good idea of stuff nobody has done yet, then

John Baez (Jul 08 2025 at 09:50):

That's probably a good place to start, anyway! In fact you'd do very well to learn about everything Amar considers important, since he's a real expert on higher categories.

Ruby Khondaker (she/her) (Jul 08 2025 at 09:50):

I'm very grateful to get advice from such an expert in the field :D

Ruby Khondaker (she/her) (Jul 08 2025 at 10:15):

Ok the paper "A Topological Perspective on Interacting Algebraic Theories" is extremely cool

Ruby Khondaker (she/her) (Jul 08 2025 at 10:18):

It seems to me that computads are just the analog of "free groups on a set", except you're taking "free (higher)-categories on a globular set"?

Ruby Khondaker (she/her) (Jul 08 2025 at 10:22):

So e.g. if I think about Leinster's algebraic definition of omega-categories, as algebras for a particular globular operad (and thus monad algebras for a particular monad), then omega-computads are precisely the free algebras

Ruby Khondaker (she/her) (Jul 08 2025 at 10:26):

i guess this would require an inherently "algebraic" notion of higher category, though

Ruby Khondaker (she/her) (Jul 08 2025 at 10:29):

e.g. to me it is unclear what computads in a "geometric" notion of higher categories would be - e.g. quasicategories aren't monadic over simplicial sets, so there's no sensible notion of a "free quasicategory" on a simplicial set, as far as I know (unless you take, like, a "discrete" version?). My assumption is that this would hold more generally - a given geometric definition of a higher category doesn't admit a "free" construction that would let you define geometric-computads

Ruby Khondaker (she/her) (Jul 08 2025 at 10:31):

so essentially it seems - given an omega-category $\mathcal{X}$, a computad is a sub-globular set $X$, together with an isomorphism between $TX$ and $\mathcal{X}$, where $TX$ is the free omega-category on $X$ (for whatever notion of "free omega-category" you like)

Ruby Khondaker (she/her) (Jul 08 2025 at 10:33):

and i'm guessing disjoint union and tensor product are analogous to the construction of the direct sum and tensor product of free vector spaces, in terms of their underlying generating sets

Ruby Khondaker (she/her) (Jul 08 2025 at 10:39):

Oh interesting, I only just realised that the categories of higher categories aren’t expected to be _cartesian_ closed, only monoidal closed, and this monoidal structure should be an appropriate “tensor product”? I guess I’ll need to be wary of that when figuring out internal homs

Ruby Khondaker (she/her) (Jul 08 2025 at 11:34):

ok nice, i think i found a way of interpreting one of Leinster's definitions that gives me what I want!!

Ruby Khondaker (she/her) (Jul 08 2025 at 11:34):

This means that it isn't new, thankfully

Ruby Khondaker (she/her) (Jul 08 2025 at 11:35):

Though I am pretty happy that I managed to kind of come up with this independently :>

Ruby Khondaker (she/her) (Jul 08 2025 at 11:35):

I'm happy to share if that's helpful

Madeleine Birchfield (Jul 08 2025 at 11:38):

Ruby Khondaker (she/her) said:

It’s a skill issue on my part of finding higher cat theory hard

John Baez said:

If.

I think everyone finds higher category theory to be hard.

As for higher category being hard, I'm waiting for a text on naive higher category theory that remains agnostic about the models of $(\infty,n)$-categories, so I don't have to deal with the complexity of the models and their equivalences with each other. Something similar to $\infty$-cosmos theory or the Cisinski et al textbook for $(\infty,1)$-category theory.

Ruby Khondaker (she/her) (Jul 08 2025 at 11:38):

What do you mean by "agnostic"?

Madeleine Birchfield (Jul 08 2025 at 11:41):

Ruby Khondaker (she/her) said:

What do you mean by "agnostic"?

It's like in real analysis, most of the time we talk about real numbers and we remain agnostic about a particular model of the real numbers (Dedekind cuts, Cauchy filters of rationals, etc).

Ruby Khondaker (she/her) (Jul 08 2025 at 11:42):

ah sure sure, so some kind of "formal" theory of omega-categories, independent of model?

Madeleine Birchfield (Jul 08 2025 at 11:42):

Yeah

Ruby Khondaker (she/her) (Jul 08 2025 at 11:42):

hm but here, we don't have anything "higher", right? the collection of all $(\infty, \infty)$ categories should form yet another $(\infty, \infty)$ category

John Baez (Jul 08 2025 at 11:43):

The idea would be to list axioms that $(\infty,n)$ categories should obey, and then develop the theory starting from those axioms. This approach is also called "synthetic". Check out Riehl and Shulman's slides on a synthetic theory of $\infty$-categories.

Ruby Khondaker (she/her) (Jul 08 2025 at 11:44):

ah yes, i've watched that talk!

Ruby Khondaker (she/her) (Jul 08 2025 at 11:44):

hm can i share what i figured out now?

Madeleine Birchfield (Jul 08 2025 at 11:45):

Sure go ahead

Ruby Khondaker (she/her) (Jul 08 2025 at 11:47):

Ok! So, I'm going off of Higher Operads, Higher Categories, in particular the last few sections. Leinster gives an "algebraic" definition of $\omega$-category through his "initial globular operad with a contraction"", which he calls $L$ - weak $\omega$-categories are then just algebras for this operad.

Ruby Khondaker (she/her) (Jul 08 2025 at 11:47):

He calls this definition of $\omega$-categories $\mathbf{L}$, as far as I can tell

Ruby Khondaker (she/her) (Jul 08 2025 at 11:48):

Then, there's a remark right near the end of the book, as he's going through various "non-algebraic" notions of $\omega$-category, of what he calls "definition $\mathbf{L}'$"

Ruby Khondaker (she/her) (Jul 08 2025 at 11:50):

The idea is that - strict $\omega$ categories are just algebras for the free strict $\omega$-category monad (on the category of globular sets), which we'll call $T$

Ruby Khondaker (she/her) (Jul 08 2025 at 11:51):

so
image.png
for $X$ a globular set (making this an algebra)

Ruby Khondaker (she/her) (Jul 08 2025 at 11:52):

you can think of this as a function which takes in a labelled pasting diagram $\hat \pi$, and assigns the unique cell of $X$ it composes to

Ruby Khondaker (she/her) (Jul 08 2025 at 11:52):

But of course, in various "geometric" notions of higher categories, composition isn't a function, right? It's a relation

Ruby Khondaker (she/her) (Jul 08 2025 at 11:52):

And there's a very natural way to represent this - a span!

Ruby Khondaker (she/her) (Jul 08 2025 at 11:53):

image.png

Ruby Khondaker (she/her) (Jul 08 2025 at 11:53):

specifically, what you want is a subobject of $TX \times X$ - here i've just "unpackaged" this into a span

Ruby Khondaker (she/her) (Jul 08 2025 at 11:54):

This gives you your composition relation, right? Given a labelled pasting diagram $\hat \pi$, and a cell $\alpha$ of $X$, it tells you whether or not $\alpha$ works as a composite of $\hat \pi$

Ruby Khondaker (she/her) (Jul 08 2025 at 11:55):

Of course, any actual algebra for $T$ trivially defines such a span, by setting $\text{Comp} = TX$ and $\pi = \text{id}$

Ruby Khondaker (she/her) (Jul 08 2025 at 11:56):

But we can consider more general spans - essentially what we want are monads in the bicategory of $T$-spans, otherwise known as $T$-multicategories. This essentially ensures our composition relation is "transitive" wrt pasting diagram composition (and "reflexive" too, in the sense that any m-cell is a composite of the trivial pasting diagram which looks like that cell)

Ruby Khondaker (she/her) (Jul 08 2025 at 11:57):

What's nice is that imposing that $\text{Comp}$ is a subobject of $TX \times X$ tells you that if it works as a monad, then the corresponding monad maps are unique - it can't be a "T-multicategory" in more than one way

Ruby Khondaker (she/her) (Jul 08 2025 at 11:58):

Does this all make sense so far?

Amar Hadzihasanovic (Jul 08 2025 at 12:23):

Ruby Khondaker (she/her) said:

It seems to me that computads are just the analog of "free groups on a set", except you're taking "free (higher)-categories on a globular set"?

They are dimensionwise free, which is a bit different...

