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Stream: theory: category theory

Topic: (n+1)-category of n-categories

Patrick Nicodemus (Feb 13 2024 at 11:19):

Recently I was looking at Leinster's Higher categories, higher operads and Trimble n-categories.
In these cases it is clear what is an $n$ category and what is a functor of $n$ categories but not a natural transformation or a modification or ...
I am wondering, for this and other entries on this page, such as opetopic sets, where is there a clear explanation of the $n$-category of functors between $n+1$ categories? Or an adjunction or adjoint equivalence between two $n$ categories?
https://ncatlab.org/nlab/show/n-category

Patrick Nicodemus (Feb 13 2024 at 11:42):

Leinster says

We do not set up a notion of weak equivalence of our weak n-categories...

the reader may feel cheated but this is the state of the art.

This was in 2003. Any updates? :D

James Deikun (Feb 13 2024 at 12:40):

Leinster $n$-categories are quite amenable to being rephrased as [[FOLDS]] theories using the technology in the appendix of Higher categories, higher operads and then you can use Makkai's notion of weak equivalence, which comes with a preservation theorem for FOLDS sentences. I don't know if anyone has spelled this out or not though.

Patrick Nicodemus (Feb 13 2024 at 12:46):

Awesome, thanks a lot.

Mike Shulman (Feb 13 2024 at 17:21):

I'm not quite sure what you have in mind, @James Deikun . Leinster and Trimble $n$-categories are algebraic, so they are not directly represented as models of any FOLDS theory. One could hope to formulate a FOLDS theory with a similar class of models, but that seems to me like a substantial amount of work that hasn't been done.

James Deikun (Feb 13 2024 at 18:45):

Well, first of all there are two definitions of $n$-category on offer in Leinster's book, although one is only a "sketch"--let's call that one Baez-Dolan-Leinster (BDL). That one is not algebraic. Then there's the usual "Leinster" (L) definition, based on a globular operad with a (specified) contraction, which is algebraic. Both of them are susceptible to this treatment. The key to doing it in a reasonable amount of work is that FOLDS model equivalence depends only on the signature, not the full content of the theory. The signature for BDL consists of simply the opetopes (there are no predicate symbols except the equality at the top dimension). The signature for L can be taken as the $n$-globes plus one relation symbol for each operation of $L_n$ (9.3.5-6). In the case of the algebraic definition L we are just looking at the restriction of equivalance to subset of FOLDS models where the relations are functional.

James Deikun (Feb 13 2024 at 18:47):

(On Trimble $n$-categories I never made any claim; it looks more difficult.)

Mike Shulman (Feb 13 2024 at 19:00):

Certainly for the opetopic definition you can do that; in fact Makkai did a lot of that himself (although he said "multitope" instead of "opetope"), it was one of his original motivations for introducing FOLDS (as you probably know).

Mike Shulman (Feb 13 2024 at 19:03):

What you describe for the Batanin-Leinster algebraic version is certainly the idea one would try to carry out, but I don't know of anyone who has actually done this, and I don't think it's entirely trivial.

(Chaitanya and I are working on a general machine that turns a presentation of a generalized algebraic theory into a FOLDS-theory. That would apply to this case, although it might be simpler than the general case.)

James Deikun (Feb 13 2024 at 19:13):

Makkai does do some FOLDSification of multitopic $n$-categories, yeah; I didn't want to claim he did Leinster's version of opetopes, though, since I am never sure when two versions of opetopes in the literature are identical. For Batanin-Leinster I'm fairly sure the idea will work out, the hardest part being sufficiently enumerating the contents of $L_n$. Of course it would be very nice to have that general machine to build FOLDS theories and have the theory too.

Mike Shulman (Feb 13 2024 at 19:15):

I'm sorry if I misinterpreted your first comment: I think I latched onto the definiteness of your first sentence that they "are" amenable to such rephrasing, and missed your last sentence saying that no one may actually have done it yet.

Mike Shulman (Feb 13 2024 at 19:17):

I don't know much about opetopes, but I generally operate on the assumption that all notions of opetope are equivalent, kind of like all notions of $n$-category are equivalent. (-: At least there's Cheng's paper https://arxiv.org/abs/math/0304277 comparing them, I don't know about others.

