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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.

David Kern (May 07 2026 at 14:33):

David Corfield said:

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

In All Segal objects are generalised monads in spans, section 6.5, I get multispans with their operadic composition as the kind of span categories coming naturally from the algebraic patterns for cyclic and modular operads, as well as plain operads (though of course, in the latter case with a distinguished output leg), but I have not thought at all of whether to get some iterated version from feeding an iterated tree construction into the machinery.

One thing that might be helpful for the issue of “forc[ing] arbitrary choices of direction or root” is the upcoming result from Barkan–Steinebrunner (cf. Theorem 3.19 in Steinebrunner's lecture notes, which also refer to Hackney's hierarchy in Example 3.6) characterising modular $\infty$-operads as the $\infty$-properads all of whose objects are dualisable. This shows that you can work with a more directed kind of structure (the algebraic pattern for properads also consists of level trees, just of cospans of pointed finite sets instead of functions of pointed finite sets) and still recover undirected objects by imposing some further conditions rather than changing the shape.

David Corfield (May 08 2026 at 08:53):

Thanks, @David Kern!

David Kern said:

...but I have not thought at all of whether to get some iterated version from feeding an iterated tree construction into the machinery.

Then you could pose my question as to whether there's an iterated (undirected) graph construction. Is there an operator (endofunctor/2-endofunctor?) on $AlgPatt$ which instead of taking $O$ to $\Omega(O)$ (trees in $O$), takes it to $\Gamma(O)$ (undirected graphs labelled in $O$), such that $\Gamma(\ast)$ is Hackney et al.'s $U$. Then iteration would give graphs of graphs...

I can see some worries with multiple levels about contraction generating loops at different levels, so that the order of contraction would matter. But that perhaps tallies with the world. When a company merges two internal roles and at the same time merges with another company, one could foresee noncommutativity.

I could also imagine that taking the decomposition space/2-Segal space condition (in some analogous graphical sense) on presheaves over the resulting algebraic pattern would be better suited to applications in say biology.

David Kern said:

This shows that you can work with a more directed kind of structure ... and still recover undirected objects by imposing some further conditions rather than changing the shape.

I believe the hope of Nils Baas was to avoid such detours, and go immediately to the undirected and its iterations.

David Corfield (May 08 2026 at 10:50):

But that's no doubt too unsubtle a strategy, and I should really be reading up about fibrous patterns, enrichable patterns and envelopes (arXiv:2208.07183, arxiv:2308.11502).

David Kern (May 08 2026 at 13:33):

One of the things that Blom–Loubaton–Ruit show in their paper (though the idea is older, from the long history of “plus constructions”) is that fibrous patterns over some base pattern $O$ are the same as complete Segal $\Omega[O]$-$\infty$-groupoids, so I don't think there is anything fibrous patterns could do that trees can't. If you want to get an equivalence with something more graphical, I imagine you would have to replace your (fibrous) patterns by something less directed than a category, whatever that could be (presumably dagger $\infty$-categories would be too “unsubtle” a replacement to do the trick...).

David Corfield (May 11 2026 at 15:29):

From some further rooting about, my best attempt at an undirected graphs of graphs construction is what Chu and Haugseng in Enriched homotopy-coherent structures would call $\mathbb{U}^{op, \natural} \wr \mathbb{U}^{op, \natural}$, constructed by taking the wreath product of their Def 3.5.6, and substituting in each place, $\mathbb{U}^{op, \natural}$, it being shown in Lemma 3.6.13 that this is an enrichable pattern.

This wreath product is iterable.

Rémy Tuyéras (May 11 2026 at 17:24):

@David Corfield Do you have an example of the kind of hierarchical structures you have in mind? I'm curious where the undirected logic becomes necessary in the structure and how it is meant to apply.

Even a simple but concrete biological example would be very helpful.

David Corfield (May 11 2026 at 18:39):

Any situation where there are distinct layers each potentially governed by symmetric-coupling and cycle-formation.