Amar Hadzihasanovic (Jul 08 2025 at 12:24):

Ruby Khondaker (she/her) said:

That would probably be a good “test” to see if what I’m doing is “just another monad tutorial” or whether it’s actually useful in practice :)

I would recommend that the workflow should be

1. Find an open problem that you want to solve,
2. See if it can be solved with existing models/techniques,
3. Only if 2. fails start thinking about new models/techniques

Amar Hadzihasanovic (Jul 08 2025 at 12:25):

The approach "develop some formalism, then search for problems that it can be applied to" is rarely successful, in my experience

Ruby Khondaker (she/her) (Jul 08 2025 at 12:25):

Mhm, of course

Ruby Khondaker (she/her) (Jul 08 2025 at 12:26):

I’ve been sharing the idea I figured out so far here btw

Ruby Khondaker (she/her) (Jul 08 2025 at 12:32):

Also, I don’t quite understand the difference between feee and dimensionwise free…

Ruby Khondaker (she/her) (Jul 08 2025 at 12:35):

Ah maybe it’s just equivalent to a free omega-category

Ruby Khondaker (she/her) (Jul 08 2025 at 12:44):

Ooh ok you probably have that every weak omega-category is anaequivalent to a strict one?

Amar Hadzihasanovic (Jul 08 2025 at 12:45):

No, one of the first things which make higher category theory hard is that this is not true :)

Ruby Khondaker (she/her) (Jul 08 2025 at 12:46):

what do you mean?

Amar Hadzihasanovic (Jul 08 2025 at 12:46):

There are weak 3-categories which are not equivalent to strict ones.

Ruby Khondaker (she/her) (Jul 08 2025 at 12:46):

what about anaequivalent though?

Ruby Khondaker (she/her) (Jul 08 2025 at 12:47):

i may have parsed this wrong but it feels like every weak $\omega$-category should be anaequivalent to a strict one, almost trivially

Amar Hadzihasanovic (Jul 08 2025 at 12:47):

I'm not familiar with the term but I assumed it meant "admit a span of weak equivalences" and that is also not true.

Ruby Khondaker (she/her) (Jul 08 2025 at 12:47):

i mean in the sense of anafunctors?

Amar Hadzihasanovic (Jul 08 2025 at 12:48):

Anafunctors are spans in which the first leg is a particular kind of equivalence, no? An isofibration iirc?

Ruby Khondaker (she/her) (Jul 08 2025 at 12:48):

specifically i expect that any weak $\omega$-category $X$ should be anaequivalent to $TX$ where $T$ is the free strict omega-category monad

Ruby Khondaker (she/her) (Jul 08 2025 at 12:48):

hm, i don't actually know much about them beyond "determine values up to unique iso"

Ruby Khondaker (she/her) (Jul 08 2025 at 12:49):

Ruby Khondaker (she/her) said:

specifically i expect that any weak $\omega$-category $X$ should be anaequivalent to $TX$ where $T$ is the free strict omega-category monad

for this the map $TX \to X$ will be composition, and the map $X \to TX$ will be unit

Ruby Khondaker (she/her) (Jul 08 2025 at 12:49):

i don't immediately see why this fails..

Ruby Khondaker (she/her) (Jul 08 2025 at 12:53):

hm lemme read up on isofibrations

Ruby Khondaker (she/her) (Jul 08 2025 at 12:54):

ok nvm this would require me to know a lot more lol

Amar Hadzihasanovic (Jul 08 2025 at 12:55):

It's a subtle thing, but I think your intuition may be that $X$ should "embed" into $TX$, and in a certain sense this doesn't destroy any data in itself, but if you treat $TX$ as a strict $\omega$-category, that is, you "forget" that it comes from some weak thing, actually it kills certain higher homotopy invariants

Ruby Khondaker (she/her) (Jul 08 2025 at 12:56):

hmm...

Ruby Khondaker (she/her) (Jul 08 2025 at 12:57):

i guess what i'm thinking is

Ruby Khondaker (she/her) (Jul 08 2025 at 12:58):

you should be able to apply $T$ to the relation to get the relation on $TX$, so that $TX$ gets the structure of a $T$-multicategory, but another compatible relation that also works is just the multiplication map on the monad

Ruby Khondaker (she/her) (Jul 08 2025 at 12:58):

i guess i would have to check whether these are really the same or not :/

Amar Hadzihasanovic (Jul 08 2025 at 13:00):

The fact that weak and strict n-categories are inequivalent for n > 2 is a well-established but non-obvious fact. I suggest you do some research on it rather than try to figure it out on your own, as it deeply confused also some very experienced and brilliant mathematicians.

Amar Hadzihasanovic (Jul 08 2025 at 13:00):

[[Simpson conjecture]] may be a good starting point...

Ruby Khondaker (she/her) (Jul 08 2025 at 13:02):

ok so cartesian monads do preserve pullbacks at least

Ruby Khondaker (she/her) (Jul 08 2025 at 13:02):

and so monos

Ruby Khondaker (she/her) (Jul 08 2025 at 13:57):

I think I am starting to see what you mean

Ruby Khondaker (she/her) (Jul 08 2025 at 14:02):

Ok I still need to think about anaequivalence but I can see this is definitely harder than I thought, haha

Ruby Khondaker (she/her) (Jul 08 2025 at 14:11):

Ok I think I understand why not every weak omega-category should be expected to be equivalent to a strict one now!

Ruby Khondaker (she/her) (Jul 08 2025 at 14:22):

Actually I think I also have a pretty good idea of why this starts failing once you get to weak tricategories

Ruby Khondaker (she/her) (Jul 08 2025 at 14:23):

It's to do with the particular properties of 3-cell composition, right? Basically the sources and targets of 3-cells are 2-cells, and it's pretty easy to cook up examples where, with a pasting diagram of 3-cells, you compose the sources and targets to non-parallel 2-cells

Ruby Khondaker (she/her) (Jul 08 2025 at 14:23):

whereas this never happens in a strict 3-category

Ruby Khondaker (she/her) (Jul 08 2025 at 14:24):

if you have a pasting diagram of 3-cells in a strict 3-category, then you have only one way to paste together the source and the target, and they're guaranteed to be parallel

Ruby Khondaker (she/her) (Jul 08 2025 at 14:26):

This also never happens in a bicategory, because for a pasting diagram of 2-cells, even though you may have many ways to paste together the source and target 1-diagrams, they'll always produce parallel 1-cells

Ruby Khondaker (she/her) (Jul 08 2025 at 14:26):

thanks for correcting me!! :)

Ruby Khondaker (she/her) (Jul 08 2025 at 14:27):

Ruby Khondaker (she/her) said:

Does this all make sense so far?

anyway i will bump this @Madeleine Birchfield in case you wanted me to continue

Ruby Khondaker (she/her) (Jul 08 2025 at 14:42):

Also I think I see why "dimensionwise" free is now important too, probably related to this parallel issue

Amar Hadzihasanovic (Jul 08 2025 at 15:17):

The phenomenon you mention is indeed a new thing that happens in dimension 3 but is not the cause of the strict-weak mismatch, since you can in fact strictify associativity.
The cause of the mismatch comes from the interplay between strict units and strict associativity producing some equations that are not "topologically sound" starting from dimension 3.

Ruby Khondaker (she/her) (Jul 08 2025 at 15:17):

hm i see

Ruby Khondaker (she/her) (Jul 08 2025 at 15:21):

ok nice i think i figured out how you define lax $(\infty, \infty)$-functors with Leinster's definition

Ruby Khondaker (she/her) (Jul 08 2025 at 15:22):

to get higher transfors though i should probably try to understand how tensor products of $\omega$-categories work

Ruby Khondaker (she/her) (Jul 08 2025 at 15:23):

luckily i think it probably suffices to figure out how to tensor an $\omega$-category with an $n$-globe (viewed as a strict $\omega$-category)

Ruby Khondaker (she/her) (Jul 08 2025 at 15:26):

since then $n$-cells of $\text{Hom}(\mathcal{X}, \mathcal{Y})$ should correspond to (weak?) functors $G_n \to \text{Hom}(\mathcal{X}, \mathcal{Y})$, which should then be functors $G_n \otimes \mathcal{X} \to \mathcal{Y}$

Ruby Khondaker (she/her) (Jul 08 2025 at 15:27):

and i guess i could try and figure out composition by mapping pasting diagrams into this...

Ruby Khondaker (she/her) (Jul 08 2025 at 15:38):

actually what's the standard definition of lax $(\infty, \infty)$ functor? might be good to check i haven't got the details wrong... is there one for theta spaces?

Ruby Khondaker (she/her) (Jul 08 2025 at 16:05):

It is pretty interesting to see how Leinster's geometric approach is in many ways different to the usual geometric approaches to $\omega$-categories

Ruby Khondaker (she/her) (Jul 08 2025 at 16:07):

as far as i can tell, the main difference is that the data of "is h a composite of f and g" is no longer witnessed "internally" by the shapes themselves, but instead witnessed "externally"

Ruby Khondaker (she/her) (Jul 08 2025 at 16:17):

so as far as i can tell, the fundamental $\omega$-groupoid of a space is a bit of a different construction to that of the usual simplicial approach

Madeleine Birchfield (Jul 08 2025 at 16:19):

Ruby Khondaker (she/her) said:

so as far as i can tell, the fundamental $\omega$-groupoid of a space is a bit of a different construction to that of the usual simplicial approach

I think those are given by two different functors from Top to $\infty\mathrm{Grpd}$.

