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Stream: community: our work

Topic: Amar Hadzihasanovic

Amar Hadzihasanovic (Apr 12 2024 at 04:34):

I have written a book on the combinatorics of higher-categorical diagrams -- free to read on the arXiv.

Here is the abstract:

This is a book on higher-categorical diagrams, including pasting diagrams.
It aims to provide a thorough and modern reference on the subject, collecting, revisiting and expanding results scattered across the literature, informed by recent advances and practical experience with higher-dimensional diagram rewriting.

We approach the subject as a kind of directed combinatorial topology: a diagram is a map from a "directed cell complex", encoded combinatorially as a face poset together with orientation data.
Unlike previous expositions, we adopt from the beginning a functorial viewpoint, focussing on morphisms and categorical constructions.
We do not tie ourselves to a specific model of higher categories, and instead treat diagrams as independent combinatorial structures that admit functorial interpretations in various contexts.

Topics covered include the theory of layerings of diagrams; acyclicity properties and their consequences; constructions including Gray products, suspensions, and joins; special shapes such as globes, oriented simplices, cubes, and positive opetopes; the interpretation of diagrams in strict omega-categories and their geometric realisation as simplicial and CW complexes; and Steiner's theory of directed chain complexes.

Amar Hadzihasanovic (Apr 12 2024 at 04:34):

And here is the table of contents:

Introduction
Acknowledgements

Elements of order theory
1.1. Closed maps of posets
1.2. Graded posets
1.3. Some operations on posets

Oriented graded posets
2.1. Orientation and boundaries
2.2. The category of oriented graded posets
2.3. Oriented thin graded posets

Molecules
3.1. Pastings and globularity
3.2. Rewrites and roundness
3.3. The inductive definition of molecules
3.4. Isomorphisms of molecules are unique

Submolecules and layerings
4.1. Submolecules and substitution
4.2. Layerings
4.3. Flow graphs and orderings
4.4. Frame dimension
4.5. Rewritable submolecules

Diagrams in strict ω-categories
5.1. Fundamentals of strict ω-categories
5.2. The ω-category of molecules
5.3. Regular directed complexes

Maps and comaps
6.1. Pushforwards and pullbacks
6.2. Maps of regular directed complexes
6.3. Comaps of regular directed complexes

Constructions and operations
7.1. Generalised pastings
7.2. Gray products
7.3. Suspensions
7.4. Joins
7.5. Duals

Acyclicity
8.1. Frame-acyclic molecules
8.2. Presenting polygraphs
8.3. Stronger acyclicity conditions
8.4. In low dimensions

Special shapes
9.1. Globes and thetas
9.2. Oriented simplices
9.3. Oriented cubes
9.4. Positive opetopes

Geometric realisation
10.1. Elements of poset topology
10.2. Face posets
10.3. Presenting regular CW complexes

Steiner theory
11.1. Directed chain complexes
11.2. Steiner complexes and acyclicity

Bibliography
Index
Index of counterexamples

Amar Hadzihasanovic (Apr 12 2024 at 04:36):

I still haven't contacted any publishers, so I'd be happy to hear from anyone who has one in mind and has had (or knows someone who has had) positive experiences. My priority is to keep at least the preprint version freely accessible.
And of course always happy to get feedback!

John Baez (Apr 12 2024 at 09:54):

Contragulations on your new book! And I'm very glad you have made it available on the arXiv.

The main thing about publishers is that you can and should negotiate with them to get what you want; they have a lot of 'standard' ways of doing things, so they won't encourage you to negotiate. If you don't negotiate, they will lead you down their standard route, which means they get everything they want and you get whatever they want you to get.

John Baez (Apr 12 2024 at 09:55):

Naive authors tend to go down the standard route, since they don't know better.

John Baez (Apr 12 2024 at 10:00):

To take a good nonstandard example, Tom Leinster has negotiated with Cambridge University Press to allow all three of his books to be freely available online.

John Baez (Apr 12 2024 at 10:02):

These appear to not be mere "preliminary versions" - they look like they have the same typeface, page numbers, etc. as the published version. It's easier to get a publisher to let you put a "preliminary version" of your book on the arXiv, and for math that is just as good iff the theorem numbers are the same and there are no significant corrections that appear only in the final version. (In the humanities, people refer to page numbers, and then if the page numbers don't match a preliminary version is much less useful!)

