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Stream: learning: questions

Topic: Baas's hyperstructures

David Corfield (Apr 24 2026 at 11:19):

Has anyone in this community taken up Nils Baas's ideas on [[hyperstructures]]? Since he was looking for a way to capture hierarchies of bonded entities without resorting to partitioning links into inputs and outputs, the idea would seem well-suited to ACT interests.

There was some interest at The $n$-Category Café, a good while ago, here, ideas worked by Urs Schreiber into this.

On Zulip, I only see reference made in this comment by @Adittya Chaudhuri here.

Adittya Chaudhuri (Apr 24 2026 at 18:31):

@David Corfield Thanks for posting this. I am also very curious about it. I have read some of the papers written by Baas, and idea of hyperstructures sounds very interesting to me.

John Baez (Apr 24 2026 at 23:06):

I never saw enough concrete work on these ideas to get me interested, I'm afraid.

David Corfield (Apr 25 2026 at 07:52):

Urs was looking to take things in the direction of iterated multispans or iterated multi-cospans. The idea of iterated spans is certainly thriving, e.g.,

• Rune Haugseng, Iterated spans and classical topological field theories, arXiv:1409.0837

But I see no sign of a multispan version.

Evan Patterson (Apr 25 2026 at 19:35):

Likely I'm missing something important here, but when I read

The point is essentially that the directedness of morphisms and — related to that — the binary notion of source and target in categories and higher categories are notions alien to these contexts, which in applications have to and are essentially removed again in a second step by adding extra structure and requiring further properties, such as various monoidal structures and dualities, which allow to change the direction of morphisms, to collect objects together, etc.

on the [[hyperstructure]] page, I think of the operadic approach to open systems, which has long been an important thread in ACT. It seems to me that the operadic approach (especially the double-operadic approach) nicely fixes the quoted problem.

What would hyperstructures add to the story? Are they about going from sets or categories of systems to higher categories of systems?

David Corfield (Apr 25 2026 at 19:57):

Right, you'd think operads were relevant. But as a prominent algebraic topologist for decades, Baas must have been totally at home with operads, and yet was he still looking for something more. Is it the iterative issue as well as the symmetry of the relata? No directedness of $n$ to $1$, and then also the hierarchical issue.

The iterated spans of Haugseng and others are pointing to higher categories. Is it that an iterated multispan construction points to $n$-tuple categories?

I need to see if I can understand it, but I was wondering if we're seeing a glimpse of this in a comment in the recent Day convolution for algebraic patterns:

David Corfield (Apr 26 2026 at 08:47):

I guess a good place to look is to cyclic and modular operads:

A modular operad is, roughly speaking, an operad-like structure where operations can have multiple inputs and outputs, but there is a duality on the set of objects that allows us to replace an input object by its dual as an output object. (Enriched homotopy-coherent structures)

Clémence Chanavat (Apr 26 2026 at 14:34):

I don't know about hyperstructures, but I can comment on this iterated tree construction if it's of any help: the same way categories are presheaves over $\Delta$ satisfying the Segal condition, there is a category $\Omega^n$ constructed by this iterated tree construction, the presheaves over it satisfying the Segal condition are the virtual n-uple categories.

Clémence Chanavat (Apr 26 2026 at 14:36):

Intuitively, the objects of $\Omega^n$ are some "cubical pasting diagrams", like for instance if n=3, the one found in the slides of Nathanael mentionned at the end of the Example

Clémence Chanavat (Apr 26 2026 at 14:42):

At the end of the day, those pasting diagrams are indeed encoded by something which could be called an iterated multispan, however each "node" has a well defined notion of input and output (they are all molecules)

David Corfield (Apr 26 2026 at 15:40):

Thanks, Clémence!

I see in

• Doherty & Hackney, Models for cyclic infinity operads,

forms of undirected composition:

cyclic dendroidal spaces admit weakly defined m-ary cyclic-operad-style compositions (see Figure 1)

Is there any chance for some iterative construction here?

Adittya Chaudhuri (Apr 26 2026 at 17:49):

@David Corfield Although I am yet to look into the technical details of cyclic infinity operads and cycle dendroiodal spaces, this diagram reminds me of a construction (that I called macroscope of process network) in my paper A mathematical framework to study organising principles in graphical representations of biochemical processes (j/w Wolkenhauer and Köhl). I am not sure whether it be very relevant to the current discussion, but somehow, a mental picture of a bond (after reading Bass's paper On the Philosophy of Higher Structures that time (2024)) kind of motivated me to define a notion which I named Process species which looks quite similar to your picture :

In this this picture, $b$ can be thought as a process, and the $k_{th}$ leg ($k_{th}$ long line segment) can be thought as a leg of type $k$. On the other hand, the short line segments attached to the $k_{th}$-leg are the entities associted to the $k_{th}$ leg. There may be no entities attached to the leg (for example the $5_{th}$ leg in the picture), and in that case , I showed rigorously that it can be safely ignored without loosing any mathematical consistency (Proposition 4.5 in my paper). Now, a process network is a collection of these process species. Formally, I defined a process network as the following:

