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Stream: learning: questions

Topic: Diagram dimensions

Kyle Wilkinson (Jul 24 2023 at 15:25):

Hello all, in a neverending effort to get a more intuitive grasp of formal content, I just noticed something and I'm wondering if there is a deeper notion behind it.

The pentagon and triangle diagrams for the associative and unital coherence properties of a monoidal category are of course 5-sided and 3-sided. Moving up one "dimension" to monoidal functors, however, the coherence diagrams become 6-sided and 4-sided, respectively.

Meaningless coincidence, trivial answer which I just don't see, or something more? Thanks

Kyle Wilkinson (Jul 26 2023 at 13:12):

So, I imagine this has seemed like a silly question. At least for me it is not an important question, just a curiousity.

But, I have found papers discussing the combinatoric (digraph) properties of commutative diagrams (conditions under which gluing diagrams can maintain commutative structures), related to the work of A. Ehressman, who takes a different approach to applying category theory (primarily cone limits of concrete diagrams rather than monoidal categories).

Closer to answering my question though is probably the idea of associahedra: https://en.wikipedia.org/wiki/Associahedron#Examples
"The two-dimensional associahedron K4 represents the five parenthesizations of four symbols, or the five triangulations of a regular pentagon. It is itself a pentagon and is related to the pentagon diagram of a monoidal category."

I don't yet understand enough to follow much in homotopy theory, which is evidently where this takes you.

Anyway, just pointing out it may not be an entirely silly question!

John Baez (Jul 26 2023 at 15:39):

I was going to say: I think it's a coincidence that the pentagon and triangle diagrams for the associative and unital coherence properties of a monoidal category are 5-sided and 3-sided, while the coherence diagrams for monoidal functors are 6-sided and 4-sided, respectively. There might be some deep reason for it, but I can't see one.

John Baez (Jul 26 2023 at 15:42):

There are a lot of nice patterns as go from monoidal categories to monoidal bicategories to monoidal tricategories and son: the pentagon gets replaced by the 3d associahedron:

.

and then the 4d associahedron and so on.

John Baez (Jul 26 2023 at 15:42):

You don't need to know any homotopy theory to study the patterns here.

• Associahedra: some (mostly on-line) resources.

John Baez (Jul 26 2023 at 15:44):

The patterns for the unit laws are less well studied (but see 'monoidahedra' and 'unital associahedra', and the laws for monoidal functors, monoidal natural transformations, monoidal perturbations etc. are even less well studied I think.

John Baez (Jul 26 2023 at 15:44):

So there's a lot to do in this realm.

Jacques Carette (Jul 26 2023 at 15:54):

@Carlos Zapata-Carratala I think these diagrams / diagram patterns are right up your alley.

Kyle Wilkinson (Jul 27 2023 at 01:21):

John Baez said:

So there's a lot to do in this realm.

Great, thanks for the resources!

Amar Hadzihasanovic (Jul 27 2023 at 09:33):

There is indeed a relation, which can be seen if you consider, for example, both the pentagon and the coherence for monoidal functors re: associators as instances of a more general coherence pattern.

A monoidal category can be seen as a pseudomonoid in the monoidal 2-category of categories functors and natural transformations, with the cartesian product as monoidal product.
A pseudomonoid is in turn a special case of a pseudomonad in a tricategory.
This is given by the data of a 0-cell $x$, a 1-cell $C: x \to x$, 2-cells $\mu: CC \to C$ and $\eta: 1_x \to C$, and invertible associator and unitor 3-cells $\alpha$, $\lambda$, $\rho$ satisfying triangle & pentagon equations. (I'm using a notation reminiscent of the monoidal category example).

Amar Hadzihasanovic (Jul 27 2023 at 09:34):

We can draw these in string diagrams; for example this is the associator:3ab1f7a4-a1ad-439c-8826-7be5b9c14f5d.jpg

Amar Hadzihasanovic (Jul 27 2023 at 09:37):

Now, before getting the “coherences” in, you can consider just the bare data up to the 3-cells, without requiring any coherence. You get a notion of “non-coherent pseudomonad” (which still decategorifies to the notion of monad).