Niles Johnson (Apr 12 2024 at 11:00):

I can second the recommendation for Cambridge University Press. Submitting a book that was previously posted to the arxiv was a non-issue.

I have a different book with the American Math Society (Surveys and Monographs series) and we had to specifically ask about keeping an online version, but they agreed without much complaint. Terry Tao has published several books with the AMS, and many of them appear to be posted online, like this one. We mentioned this during the discussion about our online version, and I think it helped.

I also have a book with Oxford University Press, but unfortunately I do NOT recommend working with them. They were very upset about the arxiv version of the book, and made us add a useless but annoying disclaimer in the arxiv "comment". The copy editing was also extremely frustrating. Editing is always a bit of a hassle, but my coauthor Donald Yau, who has much more experience with this kind of thing, described it as the worst book editing he's ever encountered.

Separately, Donald told me he had a very positive experience with Chapman and Hall/CRC, publishing a book that he had previously posted to arxiv. I wasn't involved, so I don't have any direct experience with it, but I trust his recommendation.

Amar Hadzihasanovic (Apr 12 2024 at 11:09):

Thank you John and Niles, that's very useful information!
Emily Riehl's first book with Cambridge University Press is also freely available on her page, so clearly they are more than tolerant about it. (Her second one also is, but iirc it's a bit of an “exception” in being an original textbook published by Dover).

Amar Hadzihasanovic (Apr 12 2024 at 11:13):

John Baez said:

Contragulations on your new book!

I like the word "contragulation", like a congratulation with a little bit of tragedy in it.

John Baez (Apr 12 2024 at 13:56):

Sorry, no tragedy or strangulation was implied.

Rich Hilliard (Apr 13 2024 at 14:42):

:+1: for the index of counterexamples!

Amar Hadzihasanovic (Apr 24 2024 at 06:11):

I have a new paper on the arXiv, joint work with Loubaton, Ozornova and Rovelli.
The title is A model for the coherent walking ω-equivalence.

Amar Hadzihasanovic (Apr 24 2024 at 06:13):

The idea is the following. The data of an "equivalence" in a 2-category is classified by 2-functors from the "walking 2-equivalence", which is the 2-category generated by two objects $a, b$ , with two morphisms $f: a \to b, g: b \to a$ , and two invertible 2-morphisms $gf \Rightarrow \mathrm{id}_a$ , $fg \Rightarrow \mathrm{id}_b$ .

Amar Hadzihasanovic (Apr 24 2024 at 06:15):

However this object has a problem: it is not coherent, which in this case means it is not itself 3-equivalent to the point.

Amar Hadzihasanovic (Apr 24 2024 at 06:18):

This is not good, because you would want this to be essentially a 2-categorical model of a topological "interval", which is a contractible space. It means if you attach an "interval" like this by one end to a 2-category, you end up with something inequivalent!

Amar Hadzihasanovic (Apr 24 2024 at 06:20):

So in 2-categories, you instead want to take the walking adjoint equivalence, where in addition the first 2-cell and the inverse of the second satisfy the triangle equation.
This is, indeed, a "contractible" 2-category, 3-equivalent to the point, and it also classifies equivalences by the result that every equivalence can be promoted to an adjoint equivalence.

Amar Hadzihasanovic (Apr 24 2024 at 06:22):

Now, in the case of higher $n$ -categories, or $\omega$ -categories, you can also build a "naive" walking equivalence, by just adding all higher dimensional weak inverses. And this is also non-coherent, of course.

Amar Hadzihasanovic (Apr 24 2024 at 06:23):

For abstract reasons, we knew there must be a coherent version too, but no explicit model of it was known for sure even for strict higher categories.

Amar Hadzihasanovic (Apr 24 2024 at 06:26):

There was a candidate proposed in a recent paper by my coauthors Ozornova and Rovelli: the idea being that if instead of adding a two-sided inverse of each cell, you add separately a left inverse and a right inverse, you can avoid the "non-coherent" parallel pairs.

Amar Hadzihasanovic (Apr 24 2024 at 06:26):

What we do in this paper is prove that, indeed, this object is coherent.

Amar Hadzihasanovic (Apr 24 2024 at 06:27):

Feedback is welcome!

Mike Shulman (Apr 24 2024 at 06:35):

That's cool! As you probably know, this idea of separating left and right inverses is also used in HoTT as one way to define coherent equivalences.

