## Monoidal Categories of Networks

Here are the slides of my colloquium talk at the Santa Fe Institute at 11 am on Tuesday, November 15th. I’ll explain some not-yet-published work with Blake Pollard on a monoidal category of ‘open Petri nets’:

Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. Far from mere informal tools, many of these diagrammatic languages fit into a rigorous framework: category theory. I will explain a bit of how this works and discuss some applications.

There I will be using the vaguer, less scary title ‘The mathematics of networks’. In fact, all the monoidal categories I discuss are symmetric monoidal, but I decided that too many definitions will make people unhappy.

The main new thing in this talk is my work with Blake Pollard on symmetric monoidal categories where the morphisms are ‘open Petri nets’. This allows us to describe ‘open’ chemical reactions, where chemical flow in and out. Composing these morphisms then corresponds to sticking together open Petri nets to form larger open Petri nets.

### 4 Responses to Monoidal Categories of Networks

1. Bruce Smith says:

(typo in date on first slide)

2. John Baez says:

Whoops – thanks for catching that!

3. Bruce Smith says:

On the 3rd or so slide, you say “Categories are great for describing processes of all kinds.” In this context, by “processes” you just mean “morphisms”. But some of your categories are being used to describe “actual processes”, at least implicitly (e.g. the time evolution of dynamical systems corresponding to a Petri net). That use of the term “process” (which is only implicit in your slides) has nothing to do with the explicit usage, but is a more natural use of the English word “process”. So this explicit use might be confusing to some people (IMHO).

(I’ve quibbled about that term to you in the past, since morphisms needn’t imply any notion of change or time like “process” implies; IIRC your reason for using it instead of something like “relation” or “connection” was because of the “extra info besides endpoints” a morphism can have.)

• John Baez says:

I’d already spent at least 10 minutes agonizing about this. As a mathematician I don’t feel any need for ‘processes’ to take time in the most literal sense of ‘time’: for example, I think of squaring as a process, so I don’t mind thinking of the function

$\begin{array}{rcl} f : \mathbb{R} &\to& \mathbb{R} \\ x & \mapsto & x^2 \end{array}$

as a process. The real problem, as you say, is that the morphisms in this talk often describe not ‘processes’ so much as ‘open systems’ or ‘networks’. Luckily, I can say more in words than cleanly fits on the slides—a slide packed with text is no good. I’ll try to clarify this issue verbally later. I’ll also mention that in the classic example of a category, the category of sets, an object is a set and a morphism is a function.

I considered taking out all intuitive hints as to what morphisms might be like, but since this talk is for non-mathematicians—for example, biologists—I think a statement like “Categories are great for describing morphisms of all kinds” would not be very helpful. At that point in the talk, neither categories nor morphisms have been explained, so this statement reads like “X’s are great for describing Y’s of all kinds”. Mathematicians have an amazing tolerance for sentences that express relationships between undefined terms, but ordinary people want to know what terms ‘really mean’. They’re not so used to the idea that sometimes the meaning is nothing other than the network of relationships between terms.

Once, long ago, when I gave a vaguely similar talk, someone asked me what counts as ‘process’. He seemed to want a conceptual analysis or definition of that term, and he was dissatisfied when I refused to give one: I said that ultimately I was just talking about morphisms in a category, and the category axioms are the last word on this subject!

However, in that talk I was really using morphisms to describe processes, like Feynman diagrams or spin foams. My new schtick is to use them to describe open systems.