Ruby Khondaker (she/her) (Jul 08 2025 at 16:19):

yeah exactly, cause of the different shapes

Ruby Khondaker (she/her) (Jul 08 2025 at 16:19):

presumably they are equivalent though, otherwise that'd be pretty bad

Madeleine Birchfield (Jul 08 2025 at 16:21):

I think those are represented by two different modalities in dependent type theory iirc, one is given by the shape modality, which is localization at the topological interval, while the other is given by the flat modality.

Ruby Khondaker (she/her) (Jul 08 2025 at 16:21):

ok now you've lost me, haha

Mike Shulman (Jul 08 2025 at 16:28):

I don't have anything to add to the conversation about $\omega$-categories, but I wanted to comment on this:

Amar Hadzihasanovic said:

I would recommend that the workflow should be

1. Find an open problem that you want to solve,
2. See if it can be solved with existing models/techniques,
3. Only if 2. fails start thinking about new models/techniques

That's certainly one good workflow, and I agree that it's not generally a good idea to develop a formalism first and then look for applications. But particularly for category theory, I think it's not necessary to start with open problems, at least not if by that you mean the traditional sense of "conjectures that someone has made but no one has proven or disproven yet". For instance, I think some of the best category theory comes from observing a pattern or commonality between situations arising in different areas and trying to formulate an abstract structure to describe it, even if that structure doesn't immediately solve an open problem. The spirit is the same, namely to start with an "application" and work backwards to a new definition or theory if necessary, but the starting point doesn't have to be a "problem" in the traditional sense. Indeed, it might not even be a question that anyone else has ever thought of asking before, and one of the principal contributions could be the mere asking of it (at which point it might even almost answer itself).

Madeleine Birchfield (Jul 08 2025 at 16:29):

Ruby Khondaker (she/her) said:

ok now you've lost me, haha

See for instance,

• fundamental infinity-groupoid of a locally infinity-connected (infinity,1)-topos

• cohesive (infinity,1)-topos

Ruby Khondaker (she/her) (Jul 08 2025 at 16:30):

Oh, I think that’s exactly what I was doing when I was rediscovering Leinster’s geometric definition of $(\infty, \infty)$ category! I was trying to abstract the patterns I’d seen in higher category theory and find the underlying “shape” of what they really meant, just so that it felt conceptually clear in my head.

Ruby Khondaker (she/her) (Jul 08 2025 at 16:30):

And now things feel a lot cleaner and less ad-hoc than they used to

Mike Shulman (Jul 08 2025 at 16:31):

Madeleine Birchfield said:

I think those are represented by two different modalities in dependent type theory iirc, one is given by the shape modality, which is localization at the topological interval, while the other is given by the flat modality.

Hmm, just looking at the last few messages, but I kind of doubt that's what Ruby has in mind. It sounded to me like she was describing the "same" functor at $\infty$-level, just presented using different shapes.

Madeleine Birchfield (Jul 08 2025 at 16:31):

Mike Shulman said:

Hmm, just looking at the last few messages, but I kind of doubt that's what Ruby has in mind. It sounded to me like she was describing the "same" functor at $\infty$-level, just presented using different shapes.

Well if that's the case, then never mind.

But I've always taken shape or fundamental $\infty$-groupoid or fundamental $\omega$-groupoid to be the furthest left functor of the adjoint quadruple between a cohesive $\infty$-topos and its base $\infty$-topos

Ruby Khondaker (she/her) (Jul 08 2025 at 16:32):

Yeah I think that’s what I mean - the usual one uses simplicial shapes, Leinster’s one uses globular shapes

Ruby Khondaker (she/her) (Jul 08 2025 at 16:33):

There should be a nerve-realization adjunction between simplicial sets and globular sets I think

Ruby Khondaker (she/her) (Jul 08 2025 at 16:34):

Since you can regard the n-globe as a simplicial set in a decently obvious way

Ruby Khondaker (she/her) (Jul 08 2025 at 16:35):

I would assume this can be used to show an equivalence between the corresponding functors

Mike Shulman (Jul 08 2025 at 16:39):

I haven't followed everything you're doing, but I do want to warn you that globular sets are not as well-behaved as simplicial sets in general. I think this is mainly because the mere data of a globular set doesn't include any information about "composites", whereas that of a simplicial set does (e.g. a 2-simplex characterizes the composite of two 1-simplices up to homotopy). To get a good notion of higher category out of globular sets, you generally have to either equip them with algebraic structure, or pass to some more general shapes such as $\Theta$-sets.

Ruby Khondaker (she/her) (Jul 08 2025 at 16:39):

Ah but isn’t this precisely what Leinster’s geometric approach does? You just package the data of composition “externally” rather than have it be witnessed by the actual shapes?

Ruby Khondaker (she/her) (Jul 08 2025 at 16:40):

That way you don’t need algebraic structure and you don’t need more complex shapes!

Mike Shulman (Jul 08 2025 at 16:49):

The more complex shapes are still there in what you call "external" witnesses.

Mike Shulman (Jul 08 2025 at 16:51):

You can argue about what to call what; I was just saying that, for instance, there isn't likely to be a model category structure on globular sets that presents any kind of higher category. But maybe you just meant that an adjunction between simplicial sets and globular sets could be lifted to globular sets with "$L'$-structure" and then become an equivalence.

Ruby Khondaker (she/her) (Jul 08 2025 at 16:53):

Oh yeah for sure, I’m not strongly committed to any philosophical positions here. I guess if you want to paste together n-cells, having the pasting diagrams in some capacity feels unavoidable - I suppose it’s a choice of whether they’re “external” or “internal”.

Unfortunately I don’t know much about model categories :(

But by “present a higher category” - what kind of higher category would you have in mind? What would the objects and morphisms etc be

Mike Shulman (Jul 08 2025 at 16:58):

The "lowbrow" way of saying my point is just that you can't define an $\omega$-category, or even an $\infty$-groupoid, to be a globular set with some property.

Fernando Yamauti (Jul 08 2025 at 16:59):

John Baez said:

Sure, whenever you have time to read a 529-page book it's a good time to start!

Pursuing stacks deals with a lot of other problems. The Homotopy Hypothesis part is actually very short.

Ruby Khondaker (she/her) (Jul 08 2025 at 16:59):

Ah sure, but with the geometric approach I have in mind, you have a globular set + external witnesses, which I think should let you define an omega-category? I guess, I don’t quite understand the significance of why having external witnesses is bad…

Mike Shulman (Jul 08 2025 at 17:07):

I didn't say it was, just that you need them. (-:

Mike Shulman (Jul 08 2025 at 17:07):

Sorry, I should have stayed quiet, I think I just confused you to no purpose.

Madeleine Birchfield (Jul 08 2025 at 17:07):

Ruby Khondaker (she/her) said:

Ah sure, but with the geometric approach I have in mind, you have a globular set + external witnesses, which I think should let you define an omega-category? I guess, I don’t quite understand the significance of why having external witnesses is bad…

It's not that having external witnesses is bad, it's that the term "globular set" is usually used in the literature as not having the extra structure on a globular set.

Ruby Khondaker (she/her) (Jul 08 2025 at 17:09):

Oh no no it’s fine, I definitely want to hear perspectives from lots of people! Especially since I know very little compared to everyone else here

Hmm, I understand how the terminology “globular set” would be confusing if you have something additional on it. What should I call these, then?

Fernando Yamauti (Jul 08 2025 at 17:10):

Sorry to disturb the flow here. Just something that's annoying me: what everyone here means by geometric and algebraic model?

I can't find now, but at some point someone said that algebraic models are not known to satisfy the Homotopy Hypothesis. But algebraic Kan complexes certainly do so and, if I remember correctly, they are also given by an algebraic theory (the models form an $\omega$-accessible loc pres cat). So algebraic in this thread has to mean something very different..

Madeleine Birchfield (Jul 08 2025 at 17:11):

I think we were talking about geometric and algebraic models of $n$-categories and $(\infty,n)$-categories.

Ruby Khondaker (she/her) (Jul 08 2025 at 17:11):

Ah apologies for the confusion, I’m borrowing terminology from Leinster’s Higher Operads, Higher Categories

Mike Shulman (Jul 08 2025 at 17:13):

Leinster also has a nice blog post that discusses, among other things, the "spectrum" from "algebraic" to "geometric", with algebraic Kan complexes being somewhere on the spectrum but not all the way on the algebraic end.