James Deikun (Feb 15 2024 at 09:46):

Also @Patrick Nicodemus you might be interested in:

• Richard Garner, Homomorphisms of Higher Categories (arXiv)
• Cottrell and Fujii, Hom Weak $\omega$-categories of a Weak $\omega$-category (arXiv)
• Camell Kachour, Steps toward the weak higher category of weak higher categories in the globular setting (PDF)

James Deikun (Feb 15 2024 at 23:47):

It's interesting in the first of thes how Garner repeatedly emphasizes that it's of no concern that the construction of a cofibrant replacement comonad via an AWFS depends on the set $J$ of generating cofibrations, because you usually start with the generating cofibrations (in this case the inclusions of cell boundaries) rather than the cofibrations as a class. But in this case, the cofibrations as a class are the levelwise monomorphisms, which seem to me like an even more natural pick than boundary inclusions! In fact I would pick the boundary inclusions because they form a cellular model (i.e. generate the monomorphisms).

James Deikun (Feb 15 2024 at 23:52):

But the dependence actually is super important! You can create an AWFS for the monomorphisms as cofibrations without going out of your way to pick cofibrant generators, as in Section 4.4 of Bourke and Garner, AWFS I--and instead of getting the cofibrant replacement comonad you want, you will get the identity comonad, making your weak functors the same as strict functors! And the same thing seems to apply to the construction of Leinster weak $\omega$-categories in the first place, too. This all seems very mysterious to me.

Patrick Nicodemus (Feb 16 2024 at 00:04):

I think that Garner just hadn't finalized the definition of cofibrantly generated when he wrote "Homomorphisms of higher categories." I'm sure if he wrote that paper today he would have a much different perspective on how we should think about generating cofibrations.

James Deikun (Feb 16 2024 at 00:17):

I'm pretty interested in what that perspective would be, though. It's fascinating and more than a little disturbing that something that homotopy theorists always thought of as a trifling implementation detail would have such an outsized impact here. Basically the difference is that when done in the "proper" way, you are looking at the initial object over X with the structure of a specified contraction, while when you take the shortcut you are looking at the initial object over X with the property of having a contraction, which turns out to just be X itself.

James Deikun (Feb 16 2024 at 00:24):

(Or maybe it's the initial object over X with the property of being an algebra of the partial object classifier monad? Not sure. Either way it's just X.)

Patrick Nicodemus (Feb 19 2024 at 04:19):

James Deikun said:

But the dependence actually is super important! You can create an AWFS for the monomorphisms as cofibrations without going out of your way to pick cofibrant generators, as in Section 4.4 of Bourke and Garner, AWFS I--and instead of getting the cofibrant replacement comonad you want, you will get the identity comonad, making your weak functors the same as strict functors! And the same thing seems to apply to the construction of Leinster weak $\omega$-categories in the first place, too. This all seems very mysterious to me.

James, now that I have had a bit more time to read and understand the paper, I better understand your concerns and I think they are valid. I do not yet have an answer to your question (why don't we get the identity comonad), the devil is buried somewhere in the details but I do not know where.

Recently I have been thinking about the application of the AWFS I paper to simplicial homotopy theory. I have proven, but not yet written down anywhere, that left cofibrations of simplicial sets are cofibrantly generated by the left cofibrations between simplices, natural with respect to pullback squares between them, and furthermore natural with respect to composition of monomorphisms. This is roughly what I alluded to in my previous comment, my point was that when Garner wrote the homomorphisms paper he did not yet understand that one can/should impose naturality with respect to composition in addition as is defined in the AWFS I paper. I don't have any good reason to believe this would work for higher categories, but perhaps it is worth investigating, perhaps one should choose boundary inclusions as a cellular model together with the requirement that things are natural with respect to composition.

James Deikun (Feb 19 2024 at 04:32):

Patrick Nicodemus said:

This is roughly what I alluded to in my previous comment, my point was that when Garner wrote the homomorphisms paper he did not yet understand that one can/should impose naturality with respect to composition in addition as is defined in the AWFS I paper. I don't have any good reason to believe this would work for higher categories, but perhaps it is worth investigating, perhaps one should choose boundary inclusions as a cellular model together with the requirement that things are natural with respect to composition.