“Non-coherent pseudomonads in a tricategory” form canonically a tricategory (actually in a few different ways, but I will focus on one), so you can ask what a morphism is between them.

Amar Hadzihasanovic (Jul 27 2023 at 09:43):

Well, in one of the canonical notions, a “strong” morphism from $(x, C, \mu, \eta, \alpha, \lambda, \rho)$ to $(y, D, \mu', \eta', \alpha', \lambda', \rho')$ is given by

• a 1-cell $T: x \to y$,
• a 2-cell $\sigma: CT \to TD$,
• invertible 3-cells exhibiting a “commutation of $\sigma$ wrt $\mu, \mu'$ and $\eta, \eta'$”, as this:

Amar Hadzihasanovic (Jul 27 2023 at 09:44):

659f9300-0a88-4136-9a59-e232b7f8cb4e.jpg

Amar Hadzihasanovic (Jul 27 2023 at 09:45):

Plus some coherence equations between $\sigma$ and the associators and unitors. I'll draw you the one for the associator

Amar Hadzihasanovic (Jul 27 2023 at 09:50):

Here you go: 74a9eaa2-a4cb-4b03-956a-ba7111f95309.jpg

Amar Hadzihasanovic (Jul 27 2023 at 09:51):

As you see, this coherence equation has 6 sides.

Amar Hadzihasanovic (Jul 27 2023 at 09:52):

They are given by two “apply associator” and four “commute $\mu$ through $\sigma$”.

Amar Hadzihasanovic (Jul 27 2023 at 09:56):

Now, this notion of morphism of non-coherent pseudomonads has two interesting special cases.

First special case: actions
Every 0-cell in a tricategory has the identity pseudomonad on it. This is given by $1_x$ with the structural isomorphism $1_x1_x \to 1_x$ and the identity $1_x \to 1_x$.

A morphism of (non-coherent) pseudomonads whose codomain is an identity pseudomonad is, equivalently, a left action of a (non-coherent) pseudomonad on the 1-cell $T$.
There is an "opped" version of this (in the horizontal direction) which gives you a notion of right action.

Amar Hadzihasanovic (Jul 27 2023 at 09:58):

In a left action, the 2-cell $\sigma$ goes from $CT \to T$, and in a right action it goes from $TC \to T$.

Amar Hadzihasanovic (Jul 27 2023 at 10:00):

Claim.
A non-coherent pseudomonad $(C, \mu, \alpha, \rho, \lambda)$ is, in fact, coherent if and only if $(C, \mu, \alpha, \lambda)$ determines a left action and $(C, \mu, \alpha, \rho)$ determines a right action on $C$.

Amar Hadzihasanovic (Jul 27 2023 at 10:05):

The proof is given by “erasing one 'side' of $T$ in the coherence equations that I drew before” which corresponds to instantiating $(D, \mu', \ldots)$ to an identity pseudomonad, and then instantiating $T$ to $C$ and $\sigma$ to $\mu$.

Amar Hadzihasanovic (Jul 27 2023 at 10:07):

For the one wrt the associator: 32b13a26-bbac-4ce2-bc49-62e19f364e15.jpg

Amar Hadzihasanovic (Jul 27 2023 at 10:07):

Now the one 3-cell that I circled in red becomes trivial -- so the “six-faced” equation loses one face. And that's exactly the pentagon equation!

Amar Hadzihasanovic (Jul 27 2023 at 10:08):

Similarly, for “coherence between $\sigma$ and unitors”, you would get 4-faced equations in the general case, and they will lose one of the faces in this instance, becoming the triangle equations!

Amar Hadzihasanovic (Jul 27 2023 at 10:10):

Second special case: $T$ is trivial.
If you have two (possibly non-coherent) pseudomonads on the same 0-cell $x$ (as is necessarily the case in a monoidal 2-category, which has a single 0-cell when seen as a tricategory), then you can consider the special case of morphisms where $T$ is equal to $1_x$.