Ruby Khondaker (she/her) (Jul 08 2025 at 17:14):

image_4B92B224-A384-4844-9BD1-7E957AAECED1_1751994820.jpeg

This is the relevant section I think

Ruby Khondaker (she/her) (Jul 08 2025 at 17:15):

I’ll take a look at that blog post!

Amar Hadzihasanovic (Jul 08 2025 at 17:28):

@Fernando Yamauti I see the algebraic/geometric distinction as a "loose" non-technical one, and my heuristic is that only operations that one may, conceivably, want to be preserved strictly by functors should count towards making an 'algebraic' definition

Amar Hadzihasanovic (Jul 08 2025 at 17:29):

From the point of view of this heuristic, I consider algebraic Kan complexes not at all more "algebraic" than general Kan complexes, because I cannot imagine any non-contrived example of a functor preserving all Kan fillers strictly.

Fernando Yamauti (Jul 08 2025 at 17:45):

Amar Hadzihasanovic said:

From the point of view of this heuristic, I consider algebraic Kan complexes not at all more "algebraic" than general Kan complexes, because I cannot imagine any non-contrived example of a functor preserving all Kan fillers strictly.

Hmm... but from this point of view, by the same reasoning, wouldn't Grothendieck's definition also be non-algebraic?

Ruby Khondaker (she/her) (Jul 08 2025 at 18:25):

Thanks for the blog post! I’m not entirely sure where $\omega$ categories in the sense I describe would fit

Ruby Khondaker (she/her) (Jul 08 2025 at 18:26):

As you point out, if I just use the globe category then they can’t be “presheaves with properties”

Ruby Khondaker (she/her) (Jul 08 2025 at 18:26):

But on the other hand they’re not algebras for an operad or a monad, so they’re not “presheaves with structure” in the algebraic sense

Mike Shulman (Jul 08 2025 at 18:29):

Amar Hadzihasanovic said:

From the point of view of this heuristic, I consider algebraic Kan complexes not at all more "algebraic" than general Kan complexes, because I cannot imagine any non-contrived example of a functor preserving all Kan fillers strictly.

If we define Kan fillers in the singular complex of a topological space by fixing particular retractions $|\Delta^n| \to |\Lambda^n_k|$ once and for all, then wouldn't those fillers be preserved strictly by the map of algebraic Kan complexes induced by any continuous map of topological spaces?

Ruby Khondaker (she/her) (Jul 08 2025 at 18:31):

I also wonder what $\omega$-categories internal to $\omega$ categories would be, haha

Mike Shulman (Jul 08 2025 at 18:31):

Similarly, if we have any fully algebraic notion of higher groupoid and we take its simplicial nerve, then we would construct Kan fillers by choosing some way of composing up a horn in terms of the algebraic operations. Thus, any strict algebraic functor that preserves those operations would also induce a strict Kan-filler-preserving map.

Ruby Khondaker (she/her) (Jul 08 2025 at 18:31):

$\omega$-fold categories…?

Ruby Khondaker (she/her) (Jul 08 2025 at 18:42):

alright, time to learn about the Crans-Gray tensor product!

Amar Hadzihasanovic (Jul 08 2025 at 19:09):

Mike Shulman said:

Similarly, if we have any fully algebraic notion of higher groupoid and we take its simplicial nerve, then we would construct Kan fillers by choosing some way of composing up a horn in terms of the algebraic operations. Thus, any strict algebraic functor that preserves those operations would also induce a strict Kan-filler-preserving map.

I see your point. I suppose my intuition from low-dimensional category theory is that in most examples, the existence of “natural” choices of units and composites is much more common that the existence of natural choices of weak inverses to equivalences, and that this is an indication that algebraicising inversion is somewhat less natural than algebraicising composition and units; and of course to have canonical algebraic Kan fillers in nerves of higher groupoids one needs to have algebraic inversion in addition to composition and units.

Amar Hadzihasanovic (Jul 08 2025 at 19:11):

But I cannot exclude that this may be a bias coming from the fact that, for example, the most familiar examples of bicategories are Cat-like (and tricategories are 2Cat-like); maybe other families of examples have canonical inverses but no canonical composites.

Amar Hadzihasanovic (Jul 08 2025 at 19:36):

Relatedly, I have never seen “strictification of inversion” discussed as a possible path to (semi-)strictification, not even e.g. for 2-groupoids.
Although now I'd be curious to know whether a weak 2-groupoid is equivalent to one that is weak in everything except it has algebraic inverses of 1-cells satisfying $f f^{-1} = \mathrm{id}$ and $f^{-1} f = \mathrm{id}$ strictly.

John Baez (Jul 08 2025 at 19:39):

Yes it is, but we can strictify a lot more than that for a 2-groupoid.

John Baez (Jul 08 2025 at 19:47):

I'm not finding a proof for 2-groupoids, but in my paper with Aaron Lauda on 2-groups in Proposition 39 we show that any 2-group is equivalent to a 'special' one, meaning a skeletal one where the left unitor law ℓ, the right unitor r, the unit $i \Rightarrow f f^{-1}$ and the counit $e : f^{-1} f \Rightarrow 1$ are identity natural transformations. (There's more to explain about Prop. 39, but what I'm saying is true.)

John Baez (Jul 08 2025 at 19:50):

In Proposition 45 we show any 2-group is equivalent to a 'strict' one, meaning one where the associator $a$, the left unitor law ℓ, the right unitor r, the unit $i \Rightarrow f f^{-1}$ and the counit $e : f^{-1} f \Rightarrow 1$ are identity natural transformations.

John Baez (Jul 08 2025 at 19:52):

So you can strictify everything!

And if you want your 2-group to be skeletal, you can strictify everything except associativity.

I'm pretty sure analogous results are true for 2-groupoids.

Amar Hadzihasanovic (Jul 08 2025 at 19:54):

Should your statement of Proposition 45 omit "skeletal" then?

Amar Hadzihasanovic (Jul 08 2025 at 19:55):

Thanks by the way, I somehow never learnt this result!

John Baez (Jul 08 2025 at 20:11):

Amar Hadzihasanovic said:

Should your statement of Proposition 45 omit "skeletal" then?

Yes, sorry - fixed. I was doing too much cut-and-pasting.

Ruby Khondaker (she/her) (Jul 08 2025 at 20:27):

does anyone have any advice for how i'd realise the standard n-globe as a cubical set?

Ruby Khondaker (she/her) (Jul 08 2025 at 20:28):

hm, i wonder if a kind of "gluing together of cubes" might suffice, actually..

Ruby Khondaker (she/her) (Jul 08 2025 at 20:30):

so i guess, displaying the n-globe as a "cubical complex"...?

Ruby Khondaker (she/her) (Jul 08 2025 at 22:00):

I think I have an idea for inductively realising the n-globe as a cubical set!

Ruby Khondaker (she/her) (Jul 08 2025 at 22:00):

0-globe is easy

Ruby Khondaker (she/her) (Jul 08 2025 at 22:00):

Then suppose you’ve already realised the k-globe as a cubical set

Ruby Khondaker (she/her) (Jul 08 2025 at 22:01):

Take the day convolution of that with the 1-cube (as a cubical set) to get a “cylinder space” of the k-globe

Ruby Khondaker (she/her) (Jul 08 2025 at 22:02):

Then you can include the boundary of the k-globe into this in 2 ways - either as the (boundary of) the “bottom face” or the “top face”

Ruby Khondaker (she/her) (Jul 08 2025 at 22:04):

Coequalizing these maps should “collapse” the edges of the cylinder (but leave the interior intact), giving you the (k + 1)-globe!

Ruby Khondaker (she/her) (Jul 08 2025 at 22:07):

This is great because now I should be able to set up a nerve-realization adjunction - realising globular sets as cubical sets, and taking the “globular nerve” of a cubical set

Ruby Khondaker (she/her) (Jul 08 2025 at 22:09):

Now I at least have some handle on how to take the tensor product of $(\infty, \infty)$ categories - get the underlying globular sets, realise them as cubical sets, day-convolve, and then take the globular nerve!

Ruby Khondaker (she/her) (Jul 08 2025 at 22:10):

Ofc I need to check how well this plays with the $\omega$-categorical structure

Ruby Khondaker (she/her) (Jul 09 2025 at 09:17):

Are there any “shape-agnostic” approaches to higher category theory?

Amar Hadzihasanovic (Jul 09 2025 at 09:35):

What do you mean by shape-agnostic?

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

Shapes are dual to the algebra of face operations that you have on your cells, and you must have some face operators to determine the source and target of a cell.

Fernando Yamauti (Jul 09 2025 at 09:45):

Perhaps test categories might be relevant. Though that's $(\infty, 0)$ only. I think in the past I saw an attempt to generalise that to an $(\infty, 1)$-version, but I don't remember where.