This is the magic part, though, if you choose boundary inclusions as a cellular model you get the Leinster-Batanin cofibrant replacement comonad, not the identity. But if you choose another [[cellular model]], say the universal one for toposes in the linked page, it isn't so clear what you will get or why one might be better than the other. Similar concerns arise in some current active work of mine, characterizing the canonical or "folk" model category structures for higher structures, where in the case of infinite-dimensional base categories it's necessary to pick a cellular model for the presheaf category and it gets even more involved when the base category is finite-dimensional.

James Deikun (Feb 19 2024 at 04:35):

(I have a prescription in the case of regular skeletal Reedy categories, but this doesn't cover all the examples I want to cover, nor is it clear quite where it comes from. It's the "easy" case, as in all the hard stuff for this case was already handled by Makkai and it's a matter of dotting (resp. crossing) a few i's (resp t's).)

Patrick Nicodemus (Feb 19 2024 at 05:11):

Yes. Ok. I see what you mean. I think that we could see this as an instance of the broader theme that categorification is always categorification with respect to a particular presentation of a theory. Biased and unbiased monoidal cats or bi cats are categorifications of two different presentations of the theory of monoids. Here, different presentations of the class of monomorphisms give rise to different categorifications of the notion of strict functor.

Patrick Nicodemus (Feb 19 2024 at 05:14):

I don't think there will be one that is obviously the correct presentation. One already has pseudo, lax and oplax functors and natural transformations for 2 categories and bicategories.

James Deikun (Feb 19 2024 at 12:07):

This is true. However, for biased and unbiased bicategories they end up being equivalent in a suitable sense in the end. It would be great if we could say the same with regard to cellular models, that would be a reasonable resolution of the problem. In that case the equivalents of lax/oplax would have to arise in a different way, though.

James Deikun (Feb 19 2024 at 15:59):

Okay, I think I have an idea how this could all work for Leinster-style globular (n,k)-categories. All of this is pretty speculative but here goes:

Strict (n,k)-categories and their canonical model structure

Let $-2 \le n,k \le \infty$, $\mathcal{G}_\infty$ be the category of globes and $T_\infty$ be the monad for strict omega-categories on $\widehat{\mathcal{G}_\infty}$. Let $T_{n,k}$ be the monad obtained from $T_\infty$ by $k$-truncating the input and then $n$-truncating the output. (This is notably different from what you would normally call a "strict $(n,k)$-category" but I think it works here. If it doesn't, probably the normal thing will, though.)

James Deikun (Feb 19 2024 at 15:59):

Now let's consider the class of morphisms of globular sets that are injective on $m$-globes for $m < n$. Let $\mathcal{I}$ be any algebraic cofibrant generator for this class and $(\mathsf{L}_\mathcal{I},\mathsf{R}_\mathcal{I})$ the corresponding AWFS. We (hopefully) obtain a model structure on $\widehat{G_\infty}^{T_{n.k}}$ where the cofibrations are obtained from right transfer of the above AWFS and all objects are fibrant, and where the weak equivalences are appropriate equivalences of strict $(n,k)$-categories, and perhaps even the cofibrant objects are precisely those presented by an $(n,k)$-computad and relations purely on $n$-cells.

James Deikun (Feb 19 2024 at 16:00):

Weakening a la Leinster and Garner

Call a strongly Cartesian monad on $\widehat{\mathcal{G}_\infty}$ an (n,k)-globular operad if it lies over $T_{n,k}$ by a Cartesian monad morphism. The Cartesian monad morphism is determined by its value at $1$, which we call the collection of the operad. The forgetful functor from an operad to its underlying collection is monadic; we can left transfer the AWFS generated by $\mathcal{I}$ to the slice over $T_{n,k}1$ and then right transfer it to $(n,k)$-globular operads. Then take a cofibrant replacement of the the terminal $(n,k)$-globular operad $T_{n,k}$ to get $\widetilde{T_{n,k}}$.