Now, I developed a method called macroscope of a process network (Section 4.3 in my paper), which converts a process network into a process species:

My inspiration was to think of a process network as a biochemical reaction network where each leg plays a different biochemical role like input, output, activation, inhibition, necessary stimulation, unknown influence, etc. Now, when, I apply my macroscope on a particular portion (a process subnetwork) of a process network, then, we can safely forget what is happening inside the process subnetwork, and can only focus on the what goes inside the subnetwork in what kind of way and what goes outside the subnetwork. In this way, I was trying to think of it as a mathematical Zooming-out tool that can be applied to a biochemical reaction network, for eg:

Although, I am yet to investigate how good my macroscope behaves from compositional perspective.

So, I was wondering any similarity between my macroscope construction and the $m$-ary cyclic-operad-style composition $f \circ g \circ h$ in your picture.

David Corfield (Apr 26 2026 at 18:12):

Evan Patterson said:

What would hyperstructures add to the story? Are they about going from sets or categories of systems to higher categories of systems?

I didn't address this question. Baas is looking to account for hierarchically organised structures, e.g., in On the Philosophy of Higher Structures:

A biologist who has spoken at Topos and who has spoken of his hopes for the usefulness of CT, Michael Levin, included this diagram in one of his papers:

Chis-Ciure, R., Levin, M., Cognition all the way down 2.0: neuroscience beyond neurons in the diverse intelligence era, Synthese 206, 257 (2025).

If possible to model mathematically, I'd imagine we'd need more than (double) operads, rather something more hierarchically arranged.

Adittya Chaudhuri (Apr 26 2026 at 18:48):

David Corfield said:

If possible to model mathematically, I'd imagine we'd need more than (double) operads, rather something more hierarchically arranged.

I struggle to understand how to formally define a hierarchy Level $0$, Level $1$, $\ldots$, Level $n$, such that

1. each layer can be formally characterised so that we can distinguish the elements between one layer and the other,
2. each element of level $k+1$ represents an aggreate of elements of level $k$,
3. The composition law prevelent in level $k$ (that lets us compose the elements within level $k$) differs from the composition law present in level $m \neq k$. I would like to think that the composition law in level $k+1$ is an emegent property of the composition law in level $k$.

Rémy Tuyéras (Apr 27 2026 at 00:04):

The [[biology]] page on the nLab makes the application of [[hyperstructures]] more explicit: see https://ncatlab.org/nlab/show/biology#hyperstructures

From my discussion with Nils, this was one of the intended applications of hyperstructures.

Rémy Tuyéras (Apr 27 2026 at 00:20):

Not a formal proof, but the intuitive construction can itself be encoded operadically, as outlined in this paper: https://www.biorxiv.org/content/10.1101/2021.06.25.449951v1.full.pdf

David Corfield (Apr 27 2026 at 07:57):

Adittya Chaudhuri said:

I struggle to understand how...

To understand these features jointly or separately? (1) is straightforward by itself. We see it in higher categories, such as a 2-category organised as objects (Level 0), morphisms (Level 1), 2-morphisms (Level 2). A 2-morphism is not an object nor a 1-morphism, etc.

As for (2), I'm not sure what you want "represents an aggregate to mean". Baas wants it to be about "bonding" the lower level elements. Then $k+1$-morphisms represent some kind of bonding of source and target $k$-morphism. In the double category $\mathbb{S}pan(Set)$, the loose 1-morphisms, the spans, are bonding objects, etc.

(3) Composition laws differ at different levels, e.g., span composition by pullback is not 2-cell composition in the tight direction.

As for your final sentence

I would like to think that the composition law in level $k+1$ is an emergent property of the composition law in level $k$,

I don't see why we'd want this. I don't expect what bonds teams in a league to emerge from what bonds players in a team, do I? Or what bonds letters into a word, and what bonds words in a sentence?

David Corfield (Apr 27 2026 at 09:38):

Rémy Tuyéras said:

Not a formal proof, but the intuitive construction can itself be encoded operadically, as outlined in this paper

This is very interesting, Rémy. But it still seems to me that Baas is after something more like the iterated operadic construction I mentioned above. @Clémence Chanavat pointed us to the end of @Nathanael Arkor's talk, where we see this kind of multi-level composition:

The final two slides explain the iterative process (slides 34 & 35). I think Baas is after this kind of thing with no directionality in the composition within a level.

Rémy Tuyéras (Apr 27 2026 at 10:34):

@David Corfield

[...] can have multiple inputs and outputs, but there is a duality on the set of objects that allows us to replace an input object by its dual as an output object.

[...] I think Baas is after this kind of thing with no directionality in the composition within a level.

Not sure this fully aligns with what you have in mind for modeling Nils' structures, but just to share a perspective: in category theory, the ability to turn inputs into outputs often comes from some form of arrow reversal (for example, when passing to the homotopy category).