Amar Hadzihasanovic (Jul 27 2023 at 10:15):

In this case, $\sigma$ has type $C \to D$, and what happens in string diagrams is that you “erase the wire $T$”.

In the coherence equations, you get this:

Amar Hadzihasanovic (Jul 27 2023 at 10:16):

aa1edf3d-77c7-44cd-9aa0-5b2a2b8ee881.jpg

Amar Hadzihasanovic (Jul 27 2023 at 10:17):

In the instance of “categories functors and natural transformations”, this is exactly the coherence equations saying that $\sigma$ is a strong monoidal functor.

Here there is no “trivialisation” so you get all six faces (and similarly all four faces for the “unitor” coherences).

Amar Hadzihasanovic (Jul 27 2023 at 10:18):

TL;DR:

• The pentagon equation & the coherence equation for monoidal functors wrt associators both arise as special instances of coherence equations for “strong morphisms of non-coherent pseudomonads”;
• Similarly for the triangle equations & the coherence equations for monoidal functors wrt unitors;
• The pentagon and triangle “lose” one side because it becomes trivial in this special instance.

Amar Hadzihasanovic (Jul 27 2023 at 10:26):

By the way, the notion of morphism of (non-coherent) pseudomonads is the same as for coherent pseudomonads which is why for example we don't have to think about the pentagon or triangle when defining a strong monoidal functor. Differences would only arise in higher dimensions, but here we truncate at 3.

Amar Hadzihasanovic (Jul 27 2023 at 10:28):

All these shapes of equations come from the combinatorics of the Gray tensor product. You can read about them in the second chapter of my thesis.

Amar Hadzihasanovic (Jul 27 2023 at 10:31):

In case you're not familiar with the string diagram notation I used, there's a book by @Dan Marsden and Ralf Hinze for you.

John Baez (Jul 27 2023 at 11:16):

Amar Hadzihasanovic said:

TL;DR:

• The pentagon equation & the coherence equation for monoidal functors wrt associators both arise as special instances of coherence equations for “strong morphisms of non-coherent pseudomonads”;
• Similarly for the triangle equations & the coherence equations for monoidal functors wrt unitors;
• The pentagon and triangle “lose” one side because it becomes trivial in this special instance.

Wow, that's great! So it's not a coincidence after all!

Matteo Capucci (he/him) (Jul 27 2023 at 16:04):

:clap: if this place had points, @Amar Hadzihasanovic, you would have won many with this answer

Kyle Wilkinson (Jul 27 2023 at 16:28):

Amar Hadzihasanovic said:

TL;DR:

• The pentagon equation & the coherence equation for monoidal functors wrt associators both arise as special instances of coherence equations for “strong morphisms of non-coherent pseudomonads”;
• Similarly for the triangle equations & the coherence equations for monoidal functors wrt unitors;
• The pentagon and triangle “lose” one side because it becomes trivial in this special instance.

Well, your answer is quite a lot more than I expected there to be! Very interesting and thank you.

Todd Trimble (Jul 31 2023 at 22:21):

I'd like to temporarily "unresolve" this just to point out that a lot of ideas underlying the connection were explored by Ross Street and Iain Aitchison around 40 years ago. In particular, lurking within Street's paper The Algebra of Oriented Simplexes is the combinatorics of associahedra, and the combinatorics of the polyhedral shapes associated with higher-order monoidal functors is lurking within a companion work The Geometry of Oriented Cubes by Aitchison. There are "Aitchison-Pascal" triangles which give rules for generating these polytopes. This is also connected with some old ideas of Dominic Verity for deriving parity structures on these polytopes in terms of some surface diagram theory he developed (and which I might be able to recall if I think enough about it). The PhD thesis of Sjoerd Crans is also relevant here, I believe.