But, anyways, in any similar approach, we would always need to fix one model, to be considered the standard one (in the above case, Cat with the Thomason's model structure).

Ruby Khondaker (she/her) (Jul 09 2025 at 09:45):

Like maybe some kind of synthetic approach that allows for a "category of shapes" $\mathcal{S}$ with nice enough properties, that doesn't need you to explicitly know the combinatorics of $\mathcal{S}$

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

Ooh ok test categories seem pretty relevant

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

is there a conceptual reason why they're limited to the $(\infty, 0)$ case?

Amar Hadzihasanovic (Jul 09 2025 at 09:51):

Well, basically the comparison with standard model structures for homotopy types is obtained by taking "simplicial subdivisions" of objects of the test category, and the ability to subdivide cells at will is something you only have in the $(\infty, 0)$-case.

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

Ah, this is where my lack of homotopy theory knowledge limits me :sweat_smile:

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

Adding some extra data to shapes may still allow for a similar approach in the $(\infty, 1)$-case, and I have also heard about the attempt that Fernando mentions, although I do not think anything has been published yet.
I believe that for $n > 1$ something completely different will be necessary.

John Baez (Jul 09 2025 at 10:02):

Ruby Khondaker (she/her) said:

Are there any “shape-agnostic” approaches to higher category theory?

Yes:

• Carlos Simpson, Some properties of the theory of $n$-categories.

It might be nice to try to develop "synthetic" weak $n$-category or $\omega$-category theory based on some list of axioms.

Ruby Khondaker (she/her) (Jul 09 2025 at 10:03):

Yes, that’s exactly the kind of thing I’d want!!

John Baez (Jul 09 2025 at 10:03):

Yes, Simpson was trying to work in that direction.

Ruby Khondaker (she/her) (Jul 09 2025 at 10:04):

Do you know what kinds of issues they ran into?

John Baez (Jul 09 2025 at 10:06):

Read the paper!

Ruby Khondaker (she/her) (Jul 09 2025 at 10:07):

ok!

Fernando Yamauti (Jul 09 2025 at 10:14):

John Baez said:

Yes:

• Carlos Simpson, Some properties of the theory of $n$-categories.

It might be nice to try to develop "synthetic" weak $n$-category or $\omega$-category theory based on some list of axioms.

I'm not familiar with that paper, but he seems to be choosing a model of $(\infty, 1)$ to state everything (in his case, Segal cats).

John Baez (Jul 09 2025 at 12:27):

Simpson is not choosing shapes of cells to define "$n$-category". He is simply stating expected properties of the $(\infty,1)$-category of $n$-categories. He's choosing Segal categories as his way of talking about $(\infty,1)$-categories, but the equivalence between Segal categories is various other concepts of $(\infty,1)$-category are sufficiently well-understood that this is quite benign.

In other words, there's a metalanguage thing going on here. He's talking about $n$-categories, not making any choice of shapes of cells for those - but he's talking in the metalanguage of $(\infty,1)$-categories, and that's where he's making the specific choice to work with Segal categories.

Tom de Jong (Jul 09 2025 at 12:43):

Ruby Khondaker (she/her) said:

Are there any “shape-agnostic” approaches to higher category theory?

I'm not really an expert and maybe I've misunderstood your question, but Elements of ∞-Category Theory by Riehl and Verity and Formalization of Higher Categories (in progress) by Cisinski, Cnossen, Nguyen and Walde come to mind.

Ruby Khondaker (she/her) (Jul 09 2025 at 12:48):

Tom de Jong said:

Ruby Khondaker (she/her) said:

Are there any “shape-agnostic” approaches to higher category theory?

I'm not really an expert and maybe I've misunderstood your question, but Elements of ∞-Category Theory by Riehl and Verity and Formalization of Higher Categories (in progress) by Cisinski, Cnossen, Nguyen and Walde come to mind.

ah dw, i'm certainly not an expert either :P

I haven't read Elements of $\infty$-category theory - seems like it'd be a very good starting point for me! Especially since I really seem to be lacking in knowledge of infinite-dimensional category theory :sweat_smile:

John Baez (Jul 09 2025 at 13:16):

As far as I can tell, Elements of ∞-Category Theory by Riehl and Verity gives a nice shape-agnostic treatment of $\infty$-cosmoi, which are $(\infty,2)$-categories that resemble toposes - in the metalanguage of $(\infty,1)$-category theory, where they seem to work within Lurie and Joyal's framework (also known as quasicategories).

By the way, I'm linking to a legal, free copy of their book, while Tom de Jong linked to the publisher's website.

John Baez (Jul 09 2025 at 13:17):

In case this is confusing (and it is), an object of an $\infty$-cosmos is "a thing that resembles an $(\infty,1)$-category", in just the same way that an object of a topos is "a thing that resembles a set".

John Baez (Jul 09 2025 at 13:19):

They have a footnote about the overall challenge of avoiding circularit and choosing a metalanguage in which to do do shape-agnostic higher category theory:

A less rigorous “model independent” presentation of ∞-category theory might confront a problem of infinite regress, since infinite-dimensional categories are themselves the objects of an ambient infinite-dimensional category, and in developing the theory of the former one is tempted to use the theory of the latter. We avoid this problem by using a very concrete model for the ambient (∞, 2)-category of ∞-categories that arises frequently in practice and is designed to facilitate relatively simple proofs. While the theory of (∞, 2)-categories remains in its infancy, we are content to cut the Gordian knot in this way.

Ruby Khondaker (she/her) (Jul 09 2025 at 13:25):

i guess you wouldn't run into this issue if you had a presentation of $(\infty, \infty)$ categories

Ruby Khondaker (she/her) (Jul 09 2025 at 15:06):

Ooh I've been looking into EZ categories and it seems like they capture many of the notions you'd want a "shape" category to have?

Ruby Khondaker (she/her) (Jul 09 2025 at 15:38):

I've also been looking into both complicial sets and comical sets as models for weak $(\infty, \infty)$ categories

Ruby Khondaker (she/her) (Jul 09 2025 at 15:39):

as far as i can tell, since we have gray tensor products on both, should these not let you define the weak $(\infty, \infty)$ category of weak $(\infty, \infty)$ categories in either model?

Ruby Khondaker (she/her) (Jul 09 2025 at 15:40):

for some reason i thought there wasn't yet such a theory, maybe i was wrong

Clémence Chanavat (Jul 09 2025 at 15:49):

How would you define $Cat$, the weak $(\infty, \infty)$-category of weak $(\infty, \infty)$-categories given the Gray tensor product? How I see it: given two $(\infty, \infty)$-categories $X, Y$, it is easy to define the hom $(\infty, \infty)$-category $Cat(X, Y)$ just by tensor-hom adjunction using the Gray product, but then it is less clear to how to glue those hom $(\infty, \infty)$ category into a weak $(\infty, \infty)$-category structure where 0-cells are $(\infty, \infty)$-categories

Ruby Khondaker (she/her) (Jul 09 2025 at 15:51):

I guess the “obvious” thing I would try is - 0-cells are omega categories, 1-cells are functors, 2-cells are natural transformations, and so on?

Ruby Khondaker (she/her) (Jul 09 2025 at 15:51):

Since you have an internal hom, you should have a notion of k-transfor for all k surely

Ruby Khondaker (she/her) (Jul 09 2025 at 15:52):

I’m probably missing something

Clémence Chanavat (Jul 09 2025 at 15:53):

In the $(\infty, 1)$-case, one solution is to pass via the homotopy coherent nerve, since all the hom $(\infty, 1)$ categories gather into a simplicially enriched category. Another approach is to define $Cat$ to have cells of shape $U$ some kind of fibrations over the representable cell $U$. I believe this is what Loubaton does here for the $(\infty, \infty)$-category of $(\infty, \infty)$ categories

Clémence Chanavat (Jul 09 2025 at 15:55):

Ruby Khondaker (she/her) said:

I guess the “obvious” thing I would try is - 0-cells are omega categories, 1-cells are functors, 2-cells are natural transformations, and so on?

I mean yes that what you expect, but actually constructing it is another story

Clémence Chanavat (Jul 09 2025 at 15:57):

The fibrational approach has also been done in the $(\infty, 1)$-case by Cisinski and Nguyen here. One advantage is that you essentially get the Grothendieck construction for free

Ruby Khondaker (she/her) (Jul 09 2025 at 15:58):

Ok sweet

Ruby Khondaker (she/her) (Jul 09 2025 at 15:59):

I mean I started this thread to ask about how one’s defines $(\infty, \infty)$ category theory

Ruby Khondaker (she/her) (Jul 09 2025 at 15:59):

I guess I have my answer?

