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

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.

]]>(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.)

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