I have to apologize for being so vague here; the trouble is, I've forgotten a lot of how this works and I doubt I can lay my hands on those handwritten notes by Aitchison (who is no longer in academe) that displays his Pascal triangle and his recursive rules. But it seems worth pointing out in case someone wants to take up the thread, and mining some treasure troves of all but lost ideas. My suggestion would be to get in touch with Street, who is much more organized than I am and surely still has a copy of Aitchison's diagrams. Meanwhile, if you guys know about certain secret tunnels of the "Russian underground", a doi for one of Street's papers which describes some of these ideas is doi:10.1007/BF00872948 (I'd link to it directly, but I don't know how this Zulip feels about such links).

So, it may be appropriate to mark this as unresolved after all, until I or someone else can revive the fading embers of such old thoughts...

John Baez (Jul 31 2023 at 23:10):

It's sad that one might need to resort to the "Russian underground" to get ahold of a paper published by Springer, but in this case it's not necessary since it's on Street's website:

• Ross Street, Higher categories, strings, cubes and simplex equations.

Christopher Tapo (Aug 01 2023 at 00:30):

The mention of combinatorics of the Gray tensor product has made me want to start going through a recent paper I saw on the combinatorics of the Gray cylinder by Paul Roy Lessard. There is also Christoph Dorn's thesis on associative n-categories that defines fully combinatorial models of higher categories, and conjectures that associative n-categories are the appropriate generalization of Gray categories.

Nick Gurski has also suggested to me the idea of operadic methods for categorical coherence problems. My response to this notion was to first find an appropriate decorated tree structure for things like transformations and modifications (utilizing work by Andrew Tonks, Stefan Forcey, and others). I thought it might be neat to turn some of the axioms from Gurski's book on tricategorical coherence into decorated tree versions, and included some below:

an axiom for trimodification

an axiom for tritransformation, part 1

an axiom for tritransformation, part 2

John Baez (Aug 01 2023 at 08:32):

Cool! These look like things @Todd Trimble would be interested in, since I believe he also created his definition of tetracategories with the help of trees.

Amar Hadzihasanovic (Aug 02 2023 at 07:08):

As someone who has been building on the ideas mentioned by @Todd Trimble for years, I take exception to the notion that they are "lost" or "forgotten". But I have noticed that this thought is widespread among the "first generation" who has worked on these ideas. Rather, I think that they have been passed through the reformulation developed by Richard Steiner in the 90s and 00s, at which point they may have become not immediately recognisable to their originators.

In addition to my own work, only in the past 5 years I can think of work by Ara, Maltsiniotis, Ozornova, Rovelli, Gagna, Forest, Maehara, and certainly others that I have not mentioned, which descends from Street and Aitchison. Not forgotten and very much active!

Amar Hadzihasanovic (Aug 02 2023 at 07:13):

I think one problem in communicating these developments is that, due to their combinatorial nature, they are often relegated to "technical" sections of papers whose intent is manifestly to prove some higher-categorical or homotopy-theoretic result. So it is easy to miss them!

I think combinatorially-oriented higher category theorists should more decisively invade the field of combinatorial and computational topology.

Todd Trimble (Aug 02 2023 at 10:25):

@Amar Hadzihasanovic In that case, please accept my apologies (and this is very good to hear from you). I will try to educate myself better, and particularly with regard to Richard Steiner, whom I admire but admittedly somewhat from afar (I've been meaning to learn better his work on (augmented) directed complexes, which I am told is a big improvement on earlier work by Street and Power on parity complexes and pasting schemes; they are said to be much easier to work with.

Amar Hadzihasanovic (Aug 02 2023 at 10:31):

No apology needed, I am sorry if my message came across as demanding one; there is clearly an imperfect communication of the fact that this kind of work exists, which is not the fault of anyone who "hasn't been informed".

Amar Hadzihasanovic (Aug 02 2023 at 10:37):

Steiner's work on ADCs greatly elucidated the combinatorics of Gray products which generate the "oriented cubes", in comparison to Crans's thesis; then Ara and Maltsiniotis used Steiner's theory to do the same with "oriented joins", which generate the oriented simplices.

I would like to think that the Gray product and join of "regular directed complexes" that I've worked on are a further step towards understanding these constructions, as they connect functorially the operations on n-categories to their "topological" counterparts (cartesian products and joins of CW complexes).