James Deikun (Feb 19 2024 at 16:00):

Define a model structure on $\widetilde{T_{n,k}}$-algebras where all objects are fibrant and cofibrations come again from right transfer of the AWFS generated by $\mathcal{I}$; the morphisms out of cofibrant replacements in this model structure are the weak $(n,k)$-functors.

James Deikun (Feb 19 2024 at 16:01):

Higher structure

Now, there should be a monoidal product for $\widetilde{T_{n,k}}$-algebras making this into a monoidal model structure; I have no idea at the moment how to construct it, but this should give all the expected higher transfors as appropriate weak functors $\overline{t} : G_m \otimes \mathfrak{C} \to \mathfrak{D}$ where $G_m$ is the walking $m$-cell.

James Deikun (Feb 19 2024 at 16:06):

$\quad$

It seems like all this pretty nicely resolves the problem with "why should the cofibrant replacement comonads not be the identity, and why should they be 'the same' given that they're not"--we're actually really doing the cofibrant replacements in categories where not every object is cofibrant, and we get there via right transfer of the cofibrant half of a model structure.

Patrick Nicodemus (Feb 19 2024 at 16:33):

Thank you for suggesting this approach. I will try to read it later after work. I was puzzled by the issue you pointed out with different cellular presentations giving rise to different monads but it was helpful to me to understand that merely considering the class of morphisms equippable with a coalgebra structure is forgetting a great deal of structure. Any cellular presentation of a class of maps should give at least one but potentially many different coalgebra structures on the map and the distinction between cellular presentations is reflected in the category of coalgebras and coalgebra morphisms if not in the underling class of maps reflected with a coalgebra structure. Of course this does not resolve the issue of what cell presentation to use but it helps us propagate the original differences between cell presentations down the pipeline to the differences between categories of cofibrations.

Patrick Nicodemus (Feb 19 2024 at 16:36):

Intuitively the boundaries feel correct to me because one wants to lift a functor along an acyclic fibration dimensionwise in such a way that each cell is lifted independently of others of the same dimension. If you just took arbitrary monomorphisms as cofibrations then you would be demanding to be able to lift arbitrary strict functors strictly along equivalences which are also higher grothendieck fibrations, which is implausible.

Patrick Nicodemus (Feb 19 2024 at 16:38):

It seems one wants to specify, given an n equivalence which is also an n fibration, what are the strict functors which lift strictly along such maps? Only the cell boundary inclusions have this property

Patrick Nicodemus (Feb 19 2024 at 16:39):

Other lifts can only be constructed by inductive application of these strict lifts and the resulting lift is weakly glued together out of strict lifts of cells

Patrick Nicodemus (Feb 19 2024 at 17:22):

James Deikun said:

It's interesting in the first of thes how Garner repeatedly emphasizes that it's of no concern that the construction of a cofibrant replacement comonad via an AWFS depends on the set $J$ of generating cofibrations, because you usually start with the generating cofibrations (in this case the inclusions of cell boundaries) rather than the cofibrations as a class. But in this case, the cofibrations as a class are the levelwise monomorphisms, which seem to me like an even more natural pick than boundary inclusions! In fact I would pick the boundary inclusions because they form a cellular model (i.e. generate the monomorphisms).

Wait, are you talking about the tricategory example when you say degreewise monomorphisms? @Brendan Murphy pointed out to me that the cofibrations are not degreewise monomorphisms, I think, the rightmost image is not a monomorphism in degree 3. Did you mean the oemga category example?

James Deikun (Feb 19 2024 at 17:23):

Yes, I mean the omega-category example, not the tricategory one.

James Deikun (Feb 19 2024 at 17:34):

(Note that for the tricategory example, the generating cofibrations $f_{0...4}$ generate the maps of ${\le}3$-globes that are injective on ${<}3$-globes. This is what you get by reflecting down the description above for (3,3)-categories to the category $\mathcal{G}_3$.)

James Deikun (Feb 22 2024 at 23:31):

More things of interest:

• Bourke and Garner, Algebraic weak factorisation systems II: categories of weak maps (arXiv)
• Thomas Nikolaus, Algebraic models for higher categories (arXiv)

The first one answers the question above about what Garner thinks about his weak map construction after formulating the most general form of his algebraic small object argument; the second is a curiously similar construction with a twist.