Amar Hadzihasanovic (Apr 24 2024 at 09:38):

Yes, indeed -- the non-coherence of the "naive" version is already "topological", it shows up in the homology of an associated cell complex. Conversely, I think for the "two separate inverses" version we could prove contractibility of an associated cell complex by a Hurewicz theorem argument, which presumably is what is also behind the HoTT proof.

However we did not have any way (that we know of) to use this to derive coherence in the appropriate sense for n-categories. (I do not have enough intuition for this subject to tell if there should be a way, even in principle.)

Tim Hosgood (Apr 24 2024 at 14:16):

a naive question that is maybe answered by the technical stuff you've written (but that I can't understand), is if this is related to the problem that I've come across in geometry:

you have two objects, so you draw two points. you want to say that they're homotopy equivalent in some sense, so you draw a line between them for an equivalence, so now you have something of the right homotopy type (it's contractible) but it's asymmetric: you should also draw the data of the inverse morphism. but now you have the wrong homotopy type (you've built a 1-sphere), so you should add a 2-morphism describing the invertibility (like $ff^{-1}\Rightarrow\mathrm{id}$ ). but again, right homotopy type, wrong symmetry. so you add in the other 2-morphism ( $\mathrm{id}\Rightarrow f^{-1}f$ ), but you've got the wrong homotopy type. so ...

... in the end, you are led to replacing $\Delta[1]$ with $S^\infty$

Tim Hosgood (Apr 24 2024 at 14:18):

there's a specific construction where this $S^\infty$ replacement is super helpful, but i've never really understood what formal words i should be saying here, and it sounds like your paper is sort of about this?

Tim Hosgood (Apr 24 2024 at 14:18):

(apologies if not, please do just say so!)

Amar Hadzihasanovic (Apr 24 2024 at 15:37):

Well, yes, it is, but I would say that in that paragraph there is too much ambiguity as to how the construction continues after the ellipsis :)

One interpretation would be that every time you just "add an inverse" to the cells you've just added, and then you add "a witness that it is an inverse", and you keep going. That will end up building the non-coherent walking equivalence, so it will have the wrong homotopy type after all, so it won't be your $S^\infty$ .
Another interpretation would be that you "invert cells, then fill all the new 'spheres' which appear", which will indeed almost by definition build a contractible object -- you can see this as the abstract construction of the factorisation of the functor from the 0-sphere to the point as a cofibration followed by a trivial fibration. This works but is not explicit, in particular it doesn't tell you whether the object that you build has finitely many cells in each dimension.

What we proved is that an explicit model of finite type is, in fact, a model of your $S^\infty$ .

Tim Hosgood (Apr 24 2024 at 15:46):

yeah, one of the problems I ran into was exactly how ambiguous my description was! but the thing that I intend by the "..." is the process that results in you having exactly two cells in each dimension, but maybe this is still just ill-defined vagueness?

Amar Hadzihasanovic (Apr 24 2024 at 15:55):

Ah wait I think maybe then your thing is a different object: the simplicial 0-coskeleton of the 0-sphere

Amar Hadzihasanovic (Apr 24 2024 at 15:57):

I was confused because you were assigning an orientation to your first 2-cell which is not the one of Street's oriented 2-simplex, but of course in the (non-oriented) simplicial world you do not need to think of any of these as having a specified orientation. Only if you want to interpret them as n-morphisms in a n-category you do.

Amar Hadzihasanovic (Apr 24 2024 at 16:04):

Tim Hosgood said:

yeah, one of the problems I ran into was exactly how ambiguous my description was! but the thing that I intend by the "..." is the process that results in you having exactly two cells in each dimension, but maybe this is still just ill-defined vagueness?

I think that in each context, the likely "object that you want to construct" is "a factorisation of the morphism $S^0 \to D^0$ as a cofibration followed by a trivial fibration". But what realises that construction is, of course, dependent on the model category you are in.

Amar Hadzihasanovic (Apr 24 2024 at 16:05):

So in simplicial sets with the classical model structure, a model will be given by "the 0-coskeleton of $S^0$ ".

Amar Hadzihasanovic (Apr 24 2024 at 16:07):

And that has only 2 non-degenerate simplices in each dimension.
But that construction does not help you if you are in $n$ -categories with the canonical model structure. (In fact there isn't a model which has just two non-degenerate cells in each dimension.)

