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Adittya Chaudhuri (Mar 23 2026 at 18:51):When we model an open system as a horizontal 1-cell in a double category and the morphisms of open systems as 2-moprhims in the said double category, then not only we can compose open systems along interfaces by composing the horizontal 1-cells, but also we can compose morphisms of open systems along morphisms of interfaces using the horizontal composition of 2-morphisms.
Now, on the other hand, when we model open systems by constructing suitable algebras on Spivak's opeads of directed wiring diagrams as in Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams, I think we only get a theory of how to compose open systems along interfaces prescribed by the nature of the underlying directed wiring diagram, but not how to compose morphisms of open systems along morphisms of interfaces.
Now, in this regard, I think modelling open system as a horizontal 1-cell in double category has an edge over the operadic approach. Now, since the operad of directed wiring diagram is induced from a symmetric monoidal category, this was expected.
Now, I feel there should be a notion of a "kind of double operad of directed wiring diagrams" induced from a symmetric monoidal double category where there would be a natural notion of a "morphism between wiring diagrams" and the double operad structure should tell us how to compose such morphisms between wiring diagrams. Now, I think a suitable notion of double algebra kind of structure on the double operad of directed wiring diagrams should say us how to compose morphisms of open systems along morphisms of interfaces. Then, I think this notion of double algebra on the double operad of wiring diagrams would be an operadic analogue of "modelling open systems as horizontal 1-cells in a symmetric monoidal double category".
But, I am almost certain that the above discussion is already well establised and probably has been discussed in the paper Towards a double operadic theory of systems. Although I am yet to read this paper, but I am very much interested in composing morphisms of open systems along morphisms of interfaces using an operadic approach, and would very much like to know whether the Spivak's operad's of directed wiring diagrams have already a double analogue. Before starting to read this paper, I would like to ask whether the framework in this paper is actually taking care of composing morphisms of open systems along morphisms of interfaces ?
After recenlty learning about virtual double categories, I will also not be surprised if the exisiting notion of such double operads of wiring diagrams are induced from a particular class of virtual double categories.
Evan Patterson (Mar 23 2026 at 19:59):Yes, I believe it does do that, treating directed wiring diagrams from the perspective of lens, though I'm not familiar with the contents of the paper in a detailed way.
Adittya Chaudhuri (Mar 23 2026 at 20:24):I see. Thanks very much!
Evan Patterson (Mar 23 2026 at 20:34):DOTS is a general framework, but it seems to have two really important classes of examples: the variable sharing paradigm, where the composition patterns are undirected wiring diagrams (cospans), and the machine paradigm, where the composition patterns are directed wiring diagrams (lenses).
Adittya Chaudhuri (Mar 24 2026 at 05:33):Thank you! Interesting! Earlier, I went through (although not in details) the main ideas in the paper An Algebra of Resource Sharing Machines, which unifies the two important classes you mentioned at the level of operads. Now, it seems DOTS is a double analogue of it (double operads), which also covers composition of morphisms of open systems along morphisms of interfaces.
John Baez (Mar 24 2026 at 05:49):Yes, Sophie used to be in our #practice: epidemiology group, and she unified the two classes Evan mentioned and applied this unification to stock-flow diagrams, which are a key aspect of our work.
More recently our group (which Evan has been unable to attend of late, being busy developing CatColab) spent several sessions of our weekly meeting going through Sophie's closely related paper Dependent directed wiring diagrams. This should represent yet another paradigm, and Sophie also applies this paradigm to stock-flow models!
John Baez (Mar 24 2026 at 05:52):But Evan came to our most recent meeting, which is great because he seems to have discovered that CatColab, based on double categorical doctrines, solves a lot of problems with our earlier implementation of stock-flow diagrams in CatLab.
John Baez (Mar 24 2026 at 05:55):Our group plans to keep working on these issues. Two big questions are to what extent CatColab can become the new platform for work with stock-flow diagrams in epidemiology - and if so, to what extent this is compatible for working with DOTS. Here I'm talking about actual software, not just math.
Adittya Chaudhuri (Mar 24 2026 at 06:22):Thanks!! Very interesting!!
David Corfield (Mar 24 2026 at 09:36):The DOTS approach certainly copes with these wiring styles for composition of subsystems. The tight direction is there to allow things like coarse-graining and safety guarantees, via maps out, and also detecting behaviours, via maps in. E.g., trajectories of systems are measured by maps into a system from a clock system.
Interestingly they're finding a difference between wiring styles. Symmetric monoidal loose right modules, the focus of the DOTS paper, work well when you have a global notion of time (aka synchronous); algebras for double operads work well when you have a distributed (aka concurrent or asynchronous) notion of time where you not only have to specify clocks but also combine them to form new clocks.
Matteo Capucci (he/him) (Mar 24 2026 at 11:40):You might want to check §4.3 in David's book for a breakdown on how functoriality works
Matteo Capucci (he/him) (Mar 24 2026 at 11:41):Or my notes about that
Adittya Chaudhuri (Mar 24 2026 at 12:06):David Corfield said:
The DOTS approach certainly copes with these wiring styles for composition of subsystems. The tight direction is there to allow things like coarse-graining and safety guarantees, via maps out, and also detecting behaviours, via maps in. E.g., trajectories of systems are measured by maps into a system from a clock system.
Interestingly they're finding a difference between wiring styles. Symmetric monoidal loose right modules, the focus of the DOTS paper, work well when you have a global notion of time (aka synchronous); algebras for double operads work well when you have a distributed (aka concurrent or asynchronous) notion of time where you not only have to specify clocks but also combine them to form new clocks.
Very interesting!! Thanks very much! I think now I have a lot of motivation to learn DOTS.
Adittya Chaudhuri (Mar 24 2026 at 12:07):@Matteo Capucci (he/him) Thanks very much for sharing your notes and referring the relevant portion in David's book. I will check.