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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

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

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

3. 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

4. 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

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

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

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

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

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

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

11. 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 :)

1. 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 $S^\infty$.
2. 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.