I have little biological knowledge. I'll ask Claude, and we can ignore it if it makes little sense:

David Corfield (May 11 2026 at 18:48):

Looking purely mathematically, there's an interesting array of moves to compose a network of networks, with graph substitution and edge contraction at outer and inner levels. Perhaps it wouldn't be too much of a stretch to see biological structures forming through such steps.

David Corfield (May 12 2026 at 08:00):

The wreath issue is about how to treat hierarchical levels well together. If we want a directed version of a 2-level system of systems we can opt instead of $U$ for Hackney's directed version $U_{/\mathfrak{o}}$. This as he points out in 5.1 of Categories of graphs for operadic structures is effectively the same pattern that @Joachim Kock use in his 'Whole-grain Petri nets'.

So we could have a directed network of directed networks, etc., just as easily.

John Baez (May 12 2026 at 09:06):

@Rémy Tuyéras

Do you have an example of the kind of hierarchical structures you have in mind? I'm curious where the undirected logic becomes necessary in the structure and how it is meant to apply.

Yes, I'm unable to focus on this conversation because I don't know an example of the sort of thing that David is trying to formalize.

Even a simple but concrete biological example would be very helpful.

For me any sort of example at all would be helpful. And it doesn't need to be "realistic", though that would make it more interesting.

@David Corfield

Any situation where there are distinct layers each potentially governed by symmetric-coupling and cycle-formation.

Can you describe an example of such a thing precisely enough so I can get an idea of what kind of thing you mean?

David Corfield (May 12 2026 at 11:40):

John Baez said:

Can you describe an example of such a thing precisely enough so I can get an idea of what kind of thing you mean?

As I said in my previous message the directed/undirected issue is independent from the 2-level issue. The choice of algebraic pattern determines activity within a level; the use of the wreath construction makes the level structure work properly. We can see this latter inter-level bookkeeping at work with the [[categorical wreath product]] of the simplex category providing a pattern for higher categories.

So here I'm looking to generate a levelled structure for a pattern that's not the simplex category, rather something graph-like. I was initially looking at Hackney's pattern of undirected graphs, but as I said we could also look at other related patterns, such as simply connected graphs or directed graphs.

So let's consider a situation which we might want to model by (undirected) graphs of graphs. We could have a network of companies, where each company has a network of job roles. An edge between two companies might represents some form of interaction, perhaps trading. An edge between roles within a company might represent whether collaboration is needed. At each level there could be cycles. There may also be loose edges representing potential trading opportunities of a company or potential collaboration opportunities for a company role.

The maps available in $U \wr U$ would allow us to represent various processes. An obvious example would be the merger of two trading companies. Start with outer graph a pair of nodes, trading companies $1$ and $2$, connected by an edge. Then at each node we have a graph of roles, $G_1$ and $G_2$. This is an object in $U \wr U$. Consider now a map which contracts the outer edge, merging the two trading companies. To complete the description of the map, we would also need to specify a $G_3$, roles in the new company, and maps $G_1 \to G_3$ and $G_2 \to G_3$. This would dictate how the roles of each of the two companies would be carried over to those of the newly merged company. This could generate more complex structures in the merged company, such as new cycles.

David Corfield (May 12 2026 at 12:04):

For an enthusiastic introduction to algebraic patterns from an ACT practitioner, I recommend @David Jaz Myers in a Topos Berkeley seminar and also a 2-torial. These are where I first glimpsed how powerful this technology could be for us.

There's a natural step, I know David Jaz was considering, in representing these active-inert factorizations of algebraic patterns as the loose-tight morphisms of a double category. This has been worked out in Day convolution for algebraic patterns as the concept of a double pattern. This keeps the compositional aspect neatly separate from the bookkeeping management of variables aspect.