Tim Hosgood (Apr 24 2024 at 16:30):

oh that's a super helpful explanation, thank you!

David Corfield (Apr 26 2024 at 11:38):

Sorry if this is off-topic, but I see you mention in your article Eugenia Cheng's result in An ω-category with all duals is an ω-groupoid.

I recall @John Baez talking about this result:

(The latter paper formalizes an important issue that has vexed Jim Dolan and I for a long time: it seems that only with a dimensional cutoff is the concept of ‘n-category with duals’ different from that of an n-groupoid!)

So this was something along the lines of the limit of higher versions of the walking adjunction is equivalent to the walking ω-equivalence?

Amar Hadzihasanovic (Apr 26 2024 at 12:18):

Indeed, if one only looks at it “from the inside” -- that is, one can only probe a higher category with its cells -- then there is no difference between dualisability (or "two-sided adjunction") and invertibility in the limit.

Amar Hadzihasanovic (Apr 26 2024 at 12:21):

This point of view seems to be natural among higher category theorists of “computational” extraction, because it gives a kind of operational meaning to “being an equivalence”, that is, “an equivalence is something that behaves as an equivalence”.

Amar Hadzihasanovic (Apr 26 2024 at 12:23):

However, among homotopy theorists, algebraic geometers etc. it seems that a different viewpoint is more common, in which those two notions do not coincide. I remember a conversation here on Zulip with Rune Haugseng helped me understand this.

Amar Hadzihasanovic (Apr 26 2024 at 12:27):

Basically, whereas in this “computational” tradition it is natural to think of a higher groupoid as being a special case of a higher category where “every cell behaves as an invertible cell”, in the homotopical-geometric tradition it seems to be more natural to see a higher category as structure over an underlying $\infty$ -groupoid.
In this case, an “equivalence” is a cell which belongs to the underlying $\infty$ -groupoid. So there can be “weakly invertible cells” which are not 'really' equivalences even though they behave like them. It's a somewhat more “essentialist” view of equivalences...

Amar Hadzihasanovic (Apr 26 2024 at 12:30):

The two notions are, as far as I understand, captured by the inductive (“essentialist”) and coinductive (“operational”) homotopy theories of $(\infty, \infty)$ -categories, the “only” two possibilities in the classification by Barwick and Schommer-Pries.

Amar Hadzihasanovic (Apr 26 2024 at 12:33):

In concrete models, the “real” equivalences are usually distinguished by a marking or stratification. Then one gets the “coinductive” model structure when asking that, in the fibrant objects, the marked cells coincide with the ones that behave like equivalences.

Amar Hadzihasanovic (Apr 26 2024 at 12:36):

In our paper, indeed, we derive contractibility of our walking $\omega$ -equivalence from contractibility in a coinductive model structure on marked $\omega$ -categories developed by Henry and Loubaton.

Amar Hadzihasanovic (Apr 26 2024 at 12:39):

If one truncates to some finite dimension $n$ the difference goes away, as noted in John's comment that you cited.

Amar Hadzihasanovic (Jul 10 2024 at 02:43):

My student @Clémence Chanavat and I have a new preprint on the arXiv: Diagrammatic sets as a model of homotopy types.
It's Clémence's first submission!

Diagrammatic sets are an attempt to construct the "dream model" of higher categories for those who are combinatorially/computationally inclined, i.e. like to work explicitly with cellular presentations, diagrammatic reasoning etc, while making sure that it is equivalent to the models accepted by homotopy theorists.
They can also be seen as the “topologically sound” alternative to polygraphs/computads for strict n-categories.

I had made a first attempt at this project here but I hadn't got the "shape category" quite right (too many surjections!) and got quite weak results. Meanwhile I've written a book on pasting diagrams so I've got quite a few more tools to work with.

So Clémence and I are now trying to get it all right, step by step. This first paper settles the case of $\infty$ -groupoids/homotopy types, and ones on $(\infty, n)$ -categories are in the works.
Questions and feedback are welcome.

Kevin Carlson (Jul 10 2024 at 15:12):

Congratulations to both of you! I’d love to hear more about how you see this as a “dream” computational model, in particular whether this is how you think of even the $\infty$ -groupoid case or whether more of the dreaminess is going to come out as you increase the categorical dimension. You briefly reference that it’s nice to have models of spaces without as many cells as in cubical or simplicial models in your introduction, but that’s all I caught.