From that angle, reversing directionality in compositions might correspond to equipping the underlying structure with something like a factorization system (meant here in a loose, intuitive sense rather than a formal one.)

David Corfield (Apr 27 2026 at 12:04):

Yes, there are dagger-like constructions where legs can be bent around. That's an approach to cyclic operads where one has $n+1$ legs and invariance under rotation, and chooses one of the legs to bend it round as output. But there are yet more "democratic" approaches where one sticks with certain presheaves over graph structures rather than trees.

The kind of thing defined in

• Philip Hackney, Marcy Robertson, Donald Yau, A graphical category for higher modular operads [arXiv:1906.01143]

V Slicer (Apr 27 2026 at 17:22):

David Corfield said:

Yes, there are dagger-like constructions where legs can be bent around.

Is this related to [[quantum+information+theory+via+dagger-compact+categories]] and the [[ZX-calculus]] approach by Bob Coecke and Duncan using spiders?

David Corfield (Apr 27 2026 at 20:24):

Yes, that world of [[compact closed categories]] and variants.

David Corfield (Apr 28 2026 at 09:34):

To sum up where I reached, I'm wondering whether there's a way to imitate the iterated tree construction above in the shape of an iterated (undirected) graph construction.

Is there an undirected graph of algebraic patterns construction with active-inert factorization in the realm of

• @philip hackney, Categories of graphs for operadic structures [arXiv:2109.06231] ?

Adittya Chaudhuri (Apr 28 2026 at 13:35):

David Corfield said:

I don't see why we'd want this. I don't expect what bonds teams in a league to emerge from what bonds players in a team, do I? Or what bonds letters into a word, and what bonds words in a sentence?

Thanks ! Yes, I got your point. I agree now that it may not make much sense to say: I would like to think that the composition law in level $k+1$ is an emegent property of the composition law in level $k$

Adittya Chaudhuri (Apr 28 2026 at 13:36):

David Corfield said:

To understand these features jointly or separately?

I meant to say jointly.

Adittya Chaudhuri (Apr 28 2026 at 14:15):

@David Corfield What I have vaguely in my mind is a kind of infinity category, where each level has its own composition laws, and an element in the $(k+1)_{th}$ level can be built from the composition of compatible elements in the $k_{th}$ level. I may be wrong but I would like to imagine a filler $2$-morphism as an aggregate of the two composable $1$-moprhisms.

David Corfield (Apr 29 2026 at 10:10):

@Adittya Chaudhuri Have you seen these papers (I and II) on Layered Monoidal Theories by Leo Lobski and Fabio Zanasi?

Adittya Chaudhuri (Apr 29 2026 at 21:32):

Thanks a lot for sharing the papers. No, I have not seen these papers before. I will look into them. Just a very minor comment, the second link is not working. Probably, you meant this.

David Corfield (Apr 30 2026 at 06:22):

Fixed, thanks.

David Corfield (May 05 2026 at 14:11):

From nosing about, I get the sense that what I'm after is something like the result of making the move from Segal spaces to decomposition spaces (weakening the Segal condition) but now starting with a different shape, some graphical algebraic pattern, resulting in what might be called a 'graphical decomposition space'.

Then the iteration of some such construction would allow for (undirected) networks of networks of ... for modelling hierarchical systems with mutual constraints across levels.

Rémy Tuyéras (May 05 2026 at 19:05):

Nice!

Would you say that the approach considers functors

$F:\Gamma^{\mathsf{op}} \to \mathbf{Top}$

where $\Gamma$ is the small category of finite graphs whose nodes are $[n]=\{0,1,2,\dots,n-1\}$ and whose edges are subsets of $[m]$, provided that we equip each graph with a source-target map $[m]\to [n]×[n]$ ?

David Corfield (May 06 2026 at 08:26):

I was looking more towards the category of graphs, denoted $\mathbf{U}$ in A graphical category for higher modular operads:

Variants are described in Categories of graphs for operadic structures:

David Corfield (May 06 2026 at 08:30):

$\mathbf{U}$ has a factorization of morphisms into active and inert maps which can be used to define presheaves on $\mathbf{U}$ satisfying the Segal condition. This gives rise to modular operads.

David Corfield (May 06 2026 at 08:59):

But I sense there's some iterable process which relies on some kind of decomposition space on graphs (weakening the Segal condition) that will allow for networks of networks...

In the $\Omega^n$ case mentioned above, Segal conditions work because the shapes (rooted trees) canonically encode directed composition, so global structure is uniquely determined by composing local pieces. In this potential undirected graph case, graphs don't carry a canonical notion of composition, but rather of decomposition via subgraphs and contractions, so imposing a Segal condition would force arbitrary choices of direction or root. Decomposition conditions, formulated using the inert–active factorization system, instead capture the intrinsic combinatorics of graphs by requiring compatibility across all ways of cutting them apart.

I'm hoping to arrive at a hierarchy where at each level there is no canonical direction or composition, but there is interactions between levels.