Madeleine Birchfield (Jul 09 2025 at 15:59):

Tom de Jong said:

Ruby Khondaker (she/her) said:

Are there any “shape-agnostic” approaches to higher category theory?

I'm not really an expert and maybe I've misunderstood your question, but Elements of ∞-Category Theory by Riehl and Verity and Formalization of Higher Categories (in progress) by Cisinski, Cnossen, Nguyen and Walde come to mind.

Those only talk about $(\infty,1)$-categories iirc, while @Ruby Khondaker (she/her) seems to be interested in $(\infty,n)$-categories and $(\infty,\infty)$-categories as well.

Ruby Khondaker (she/her) (Jul 09 2025 at 16:08):

Mhm! My idea was, if I can understand $(\infty, \infty)$ categories, then all that’s left is to understand how to truncate appropriately

Ruby Khondaker (she/her) (Jul 09 2025 at 16:35):

Now I can finally go back to learning algtop

Mike Shulman (Jul 09 2025 at 16:39):

One might argue that a different meaning of "shape-agnostic" would be to work with a category of shapes that includes all possible shapes at once. This was one of the original motivations for higher computads. Unfortunately $n$-computads aren't a presheaf category when $n>2$, but there are various refinements of them like positive computads or weak computads that are. It seems that diagrammatic sets are based on a similar intuition, but I don't remember (if I ever knew) how exactly they are related to computads.

Ruby Khondaker (she/her) (Jul 09 2025 at 16:39):

oh i think amar was talking about them over in #learning: questions > Cubical-to-Globular translation

Amar Hadzihasanovic (Jul 09 2025 at 16:52):

The relation is quite subtle. Up to dim 3 one can fairly think of diagrammatic sets as "computads with weak units" but in higher dimensions they are also stricter in some ways than computads for strict n-categories (while still having "weak units").

Amar Hadzihasanovic (Jul 09 2025 at 16:53):

But certainly I think that both computads and diagrammatic sets are aiming to capture the same intuition, which I'd say is "directed CW-complexes".

Ruby Khondaker (she/her) (Jul 09 2025 at 22:54):

Ok so I've been reading up on model category structures, think I have a better grasp for them now

Ruby Khondaker (she/her) (Jul 09 2025 at 22:54):

However, all the references i can tell say let you present an $(\infty, 1)$-category

Ruby Khondaker (she/her) (Jul 09 2025 at 22:55):

so in particular, if you have a notion of $\omega$-category, then a model structure on this would give you the $(\infty, 1)$-category of such $\omega$-categories?

Ruby Khondaker (she/her) (Jul 09 2025 at 22:55):

is there a way to get the $\omega$-category of such $\omega$-categories from this though

Ruby Khondaker (she/her) (Jul 09 2025 at 22:58):

Amar Hadzihasanovic said:

One reason that geometric definitions are more used is that often they fit into model categories where all objects are cofibrant, whereas typically in algebraic definitions the cofibrants should be the free (computad-like) objects.
It just so happens that a lot of model category theory has been developed in the special case of “all objects are cofibrant” because, well, geometric definitions are very natural for the homotopical/homological purposes that model category theory was created for :)

i'm also trying to understand this remark from Amar - is there a reason why cofibrant objects in "algebraic" theories have to be free? and more generally, if you have a "presheaf with structure" definition of $\omega$-category, are you going to fail to make all objects cofibrant?

Mike Shulman (Jul 09 2025 at 23:45):

Ruby Khondaker (she/her) said:

so in particular, if you have a notion of $\omega$-category, then a model structure on this would give you the $(\infty, 1)$-category of such $\omega$-categories?

Yes.

Mike Shulman (Jul 09 2025 at 23:49):

Ruby Khondaker (she/her) said:

is there a way to get the $\omega$-category of such $\omega$-categories from this though

Not as easy a one. If a model category $C$ is suitably enriched over some other model category $V$ which presents some $(\infty,1)$-category $hV$, then its presented $(\infty,1)$-category $hC$ will be enriched over $hV$. So if you can pick an appropriate $V$ such that an $hV$-enriched $(\infty,1)$-category can be viewed as an $\omega$-category, you can use a $V$-enriched model category $C$ to present an $\omega$-category.

Mike Shulman (Jul 09 2025 at 23:52):

Ruby Khondaker (she/her) said:

is there a reason why cofibrant objects in "algebraic" theories have to be free? and more generally, if you have a "presheaf with structure" definition of $\omega$-category, are you going to fail to make all objects cofibrant?

In a 1-category whose objects are equipped with algebraic structure, the morphisms generally preserve that structure strictly. But the morphisms in the presented $(\infty,1)$-category should generally only preserve the algebraic structure up to equivalence. A cofibrant object is one that's sufficiently "free" so that every "weak morphism" out of it is equivalent to a strict one, so that the strict morphisms in the 1-category with cofibrant domain suffice to represent all weak morphisms. For instance, if a 2-category $C$ has an underlying 1-category that is freely generated by a graph, then every pseudofunctor $C\to D$ is equivalent to a strict 2-functor.

Ruby Khondaker (she/her) (Jul 09 2025 at 23:54):

is it fair to say that a functor from C -> D which is “full, faithful and essentially surjective” is a “weak homotopy equivalence” of categories?

Ruby Khondaker (she/her) (Jul 09 2025 at 23:54):

and then you get a whitehead theorem saying that these are the same as “strong homotopy equivalences”

Mike Shulman (Jul 09 2025 at 23:56):

Yeah, that's reasonable to say. Although in the case of categories I would usually just say "weak equivalence" and "strong equivalence".

Mike Shulman (Jul 09 2025 at 23:56):

(Note that the whitehead theorem requires the axiom of choice.)

Ruby Khondaker (she/her) (Jul 09 2025 at 23:57):

Right right, same with the theorem in Cat!

Ruby Khondaker (she/her) (Jul 09 2025 at 23:57):

Essentially what I’m trying to see is a “trivial” case of model categories

Ruby Khondaker (she/her) (Jul 09 2025 at 23:58):

If you take the 1-category whose objects are small categories and whose morphisms are functors

Ruby Khondaker (she/her) (Jul 09 2025 at 23:58):

Then you can define a notion of “weak equivalences” here, by functors that are full, faithful and essentially surjective on objects

Ruby Khondaker (she/her) (Jul 09 2025 at 23:58):

So in principle you could try to get a model structure and find fibrant/cofibrant objects etc

Ruby Khondaker (she/her) (Jul 09 2025 at 23:59):

But actually, assuming choice, you get a whitehead theorem - every “weak equivalence” is a “strong equivalence”

Mike Shulman (Jul 09 2025 at 23:59):

Yep, the [[canonical model structure on Cat]]. Although in this case it's sufficiently trivial (assuming AC) that all objects are fibrant and cofibrant.

Ruby Khondaker (she/her) (Jul 09 2025 at 23:59):

Right, yeah!

Ruby Khondaker (she/her) (Jul 10 2025 at 00:00):

So I guess - if you had some kind of notion of weak omega-category, and you could prove an analog of the whitehead theorem

Ruby Khondaker (she/her) (Jul 10 2025 at 00:00):

Then you wouldn’t actually need model categories for much? Since weak equivalences are the same as strong ones anyway

Ruby Khondaker (she/her) (Jul 10 2025 at 00:02):

Like, for quasicategories, a “weak equivalence” between quasicategories is the same as a strong one, right?

Ruby Khondaker (she/her) (Jul 10 2025 at 00:02):

With choice assumed

Mike Shulman (Jul 10 2025 at 00:26):

For dimensions >1, it's generally not the case that all objects are fibrant and cofibrant any more, and whitehead theorems only hold for morphisms between objects that are both fibrant and cofibrant. For a geometric definition like quasicategories, generally all objects are cofibrant but not all objects are fibrant: in the model structure on simplicial sets for quasicategories, the fibrant objects are quasicategories, and so the whitehead theorem holds for all maps between quasicategories. For an algebraic definition, generally all objects are fibrant but not all objects are cofibrant, and the whitehead theorem only holds for maps between cofibrant objects. For instance, as I mentioned, the cofibrant 2-categories are those whose underlying 1-category is free, and only between such 2-categories does the whitehead theorem hold.

Ruby Khondaker (she/her) (Jul 10 2025 at 00:27):

hm so in general, for a "good" notion of weak $\omega$-category, should the whitehead theorem hold for maps between such weak $\omega$-categories?

Mike Shulman (Jul 10 2025 at 00:28):

Only for a nonalgebraic definition.

Ruby Khondaker (she/her) (Jul 10 2025 at 00:29):

awesome!