For me, one of the key difficulties of explicit combinatorial topology is that you get an unmanageable number of new cells when you (explicitly) invert a morphism, relative to what happens with eg CW complexes, where everything is inherently inverted already. This model structure doesn’t do anything about this challenge, right? Should I basically think about its advantage for $\infty$ -groupoids as that it makes all in-some-sense “reasonable” cell shapes representable?

And, if you’ll pardon the question revealing I haven’t read your book at all, how general should I imagine the homotopy types modeled by arbitrary regular directed complexes to be? All the finite CW-complexes?

Amar Hadzihasanovic (Jul 10 2024 at 15:31):

Hi, thanks for your comments!
Of course this stuff about the "dream" model is the sort of vague motivation that I would only share informally like this, in the paper introduction we kept everything very factual :)

I would say that, yes, the advantage is supposed to become clearer in the higher-categorical case than the higher-groupoidal one, and I see it mainly in the fact that

pasting diagrams should “just work”, i.e. you should have a model with a “native” strong pasting theorem,
various constructions that have a simple description at the cellular level should likewise admit simple, workable definitions, in particular things like Gray products which are very tricky to define even in strict models;

these are not-so-evident problems in the “higher groupoid” case, (1) because the undirectedness of cells makes it possible to always subdivide until cells have simplicial/cubical/... shapes without worrying about orientation, which is not possible for directed cells (at least not while preserving computational properties) (2) Gray products and cartesian products “coincide” for higher groupoids at least up to equivalence, and the latter are not so problematic

Amar Hadzihasanovic (Jul 10 2024 at 15:35):

On the other hand, while this is not something that I have either explored, or know how to “quantify”, I do see a use case in computational topology in the fact that constructing a CW complex as a finite diagrammatic set equips its cells with a good orientation for computing its cellular homology -- the “cellular chain complex” is a functor from diagrammatic sets to (augmented) chain complexes of abelian groups, which is not the case for undirected CW-complexes because there is no canonical orientation (you need to pass to homology to have a functor)

(The "cellular chains" functor from diagrammatic sets is not in the paper but it is an obvious extension of the one from regular directed complexes defined in the book)

Amar Hadzihasanovic (Jul 10 2024 at 15:39):

And the existence of “globular” orientation makes it possible to specify the pasting of cells algebraically. So I think that it can actually be advantageous to try to “design” a cell complex as a directed cell complex, when possible.

To answer your last question, however, I do not know a characterisation of the cell complexes that can be constructed as directed cell complexes, not even in the regular case.

Amar Hadzihasanovic (Jul 10 2024 at 15:42):

(Of course by “constructed” I mean “constructed with the same set of generating cells”.)
Edit: Actually I see that your question was more specifically about “what are the homotopy types that can be realised as regular directed complexes”, and I also don't have an answer to that.

Amar Hadzihasanovic (Jul 10 2024 at 15:45):

I think a lot of interesting cell complexes can be constructed as directed cell complexes without ever inverting any cells, by cleverly distributing n-cells between the input and output boundary of an (n+1)-cell. Perhaps trying to do this can be a good design heuristic.

The existence of "directed counterparts" to products, joins, and suspensions at least tells us that the cell complexes that can be constructed in this way are also closed under these operations.

Kevin Carlson (Jul 10 2024 at 15:59):

Thanks Amar! The cellular complex functor is a very interesting point.

Amar Hadzihasanovic (Sep 03 2024 at 15:55):

One small update: yesterday, my joint paper with Diana Kessler, Acyclicity conditions on pasting diagrams, has appeared on the arXiv. It will also appear in Applied Categorical Structures.

The material in this paper, which Diana and I developed together as technical background to our work on algorithmic subdiagram matching, is also (mostly) covered in my pasting diagrams book.
However, we thought it would be good to have an independent research paper on it, because the role of acyclicity in the theory of pasting is particularly subtle and misunderstood.
I have also given a talk on this point at CT 2024.

Amar Hadzihasanovic (Oct 02 2024 at 06:02):

Today the 2nd paper in the planned series by @Clémence Chanavat and me dropped on the arxiv -- it is called Equivalences in diagrammatic sets.

