Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: How to compose systems of different `types' using ACT ?

Adittya Chaudhuri (Jan 27 2026 at 15:24):

It may be possible that we may need to compose system of different types, for example composing a Petri net with a monoid labeled graph, composing a differential equation based dynamical system with a Boolean dynamical system, etc. Below, I briefly describng a scenario where such composition can be natural.

Let us assume we are studying interaction among large number of biochemical entities. It may happen that due to the availability-unavaibility of data, some of the interactions are possible to be modeled as reaction networks with rates and some are possible to be modeled only as signed/monoid labeled graphs. Similarly, at the level of dynamics, we may need to model the dynamics of some interactions using logic and some using O.D.E. Thus, the whole system can be seen as a hybrid system : a mix of Petri nets and signed graphs, and a mix of Boolean networks and O.D.E's.

In ACT frameworks, I think usually we comopose systems of similar nature i.e

• Petri nets with rates with Petri nets with rates
• Petri nets with Petri nets
• Signed graphs with signed graphs
• O.D.E with O.D.E
• .
• . so on

My question is the following:
Are there existing ACT-based frameworks which allow composing networks of different types? In that case, it seems like we need to compose a morphism $f$ of a categoy $C$ with a morphism $g$ of category $D$. More broadly, how to compose models of different types using ACT ?

Chad Nester (Jan 27 2026 at 15:46):

While there seems to be an obvious answer, I'm not sure it's what you're looking for. In any case:

Let's say that our "types of system" are each some sort of category, and that our individual systems of some type are morphisms in that category. For example, $f$ a morphism of $\mathbb{A}$ and $g$ a morphism of $\mathbb{B}$. One way to reason about $f$ and $g$ together is to embed them into some common domain $\mathbb{C}$ via functors $A : \mathbb{A} \to \mathbb{C}$ and $B : \mathbb{B} \to \mathbb{C}$.

Chad Nester (Jan 27 2026 at 15:48):

Of course, for this to work you need a suitable $\mathbb{C}$, being a "type of system" that is flexible enough to accommodate both $\mathbb{A}$ and $\mathbb{B}$. In a perfect world you would also have that $A$ and $B$ have nice properties. For example, it would be very nice if they were faithful.

Kevin Carlson (Jan 27 2026 at 17:22):

Yes, and there is a particularly common case where $\mathbb{C}=\mathbb{A}$ or $\mathbb{C}=\mathbb{B}.$

Adittya Chaudhuri (Jan 27 2026 at 17:36):

Thanks @Kevin Carlson @Chad Nester !! I got the point. Then, by a nice functorial relationship between the corresponding categories of two different systems (more generally, with respect to another common category where the corresponding categories can be embedded) we can build a composable framework for the hybrid system.

Adittya Chaudhuri (Jan 27 2026 at 17:52):

But, what if in reality we can not find suitable $A \colon \mathbb{A} \to \mathbb{C}$ and $B \colon \mathbb{B} \to \mathbb{C}$?

For example, a common hybrid system that is used in the modelling of biological networks is the hybrid of Boolean model and O.D.E based model. In that case, I am not sure how to find suitable $A \colon \mathbb{A} \to \mathbb{C}$ and $B \colon \mathbb{B} \to \mathbb{C}$.

Kevin Carlson (Jan 27 2026 at 17:57):

Hybrid dynamical systems can often already be placed in their own category in a perfectly natural way, depending on the formalism, which you haven't quite specified. The simplest case is of simply allowing differential equations with some discontinuities, as in the bouncing ball system. There is no big problem to building $\mathbb{C}$ as the category of all such differential equation systems. But you may mean something quite different. There's no general answer to constructing $A$ and $B$ that doesn't look at the details of a given case without degrading into triviality.

Adittya Chaudhuri (Jan 27 2026 at 18:52):

Thanks!! I got your point.

John Baez (Jan 27 2026 at 21:46):

In my talks these days I always advertise composing systems of different types by mapping two (double) categories to a common (double) category, as Chad suggests. Following @Evan Patterson, I call this 'interoperability'. My favorite example is composing Petri nets with rates and stock-flow diagrams by mapping them both to dynamical systems:

• Modeling with applied category theory (page 30 of 31).

John Baez (Jan 27 2026 at 21:47):

Though this should be rather easy in the AlgebraicJulia framework, I don't know of anyone actually doing it there yet!

Adittya Chaudhuri (Jan 28 2026 at 05:57):

John Baez said:

In my talks these days I always advertise composing systems of different types by mapping two (double) categories to a common (double) category, as Chad suggests. Following Evan Patterson, I call this 'interoperability'. My favorite example is composing Petri nets with rates and stock-flow diagrams by mapping them both to dynamical systems:

• Modeling with applied category theory (page 30 of 31).