Ruby Khondaker (she/her) (Jul 10 2025 at 00:29):

hm, so maybe what i could try to do as a little experiment is

Ruby Khondaker (she/her) (Jul 10 2025 at 00:29):

take $\mathbf{L}'$, try to define a notion of "weak functor" and "weak equivalence", and see if the whitehead theorem holds

Ruby Khondaker (she/her) (Jul 10 2025 at 00:30):

if it does, then that notion of "weak functor" and "weak equivalence" passes at least one test for being a good notion of $\omega$-category

Amar Hadzihasanovic (Jul 10 2025 at 04:28):

Some work in this direction has been done, this is for strict functors that are weak equivalences

Mike Shulman (Jul 10 2025 at 04:33):

Isn't that for fully algebraic Batanin-Leinster $\omega$-categories?

Amar Hadzihasanovic (Jul 10 2025 at 04:38):

Oh, I'm not so familiar with this model, what is the difference?

Amar Hadzihasanovic (Jul 10 2025 at 04:40):

I kind of assumed L' was Leinster's name for what others call Leinster $\omega$-categories :)

Mike Shulman (Jul 10 2025 at 04:40):

No, that's L. (-:

Mike Shulman (Jul 10 2025 at 04:42):

If $T$ is the monad such that Leinster $\omega$-categories are $T$-algebras, then an $L'$-category is a $T$-multicategory whose source map is contractible.

Amar Hadzihasanovic (Jul 10 2025 at 04:55):

In that case, yes, the paper is about L and not L'.

Ruby Khondaker (she/her) (Jul 10 2025 at 06:33):

Yeah essentially $L’$ feels conceptually cleanest to me, so I’m on a “quest” to see if I can show that it really does give a good notion of $\omega$-category

Ruby Khondaker (she/her) (Jul 10 2025 at 06:35):

And hopefully I can learn some higher category theory along the way!

Amar Hadzihasanovic (Jul 10 2025 at 06:54):

Yeah that sounds good! But don't make the mistake of attributing an absolute value to "conceptual cleanliness" beyond your own perception, as I think that is an instance of the monad tutorial fallacy.

Amar Hadzihasanovic (Jul 10 2025 at 06:56):

Especially in a subject that has so many facets (algebraic, homotopy-theoretic, logical, computational...) it is unlikely that any single model will appear "conceptually transparent" from all angles.

Ruby Khondaker (she/her) (Jul 10 2025 at 07:08):

Oh of course of course, I’m not trying to say something like “hey everybody stop what you’re doing and use L’”, this is more just a hobby project for me :P

Ruby Khondaker (she/her) (Jul 10 2025 at 07:13):

Do you have favourites for each type, by the way? In terms of a model that’s best algebraically, one that’s best homotopy-theoretically, one that’s best logically, one that’s best computationally…?

Amar Hadzihasanovic (Jul 10 2025 at 07:18):

No, I don't think I can put a "thinking cap" on for any kind of mathematician that I am not.

Amar Hadzihasanovic (Jul 10 2025 at 07:21):

I have been working on models that are natural from the perspective of combinatorial and computational topology which is what fits my modes of intuitive reasoning. I cannot tell what is natural from other perspectives because I cannot access other minds.

Kevin Carlson (Jul 10 2025 at 07:22):

Is “monad tutorial fallacy” an established phrase?

Ruby Khondaker (she/her) (Jul 10 2025 at 07:23):

Yes I think so? I saw it here.

Amar Hadzihasanovic (Jul 10 2025 at 07:24):

Kevin Carlson said:

Is “monad tutorial fallacy” an established phrase?

Sort-of :) we discussed it earlier in this same thread. See Ruby's link and this post by John Baez that discusses it.

Ruby Khondaker (she/her) (Jul 10 2025 at 07:24):

Also I think I’m even less likely to be able to figure out what’s natural for other mathematicians, since I’m a physicist!

Ruby Khondaker (she/her) (Jul 10 2025 at 07:25):

In a sense L’ feels natural to me from a physics standpoint

Ruby Khondaker (she/her) (Jul 10 2025 at 08:20):

Ok I definitely need to work through the details of this but I can’t see an obvious reason why the “whitehead theorem” should fail for $\mathbf{L}’$

Ruby Khondaker (she/her) (Jul 10 2025 at 08:21):

Since the space of composites of any pasting diagram is contractible, so “unique enough”, and so you should probably just be able to copy the proof that full, faithful + essentially surjective => equivalence for Cat?

Ruby Khondaker (she/her) (Jul 10 2025 at 08:22):

Of course you need a suitable notion of weak functor between L’ categories but since it’s a geometric definition this shouldn’t be bad

Ruby Khondaker (she/her) (Jul 10 2025 at 08:23):

And the inclusions of the boundaries of the standard n-globe into the n-globe should form a class of generating cofibrations

Ruby Khondaker (she/her) (Jul 10 2025 at 08:24):

In fact you probably just literally have an interval object by tensoring with the 1-globe

Amar Hadzihasanovic (Jul 10 2025 at 08:49):

“Tensoring” is definitely something that has not been constructed for this class of models.

Ruby Khondaker (she/her) (Jul 10 2025 at 08:50):

Amar Hadzihasanovic said:

“Tensoring” is definitely something that has not constructed for this class of models.

oh, why not? what about those papers on complicial or comical sets that have a tensor product?

Amar Hadzihasanovic (Jul 10 2025 at 08:51):

...they are different models?

Ruby Khondaker (she/her) (Jul 10 2025 at 08:51):

hm i don't quite understand the issue :(

Amar Hadzihasanovic (Jul 10 2025 at 08:53):

There is no comparison between L' and the complicial and comical models of higher categories. Nobody has proven that they present the same theory.

Ruby Khondaker (she/her) (Jul 10 2025 at 08:53):

ohhh right ok

Ruby Khondaker (she/her) (Jul 10 2025 at 08:53):

so what you're saying is that i'd need to figure out tensoring for L' for this to work

Amar Hadzihasanovic (Jul 10 2025 at 08:54):

And anyway definitions of the Gray product in different models can be wildly different, and then one also needs to prove that they present the same model-independent tensor product.

Ruby Khondaker (she/her) (Jul 10 2025 at 08:54):

mhm mhm

Ruby Khondaker (she/her) (Jul 10 2025 at 08:54):

maybe if i had something like L' but for cubical shapes instead, and then used the natural tensor product there, things could work

Ruby Khondaker (she/her) (Jul 10 2025 at 08:54):

but i'd have to see

Amar Hadzihasanovic (Jul 10 2025 at 08:56):

The Gray product is a bit of a technical nightmare for strict $\omega$-categories and one can expect that for other algebraic models based on “weakenings” of the monad for strict $\omega$-categories the situation is only worse.

Ruby Khondaker (she/her) (Jul 10 2025 at 08:56):

oh i didn't realise it was a nightmare

Ruby Khondaker (she/her) (Jul 10 2025 at 08:57):

i wonder how bad this mixed algebro-geometric approach would fare

Ruby Khondaker (she/her) (Jul 10 2025 at 08:58):

hm does the "free strict cubical omega-category" monad commute with tensor products of cubical sets?

Amar Hadzihasanovic (Jul 10 2025 at 08:59):

Personally, having been interested in Gray products for many years, I have eventually taken this to be evidence against these “globular” combinatorics as the right foundation for algebraic higher categories.

Ruby Khondaker (she/her) (Jul 10 2025 at 08:59):

mhm mhm, hence why i've been looking more into cubical stuff recently

Amar Hadzihasanovic (Jul 10 2025 at 09:00):

Well it depends on how cubical is used, cubical can be “globular in disguise”.

Ruby Khondaker (she/her) (Jul 10 2025 at 09:00):

let's say "actually cubical"

Ruby Khondaker (she/her) (Jul 10 2025 at 09:01):

basically i want to see why the following doesn't work

John Baez (Jul 10 2025 at 09:01):

Does the "nightmare" come from the combinatorics of taking a product of globes and having to figure out how to express it in terms of globes? For simplices that process gives the combinatorics connected to standard proofs of the Eilenberg-Zilber theorem, which is one of those theorems in homology theory that's actually tiresome to prove.

Ruby Khondaker (she/her) (Jul 10 2025 at 09:01):

image.png
here X and Y are cubical sets, and T is the "free strict cubical omega-category" monad

Ruby Khondaker (she/her) (Jul 10 2025 at 09:02):

but of course you need a relation between $T(X \otimes Y)$ and $TX \otimes TY$ for this

Ruby Khondaker (she/her) (Jul 10 2025 at 09:03):

maybe this is that "monoidal monad" phrase i've seen used in the literature

Amar Hadzihasanovic (Jul 10 2025 at 09:05):

John Baez said:

Does the "nightmare" come from the combinatorics of taking a product of globes and having to figure out how to express it in terms of globes? For simplices that process gives the combinatorics connected to standard proofs of the Eilenberg-Zilber theorem, which is one of those theorems in homology theory that's actually tiresome to prove.

