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Stream: theory: applied category theory

Topic: Graphical composition

David Corfield (May 20 2026 at 13:48):

I just wanted to record some findings from my recent search for a form of undirected graph composition in the hyperstructures thread. I was complicating this theme by simultaneously looking for a hierarchical structure (networks of networks). Better to treat them separately. (Yet a further theme concerned which Segal conditions to apply, which again can be put to one side.)

I suspect that there's something of use to ACT systems theory in all of this.

1. [[Algebraic patterns]] provide an excellent framework for different modes of composition: categories, operads, properads, ...
2. A key aspect of an algebraic pattern is the factorization of any morphism into an inert and an active morphism.
3. Inerts play a bookkeeping role, tracking variables, whereas actives capture composition. (To see them at work for the simplicial pattern on $\Delta$, i.e., the composition of categories, you can watch @David Jaz Myers in a Topos Berkeley seminar and also a 2-torial.) $\Omega$, the dendroidal category, provides the pattern for operads.
   
   
4. There are algebraic patterns for undirected and directed connected graphs. These are the $U$ and $U_{/\mathfrak{o}}$ of Philip Hackney's Categories of graphs for operadic structure. These provide patterns for modular operads and wheeled properads.
5. Elsewhere, (Question 5.9 of Segal conditions for generalized operads), Philip asks for nonconnected versions of these to be able to approach (wheeled) PROPs, non-connected modular operads, etc.
6. The $\Xi^{\times}$ of Sophie Raynor's Modular operads, iterated distributive laws and a nerve theorem for circuit algebras, Theorem 8.4 looks like it answers some of what's wanted in the undirected case, giving rise to an algebraic pattern on which Segal presheaves are circuit algebras. The category of circuit algebras is equivalent to the Eilenberg-Moore category of algebras for a monad on Joyal and Kock’s graphical species category.
7. Morphisms in $\Xi^{\times}$ possess ternary factorization: composition (graphs of graphs); deletion; and étale maps. These allow for relevant network operations : refinements of nodes into graphs; forgetting substructure; locating subgraphs. An ordinary factorization arise from counting the first as the actives and the second and third together as inerts. There is then an algebraic pattern structure on $\Xi^{\times}$.
8. Constructing a directed version of $\Xi^{\times}$ would relate the apparatus of algebraic patterns to Joachim Kock's whole-grain approach to Petri nets.

That's about as far as I've reached in that direction. There's surely some potential here to bring parts of categorical systems theory into relation with interesting topics in mathematics and mathematical physics. As for the hierarchical structure that Nils Baas sought, there's a wreath ($\wr$) composition available for algebraic patterns.

David Corfield (May 21 2026 at 09:10):

I should read more closely. The directed version is already there.

9. There is a directed or oriented version of $GS$, $\Xi^{\times}$, etc. in Modular operads, iterated distributive laws and a nerve theorem for circuit algebras derived by working in the slice over $Di$, where this $Di$ provides the orientations. So there are $OGS$ and $O\Xi^{\times}$, etc. Segal presheaves on the latter are wheeled props (Corollary 8.6).

David Corfield (May 21 2026 at 12:59):

10. The monad $\mathbb{O}$ on $GS$, the category of graphical species, is such that its EM-algebras are circuit algebras. $\mathbb{O}$ factorizes into three monads, $\mathbb{O}=\mathbb{L}\mathbb{D}\mathbb{T}$. In the oriented version, $\mathbb{O}^{or}$ on oriented graphical/digraphical species, the EM-algebras are oriented circuit algebras (equivalently, wheeled props). $\mathbb{O}^{or}$ factorizes as $\mathbb{L}^{or}\mathbb{D}^{or}\mathbb{T}^{or}$. A Petri net, $P$, is a flat digraphical species. The application of the monads successively adds: all sequential firings, units (null firings), concurrent firings.

David Corfield (May 21 2026 at 15:20):

11. As for what this technology could bring to systems theory, one suggestion from Raynor, "Given a circuit algebra $A$, the bar construction for $\mathbb{O}=\mathbb{L}\mathbb{D}\mathbb{T}$ and $A$ may be viewed as a rulebook for zooming in and out of networks decorated by $A$" (p. 3). Presumably this carries over to the oriented case.