Adittya Chaudhuri (May 13 2026 at 07:15):

@David Corfield Hi! I am not fully sure whether this example (based on regulatory netoworks) could be relevant here. Recently, I came acorss an interesting paper Gene regulatory networks: from correlative models to causal explanations, where authors talked about three levels of descriptions (claimed to be inspired from Marrs' levels of analysis Vision and From Understanding Computation to Understanding Neural Circuitry) for a how GRN process the signals it receives from the signal transduction network for the tissue morphogenesis. Out of the three levels, I want to talk about the first two levels, which I am finding to be possibly closely relevant to your example.

1. Implementation level: Here authors started with a directed signed graph representation of GRN and partioned the vertex set into modules, which allows us to decode how these modules of genes are regulating each other by noting how the genes present in one module are regulating the genes present in the other modules.
2. Representation level: Here, they are building a logical model (using logic gates), where the components are open modules (modules derived in the implementation level equipped with input and output), incoming signals from the signal transduction pathway and the outgoing signals (signalling molecule). The thing that I find interesting is that the interconnection pattern in this level is substantially distinct in flavour from that of the implementation level. Although formalisation of these concepts are not present in that paper, I am trying to think of it as map $\mathsf{Syntax} \to \mathsf{Semantics}$, where $\mathsf{Syntax}$ is the Implementation level and $\mathsf{Semantics}$ is the representation level. In natural way, I think this argument can be generalised to more than two levels.

The description that I discussed on levels is in the Figure 3 of Gene regulatory networks: from correlative models to causal explanations, and some open questions in this direction is discussed in Box 1, some of which I find quite interesting to think about.

David Corfield (May 14 2026 at 08:25):

@Adittya Chaudhuri I wonder if perhaps the technology of Layered Monoidal Theories, I and II, might be better suited. Layers there correspond to different 'theories', which seems close to the idea in the paper you mention:

In the papers I mentioned, we're seeing something like that, e.g., in the Electric Circuits of Example 5.2, here's a diagram of interlevel representation:

However, it seems they're not considering multiple realisability of macrostate by microstate (I, p. 45). So it's more a question of refining a theory:

Adittya Chaudhuri (May 14 2026 at 10:18):

Thanks!

philip hackney (May 14 2026 at 18:20):

since there's some interest in generalizations of decomposition spaces here, I wrote up this little post after talking with @Brandon Shapiro and @Eleftherios Chatzitheodoridis this week: against active-inert pushouts. interestingly it's not always the case that segal => decomposition, which I find pretty interesting!

David Corfield (May 15 2026 at 08:21):

Thanks, @philip hackney! I thought I'd come across this issue in a recent paper. Did I glean correctly from Burkin's Twisted arrow categories, operads and Segal conditions that for a pattern describable as the twisted-arrow category of an operad, then if that operad is palatable then, segal => decomposition for the pattern?

David Corfield (May 15 2026 at 10:21):

As for what I was hoping for from generalized decomposition spaces, such as graphical decomposition spaces, it's all rather exploratory. Something that has struck me is the connection between your oriented graphical algebraic pattern, $U_{/\mathfrak{o}}$, and the work of @Joachim Kock on 'whole-grain Petri nets'. Petri nets are at the heart of many ACT approaches. It's led me to wonder how we might bring over algebraic pattern technology to the applied category theoretic study of systems. Perhaps other Segal conditions might feature in applications.

Another line of generalization occurred to me yesterday. A few months ago we had here a lengthy discussion on Symmetries of Living Systems, which makes great play of a certain kind of directed graph morphism they call symmetry fibration. A morphism of this kind identifies nodes with isomorphic input trees, e.g,

It looks like this could be seen as a special kind of active morphism in your $U_{/\mathfrak{o}}$. That got me wondering whether it might be possible to factor general active directed graph morphisms into symmetry fibrations and some other class. But, piling speculation upon speculation, since we certainly see [[ternary factorization systems]] in nature, might there be one here? The nLab page [[category of simple graphs]] describes a ternary factorization in the domain of graphs.

That raises a further question as to whether there might be an algebraic pattern analogue for ternary factorizations.

philip hackney (May 15 2026 at 10:22):

David Corfield said:

Thanks, philip hackney! I thought I'd come across this issue in a recent paper. Did I glean correctly from Burkin's Twisted arrow categories, operads and Segal conditions that for a pattern describable as the twisted-arrow category of an operad, then if that operad is palatable then, segal => decomposition for the pattern?