Here's the abstract:

We show that diagrammatic sets, a topologically sound alternative to polygraphs and strict ω-categories, admit an internal notion of equivalence in the sense of coinductive weak invertibility. We prove that equivalences have the expected properties: they include all degenerate cells, are closed under 2-out-of-3, and satisfy an appropriate version of the "division lemma", which ensures that enwrapping a diagram with equivalences at all sides is an invertible operation up to higher equivalence. On the way to this result, we develop methods, such as an algebraic calculus of natural equivalences, for handling the weak units and unitors which set this framework apart from strict ω-categories.

I am quite proud of this paper, because at first we were trying to define just enough machinery for handling "weak units" to emulate an older proof about strict ω-categories in our non-strict setting.
But in the process, we ended up discovering some really natural concepts that led us to a new, more high-level and in my opinion more elegant proof. So here's a case where taking the time to think about what works in a proof beyond the strict context really improved it.

Amar Hadzihasanovic (Oct 28 2024 at 03:13):

My 3rd paper with Clémence is now on the arXiv:

Model structures for diagrammatic $(\infty, n)$ -categories

Here we present our promised model of $(\infty, n)$ -categories and develop the surrounding homotopy theory, showing it fits in the “right” place in a model structure and has the “right” weak equivalences.

First on my own and then with my students, I have been perfecting this model through several iterations and only recently I think we got it “just right”. To the extent that there even is such a thing as a “natural” model of higher categories, I think this comes close; working with it genuinely feels like a generalisation of 1- and 2-category theory.
At least technically I would even say it is a simpler generalisation to work with than strict n-categories (besides being “homotopically correct” which those aren't), because we took a lot of time thinking about what works “strictly”, what doesn't, and what is its right generality.
This is particularly true for the purposes of diagrammatic reasoning/diagram rewriting in higher dimensions, so I encourage anyone with an interest in this to get acquainted with the model.

Amar Hadzihasanovic (Oct 28 2024 at 03:18):

Of course a model is only as good as what it does, and we are working hard on tackling some unsolved problems in higher category theory through it.

But one first application that's already in the paper is that we were able to prove the homotopy hypothesis (in the strongest possible terms) for a definition of $\infty$ -groupoids which is actually a direct generalisation of the algebraic characterisation of 1-groupoids within 1-categories, which we think is a first.
That is, in the model, an $\infty$ -groupoid is defined as an $(\infty, \infty)$ -category whose cells in dim > 0 are all invertible up to a higher invertible cell. This is formulated entirely in the natural “algebraic” language of cells, units, and pasting within the model, with no reliance on extraneous data or some underlying notion of space or homotopy.

John Baez (Oct 28 2024 at 04:56):

Sounds great!

David Corfield (Oct 28 2024 at 07:41):

Amar Hadzihasanovic said:

That is, in the model, an ∞-groupoid is defined as an (∞,∞)-category whose cells in dim > 0 are all invertible up to a higher invertible cell.

So in line with the former "computational" model of the two you discuss here.

Matteo Capucci (he/him) (Oct 28 2024 at 07:43):

maths aside, this latexing hits hard :fire:
image.png

Matteo Capucci (he/him) (Oct 28 2024 at 07:47):

out of curiosity, in the introduction you say:

This multiplicity of models has been brought about by the failure of strict ω-categories [ABG+23] [...] to account for the geometric aspect of higher categories; that is, the failure to satisfy the homotopy hypothesis

I didn't know that! What goes wrong?

Amar Hadzihasanovic (Oct 28 2024 at 07:59):

David Corfield said:

Amar Hadzihasanovic said:

That is, in the model, an ∞-groupoid is defined as an (∞,∞)-category whose cells in dim > 0 are all invertible up to a higher invertible cell.

So in line with the former "computational" model of the two you discuss here.

Yes, it's in that style!
But we also give a model in the “essentialist”/“inductive” style, which is equivalent for $n < \infty$ , which we hope will facilitate the comparison with other geometric models.

Amar Hadzihasanovic (Oct 28 2024 at 08:11):

Matteo Capucci (he/him) said:

out of curiosity, in the introduction you say:

This multiplicity of models has been brought about by the failure of strict ω-categories [ABG+23] [...] to account for the geometric aspect of higher categories; that is, the failure to satisfy the homotopy hypothesis

I didn't know that! What goes wrong?