Describing the products is easy enough, what is harder is having a concrete model of the full subcategory of $\omega\mathrm{Cat}$ generated by these.

Ruby Khondaker (she/her) (Jul 10 2025 at 09:11):

Ruby Khondaker (she/her) said:

maybe this is that "monoidal monad" phrase i've seen used in the literature

Ah so it seems like, to make this work, i need only construct compatible left and right strengths

Amar Hadzihasanovic (Jul 10 2025 at 09:12):

Anyway, Ruby, in everyone's experience, comparison of models is one of the hardest problems in higher category theory, so you cannot keep shifting the goalposts; you say you would like to prove something in one model, I tell you "well that will be difficult because of this", you respond "oh well then I can just do it in this other model (which has not yet been defined)", then after a few posts you are back talking about the original model...
To be honest this is now making the conversation frustrating and I may switch off.

Amar Hadzihasanovic (Jul 10 2025 at 09:13):

Whatever you prove in one model is very non-trivial to transport to another model!

Ruby Khondaker (she/her) (Jul 10 2025 at 09:13):

oh sorry, i really don't mean to be frustrating..

Ruby Khondaker (she/her) (Jul 10 2025 at 09:16):

i'm really just exploring very informally here, that's why i keep "shifting the goalposts" so to speak - i promise it's not out of malicious intent

Amar Hadzihasanovic (Jul 10 2025 at 09:44):

I don't think it's malicious, but it comes across as dismissive, like the conversation is

• I want to do X in Y.
• Doing X in Y is actually pretty hard...
• Oh what's the problem? You can just do X in Y' then, that seems easy.

Ruby Khondaker (she/her) (Jul 10 2025 at 09:45):

Oh I’m very sorry, I’ll try to correct that habit

Ruby Khondaker (she/her) (Jul 10 2025 at 09:45):

On my end it’s just me being naive

Ruby Khondaker (she/her) (Jul 10 2025 at 09:46):

I’m not saying “wait why haven’t you thought of this, are you stupid?”, but more “hmm ok thanks, I wonder if trying this will work or whether I’ll run into problems”

Amar Hadzihasanovic (Jul 10 2025 at 09:56):

I think that if you have a goal which is “model-independent” then it makes sense to go “shopping for models”. But I went with your statement earlier that you would set yourself the goal of developing some aspect of the theory of L'-categories. If your goal is model-specific, then it is inconsistent to be model-shopping.

Ruby Khondaker (she/her) (Jul 10 2025 at 09:56):

Ah yeah I think my goal is quite poorly-defined atm :P

Ruby Khondaker (she/her) (Jul 10 2025 at 09:57):

Again this is really just a hobby for me

Ruby Khondaker (she/her) (Jul 10 2025 at 09:57):

I like L’ categories but I’m not married to them, if that makes sense

Amar Hadzihasanovic (Jul 10 2025 at 11:14):

Sure, I just don't feel like answering your questions is worth my time if the typical response is "actually now I want to do something else entirely, and anyway this is just a pastime for me". Good luck with your hobby but I'll switch off for the time being.

Ruby Khondaker (she/her) (Jul 10 2025 at 12:19):

Hm so i've been thinking about what the $\omega$-category of $\omega$-categories should look like

Ruby Khondaker (she/her) (Jul 10 2025 at 12:19):

would the natural shape for this be "globular"?

Ruby Khondaker (she/her) (Jul 10 2025 at 12:20):

e.g. with the category of categories, you have categories, functors, and natural transformations between parallel functors

Ruby Khondaker (she/her) (Jul 10 2025 at 12:20):

now of course, you can instead say a category is a simplicial set with a property (unique inner horn filler i think?)

Ruby Khondaker (she/her) (Jul 10 2025 at 12:20):

but you don't describe the category of categories as a simplicial set with a unique horn filler, as far as i can tell

Ruby Khondaker (she/her) (Jul 10 2025 at 12:21):

so e.g. with complicial sets or comical sets, you describe individual $\omega$-categories as presheaves on the appropriate shape categories with properties

Ruby Khondaker (she/her) (Jul 10 2025 at 12:22):

but for the $\omega$-category of $\omega$-categories, it would seem a little strange to me to describe it as a presheaf on the appropriate shape category with properties

Ruby Khondaker (she/her) (Jul 10 2025 at 12:22):

now you have to consider simplices or cubes whose vertices are omega-categories, whose 1-cells are functors, 2-cells natural transformations..

Ruby Khondaker (she/her) (Jul 10 2025 at 12:23):

maybe this is silly of me but it feels more "natural" to say that the omega-category of omega-categories should have 0-cells be omega-categories, 1-cells be functors, 2-cells be natural transformations between parallel functors, etc etc, which would be a globular shape

Ruby Khondaker (she/her) (Jul 10 2025 at 12:23):

does that make any sense...?

Ruby Khondaker (she/her) (Jul 10 2025 at 12:24):

also, since Cat is a strict 2-category, would we expect the omega-category of omega-categories to also be strict?

Notification Bot (Jul 10 2025 at 12:48):

163 messages were moved from this topic to #learning: questions > The n-globe as a cubical set by Morgan Rogers (he/him).

Morgan Rogers (he/him) (Jul 10 2025 at 13:09):

Ruby Khondaker (she/her) said:

but you don't describe the category of categories as a simplicial set with a unique horn filler, as far as i can tell

Normally one considers a 2-category of 1-categories; omega-cats are special in that assembling them into a kind of category one dimension higher should (in principle) land you at the same level. However, this does seem like it should run into some form of Russell's paradox if presented naively (without controlling for size issues, say).
Note that a 2-category is just one of the ways that one can arrange 1-categories into a higher structure. Another important one is [[virtual double category]].

Ruby Khondaker (she/her) (Jul 10 2025 at 13:50):

Right yes - so it might be that the collection of all omega-categories doesn’t “naturally” form an omega category, but some other kind of categorical structure!

Ruby Khondaker (she/her) (Jul 10 2025 at 13:51):

Like how proarrow equipments let you do “formal category theory” but in a way that gives the correct notions of weighted limits etc, in a way that 2-categories alone don’t

Mike Shulman (Jul 10 2025 at 15:00):

Morgan Rogers (he/him) said:

Normally one considers a 2-category of 1-categories; omega-cats are special in that assembling them into a kind of category one dimension higher should (in principle) land you at the same level. However, this does seem like it should run into some form of Russell's paradox if presented naively (without controlling for size issues, say).

Issues of categorical dimension are unrelated to issues of set-theoretic size. The 2-category of 1-categories has to control for size issues in exactly the same way.

Morgan Rogers (he/him) (Jul 10 2025 at 15:09):

Good point! In particular this means that whatever model you use to present your omega-categories, unless you incorporate some way to manage size into your foundations (e.g. a universe of sets which is a member of a universe of classes) you cannot hope for "the" omega-category of omega-categories to fit inside the model of omega-categories, even if you decide it's the right shape to intuitively count as an omega-category.

Mike Shulman (Jul 10 2025 at 15:11):

Well, it'll just fit inside the same thing defined at the next universe level. (-:

Ruby Khondaker (she/her) (Jul 12 2025 at 04:34):

I think I’ve realised that what I like about $\mathbf{L}’$ is actually essentially the same thing I like about space-based models?

For L’ you encode composition as a relation (roughly), between a pasting diagram and a n-cell, such that, if you fix boundary conditions, the resulting globular set of composites is contractible. This works cause globular sets can model “spaces” ($\infty$-groupoids), and so it makes sense to ask for a globular set to be contractible.

For space-based models, it seems like much the same idea - you have a pasting diagram that you want to compose to a n-cell. The requirement is that, if you choose a map from the boundary of the n-cell to the boundary of the pasting diagram, then the induced map from the pasting diagram to the associated (homotopy) pullback of n-cells with specified boundary has contractible fibers! I think.

So really, in both cases I liked the idea of encoding everything in terms of contractibility - I loved how, in Leinster’s approach, both composition and coherence arose as two sides of the same coin! I think it took me a while to realise that space-based models were really doing the same kind of thing. :)

Mike Shulman (Jul 12 2025 at 04:38):

That is certainly a similarity, modulo that globular sets don't "model" $\infty$-groupoids in the same sense that (say) simplicial sets do, i.e. there isn't a model structure on the category of globular sets that presents the homotopy theory of $\infty$-groupoids.

However, an essential component of space-based models is that they also have a space of objects, which L' doesn't: it only has a set of objects. Having a space of objects is what gives rise to most of the advantages of space-based models that I talked about before.