Thanks a lot. This is exactly what I was asking. I was not aware of this natural idea of interoperability in ACT before.

John Baez (Jan 28 2026 at 06:49):

I wish people would start actually doing it!

David Corfield (Jan 28 2026 at 09:47):

Has much been done on theory or operad mappings? In Operadic Modeling of Dynamical Systems: Mathematics and Computation, the authors consider three systems theories:

image.png

They discuss a functor from $Th(DWD)$ to $Th(CPG)$, engendering an adjoint triple between $[Th(DWD), Set]$ and $[Th(CPG), Set]$. There's a corresponding operad $CPG$ which is a suboperad of $DWD$.

Has anything been done at a general level about the collection of such theories/operads? When are there colimits of such things, etc.?

David Corfield (Jan 28 2026 at 09:54):

Could we see @Sophie Libkind's amalgamation of the directed and undirected settings in An Algebra of Resource Sharing Machines in terms of a colimit?

David Corfield (Jan 28 2026 at 10:14):

Obviously people have studied things like the category of operads (for some flavour) and can prove things like:

the category Operad is small complete and small cocomplete, (Weiss, Operads)

and also a symmetric closed monoidal category .

But I'm wondering more about in the ACT context of systems.

Chad Nester (Jan 28 2026 at 14:18):

A related question: I wonder how much is known about sorts of structured category for which the amalgamation property holds.

Chad Nester (Jan 28 2026 at 14:24):

I guess the higher-level question of when we can "amalgamate" different sorts of structure on a category is also relevant here. For example, if our $\mathbb{A}$ and $\mathbb{B}$ are each different sorts of structured category (say in the sense of KZ-monads), and their structures are "amalgamable", and also this amalgamated sort of structure (call it $T$) is such that $T$-structured categories have some sort of amalgamation property, then there should be a $T$-structured category into which both $\mathbb{A}$ and $\mathbb{B}$ embed.

Chad Nester (Jan 28 2026 at 14:27):

So you could probably study this kind of thing with "system interoperability" as your motivation.

John Onstead (Jan 28 2026 at 20:14):

Chad Nester said:

A related question: I wonder how much is known about sorts of structured category for which the amalgamation property holds.

For mathematical structures that form 1-categories, the structure possesses the amalgamation property precisely when the 1-category of those structures has all pushouts (or at minimum all pushouts of monos) and those pushouts preserve monos. This is easy to see: if you have objects $A, B, C$ and $A$ embeds into $B$ and $C$ via having monos into them, all you need to do is just take the pushout. This "glues together" $B$ and $C$ along $A$, and by our assumption of mono preservation, the projections of $B$ and $C$ into the pushout are monos. The pushout square is automatically commutative, and thus amalgamation is satisfied. Common cases of categories where this is true are adhesive categories.

So you'd just need to define a 2-categorical version of this property, and then check if your target 2-category of structured 1-categories satisfies it.

James Deikun (Jan 29 2026 at 15:08):

Saying "precisely when" is wrong. For example, the finite linear orders (forming the augmented simplex category) have amalgamation, but they don't have pushouts.

Morgan Rogers (he/him) (Jan 29 2026 at 17:38):

Agreed: pushouts provide a universal choice of amalgamation when they exist, but they needn't!

Kevin Carlson (Jan 29 2026 at 17:39):

Fields are another classic example of this problem. You study the "compositum" of two subfields of a common extension in Galois theory, but this compositum can vary based on the common extension, and so is not a pushout.

John Onstead (Jan 29 2026 at 22:13):

You're right, my apologies, I only showed the connection in one direction. I probably should have said "when" instead of "precisely when". From a practical perspective, proving that pushouts exist and preserve monos for a category might in many cases be the simplest way to show the amalgamation property holds (since pushouts are universal constructions we can do good computations with), but if it doesn't work then whether amalgamation is satisfied remains inconclusive.

John Onstead (Jan 29 2026 at 22:13):

Not to get too far off the original topic, but I did think of another way to do a category-theoretic interpretation of amalgamation I think works better. Define $\mathrm{Sub}^{op}(A)$ of an object $A$ to be the subobject poset of monos out of the object $A$, which is reverse to how things are normally done with the usual subobject poset of monos into $A$ (and not to be confused with the opposite of the usual subobject poset!). Then amalgamation appears to be precisely the property that for every $A$ in the category, this poset is directed (every pair of subobjects out of $A$ share an upper bound, which would be an amalgamation). That said while this is a more precise category theoretic interpretation, it's probably much harder to computationally/practically show than the pushout property.

Kevin Carlson (Jan 29 2026 at 22:19):

Yes, that's a more refined formulation. Less convenient but quite important, as preservation of monos by pushout is an exceptionally special property.