David Corfield Categorical systems theory

Firm points

1. [Algebraic patterns]?](https://ncatlab.org/nlab/show/algebraic+pattern) 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](https://www.youtube.com/watch?v=52AsVTxHXU0) and also a [2-torial](https://www.youtube.com/watch?v=3VP3ZfgfpTg).)$$\Omega$$, 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.
9. There is a directed or oriented version of $$GS$$,$$\Xi^{\times}$, etc. in [Modular operads, iterated distributive laws and a nerve theorem for circuit algebras](https://arxiv.org/abs/2412.20262) derived by working in the slice over $Di$$, 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).
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.
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.

Speculation

Boateng and Marcolli, Formal Languages and TQFTs with Defects, a pointer in this direction:

4.7. Further directions: higher dimensional automata and TQFTs with defects.

There is a notion of higher dimensional automata (see [8]), based on James Roger’s “higher dimensional trees” [19], and allowing for a rich categorical formulation and for algebra and coalgebra structures. It seems then natural to investigate the question of possible relations between these higher dimensional automata and TQFTs with defects, especially in the 2-dimensional case where ordinary TQFTs admit an algebraic description in terms of Frobenius algebras, and versions with defects have been analyzed, for instance, in [2]. We will return to this question in future work.