Well, you know how in a space, higher homotopy groups are abelian; the commutativity is witnessed by some “braiding” homotopies, and usually there is a lot of structure in the space of these braidings. Some of this is captured by an invariant called the Whitehead product or Whitehead bracket. Anyway, in strict $\omega$ -categories the braidings are all identical so this structure is always flat. This prevents them from modelling all homotopy types (the first counterexample is already $S^2$ ) in the way that the homotopy hypothesis prescribes.

Matteo Capucci (he/him) (Oct 28 2024 at 08:12):

I see, interesting!

Amar Hadzihasanovic (Oct 28 2024 at 08:14):

The standard reference is Section 4.4 of Carlos Simpson's book

David Michael Roberts (Oct 28 2024 at 08:25):

@Matteo Capucci (he/him) you are in good company: Voevodsky thought he and Kapranov had sidestepped the issue, but this is what Simpson found: already no strict 3-groupoid models the homotopy 3-type of S^2, so forget about the whole homotopy type. But the precise error in the KV paper is not trivial, and not found before a few years ago (if at all).

Amar Hadzihasanovic (Oct 28 2024 at 08:33):

Yes, it is quite easy to see that the “obvious” cellular model of $S^2$ in strict $\omega$ -categories doesn't have the right type, but it is not at all trivial that there isn't some “clever” realisation functor of strict $\omega$ -categories which would realise $S^2$ from some other object (which may already be seen as “cheating” but there's enough vagueness in the homotopy hypothesis to brush it off).

So instead Simpson showed that there is no functor at all satisfying some minimal properties that would be expected from a homotopy hypothesis worth its name.

Amar Hadzihasanovic (Oct 28 2024 at 08:35):

(Meaning that any functor satisfying such properties is going to “miss” $S^2$ )

Nathanael Arkor (Oct 28 2024 at 08:57):

Do you know how your definition of $(\infty, n)$ -category compares to the classical notions of bicategory and tricategory when you truncate?

Amar Hadzihasanovic (Oct 28 2024 at 09:12):

We haven't constructed an explicit nerve functor for either bicategories or tricategories.
We do have some adjacent constructions:

In our previous paper, Section 4, we showed that for every $n$ and every diagrammatic set $X$ (not necessarily an $(\infty, \infty)$ -category) there is a strictly associative bicategory of

round $n$ -diagrams in $X$ ,
round $(n+1)$ -diagrams in $X$ ,
round $(n+2)$ -diagrams in $X$ up to equivalence
which we did mainly to show that one can safely use string diagrams for proofs, or import other bicategorical results for proofs that involve only 3 consecutive dimensions.

In my paper from a few years ago, which uses some outdated definitions (but not outdated in an essential way) I also constructed a faithful nerve from Gray-categories to diagrammatic sets, and I am confident that the nerve of a Gray-category is an $(\infty, 3)$ -category in the sense of this paper.

Matteo Capucci (he/him) (Oct 28 2024 at 09:15):

David Michael Roberts said:

Matteo Capucci (he/him) you are in good company: Voevodsky thought he and Kapranov had sidestepped the issue, but this is what Simpson found: already no strict 3-groupoid models the homotopy 3-type of S^2, so forget about the whole homotopy type. But the precise error in the KV paper is not trivial, and not found before a few years ago (if at all).

I'm definitely the odd one out in that triplet :laughing: I guess this counterexample also shows why there is no full strictification of tricategories

Amar Hadzihasanovic (Oct 28 2024 at 09:18):

I don't think there's more that can be done with respect to this model for comparison with bicategories/tricategories than looking at nerve-realisation pairs of functors, since it is non-algebraic and there is not going to be any canonical way to “choose” a bicategory/tricategory structure directly on one of our $(\infty, 2)$ or $(\infty, 3)$ -categories, even after truncation.

Amar Hadzihasanovic (Oct 28 2024 at 09:20):

(In the first construction above, we kind of bypass this by looking at round diagrams instead of cells which does not involve any composition operation hence any choice -- you just paste the diagrams together -- and that's why you can do it with any object in our category and not just the fibrant objects)

Amar Hadzihasanovic (Oct 31 2024 at 08:37):

A couple of small updates:

I uploaded a minor revision of the pasting diagram book. Mostly it corrects typos, there was one incorrectly stated lemma which has been fixed, and there is a single new result that wasn't in v1 (it's “fundamental” enough but I hadn't realised it until recently.)
Last week I gave a talk on diagrammatic $(\infty, n)$ -categories at the Atlantic Category Theory seminar; the slides are on my website.