Yes, that's right. Palatable is defined in that paper exactly as Segal => decomposition for the associated pattern.

In hindsight of course one wouldn't expect Segal => decomposition to be true generally, since choosing every object to be elementary then makes every presheaf Segal. (algebraic patterns are very flexible) But the failure in the "double categories" case seems meaningful to me.

philip hackney (May 15 2026 at 11:06):

David Corfield said:

As for what I was hoping for from generalized decomposition spaces, such as graphical decomposition spaces, it's all rather exploratory. Something that has struck me is the connection between your oriented graphical algebraic pattern, $U_{/\mathfrak{o}}$, and the work of Joachim Kock on 'whole-grain Petri nets'. Petri nets are at the heart of many ACT approaches. It's led me to wonder how we might bring over algebraic pattern technology to the applied category theoretic study of systems. Perhaps other Segal conditions might feature in applications.

This would be very cool.

Another line of generalization occurred to me yesterday. A few months ago we had here a lengthy discussion on Symmetries of Living Systems, which makes great play of a certain kind of directed graph morphism they call symmetry fibration. A morphism of this kind identifies nodes with isomorphic input trees, e.g,

It looks like this could be seen as a special kind of active morphism in your $U_{/\mathfrak{o}}$. That got me wondering whether it might be possible to factor general active directed graph morphisms into symmetry fibrations and some other class.

A quick glance at the book and the picture makes me feel that this could be considered as an active map (in the reverse direction) in a non-connected version of $U_{/\mathfrak{o}}$. (The non-connectedness is about the fibres of nodes not being connected).

Maybe if your category consists of graphs equipped with an "equitable partition" then these symmetry fibrations would just be the active maps.

But, piling speculation upon speculation, since we certainly see [[ternary factorization systems]] in nature, might there be one here? The nLab page [[category of simple graphs]] describes a ternary factorization in the domain of graphs.

That raises a further question as to whether there might be an algebraic pattern analogue for ternary factorizations.

Many of the algebraic patterns I know are part of a ternary factorization system. But in those cases the second factorization system is the one for a generalized Reedy category, which doesn't seem to have the kind of flavor you're looking for.

David Corfield (May 15 2026 at 13:31):

philip hackney said:

a non-connected version of $U_{/\mathfrak{o}}$.

So this is $(U_{/\mathfrak{o}})^{\coprod}$ from Enriched homotopy-coherent structures, Sec. 3.5?

philip hackney (May 15 2026 at 18:54):

I don't think that's the same – you do get disconnected graphs as your objects, but the morphisms are not quite right. You're not able to send a node to a disconnected graph in $(U_{/\mathfrak{o}})^{\coprod}$. (This disconnected version of $U_{/\mathfrak{o}}$ I'm thinking about here doesn't appear anywhere, yet, but is analogous to what's described in Question 5.9 of Segal conditions for generalized operads.)

Also looking back at the picture you posted from Symmetries of Living Systems, I was too quick to say that its reverse is in this disconnected graphical category. A fundamental feature of all of these "operad-y" graph categories is that they preserve cardinalities of input / output edges. This isn't the case for this example (comparing the blue vertices on the bottom).

David Corfield (May 16 2026 at 18:27):

Thanks, Philip. Time for me to knuckle down and do some reading.

David Corfield (May 18 2026 at 09:50):

How close is Sophie Raynor's $\Xi^{\times}$ from Modular operads, iterated distributive laws and a nerve theorem for circuit algebras to providing an answer to your Question 5.9 for $\mathbf{U}_{\textrm{dis}}$, @philip hackney? It's a category of undirected graphs, which is an extendable algebraic pattern (p. 55) and "a presheaf $P$ on $\Xi^{\times}$ is equivalent to the nerve of a circuit algebra if and only if it is “Segal” (Theorem 8.4).