Naturalness in Category Theory is a difficult notion to pin down. Here is some discussion about naturalness.

I try to study and generate applications of Category Theory to the natural world. I believe that naturalness has been a guiding principle in my work over the years. Another principle that has guided me is Occams’s razor. I get the feeling they may be related, but I have never seen this written about. Does anyone have any thoughts on this?

I have been developing ideas about science couched in the language of monads. They seemed rather natural to me. I am now discovering sketches, but have a very limited understanding of both of these. They both seem like a way to have natural presentations of theories or analysis of ideas. Is one more natural than the other? I feel that monads are more natural, but that might just be because I don’t understand sketches very well. I think monads are more natural because there is this effect that, when you are looking for a given monad, it may only exist on a particular category. It forces you to accept the particular site where everything that ought to be true, is true. Cleverness, or brittle constructions are not as useful.

I suppose the counter argument could be made that sketches work by choosing the kinds of limits you are going to use and then you can achieve various presentations of theories given that kind of limits you chose. So, you work in the universe or logic of particular limits and all things that ought to be true with those limits are true.

It has been put forth that monads and sketches are not related, in that they don’t present the same kinds of things (or theories). Taking a look at Barr and Well’s CT for computing science text, the first example they give is the “free monoid construction”. How is that any different from the list monad, who’s category of algebras is the category of monoids? Also, we know that you can generate monads from sketches. They are clearly related, and I would argue, presentations of the same things.

Here we see a publication where the authors assert the following

“Lawvere theories and monads have been the two main category theoretic formulations of universal algebra”

I believe Lawvere Theories are just sketches restricted to finite products.

The fact that sketches tend to be more like complicated constructions, would suggest to me that they are less natural than monads.

Why is this all important? Look at this quote in Spivak’s latest paper

“Previous work has shown how to allow collections of machines to reconfig- ure their wiring diagram dynamically, based on their collective state. This notion was called “mode dependence”, and while the framework was compositional (forming an operad of re-wiring diagrams and algebra of mode-dependent dynamical systems on it), the formulation itself was more “creative” than it was natural.”

Naturalness mattered to the tool he choses in his paper, namely the category of polynomial endofunctors on set and their monads

If you read this book by Hosseenfelder, you see that what physics may be in need of now is a new guiding principle, other than the standard understanding of beauty. I am proposing that the new principle which we should be looking for is the categorical notion of naturalness.

]]>