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.
]]>The classic way to encounter the theory of categories is via Set Theory via the typical definition we see for categories. We see all kinds of categories that are equivalent to the category of small categories. I wonder about presentations of the theory of categories. To facilitate a discussion, we may need to define what a presentation of a theory is. It may consist of a logical language or even the standard presentations of algebraic structures. For instance, a presentation of the theory of partial monoids would count as a presentation of Categories. The presentation should come with enough structure to analyze all small Categories.
I saw Marsden put together a presentation of categories in terms of string diagrams.
I like to think that string diagrams can be seen as containers. This is a paper about containers. So the idea is that you have a (co)monad that encodes the container for the theory of categories. Could this work?
]]>I feel that, while the concept of algebra over a monad is of course an instance of the more general module over a monad, it could benefit from having its own dedicated page, with motivation from other fields and dedicated examples.
If I get a green light from enough people here, I can create the page myself (and replace the redirection by links both ways).
]]>I would like to add content to the nLab about the costrength of a monad. I could either (significantly) modify the strong monad page, or create a separate dedicated page. Any thoughts?
]]>I believe commutative monad should redirect to monoidal monad and not to commutative algebraic theory (but of course have a link to the latter). Is anyone disagreeing?
]]>Hello. I’ve been reading the article “Stuff, structure, property”, and related entries. I love the idea, and the “homotopical flavour” to it.
Here are anyway some things that are not clear to me:
So, it looks to me that the lowest (nontrivial) level is always some kind of “property”, but the highest level always looks like some kind of “stuff”, and the things that change are in between (I would call them “structure”, “2-structure”, and so on, and properties “0-structure”.) Am I looking at this wrong? Or what would be examples where it’s clear that:
Can anyone answer? I hope that the question is clear, please tell me if I should explain more, and forgive my possible mistakes!
]]>Is there a term to denote collectively all the constructions of “theories” of some sort, without giving rise to ambiguities?
What I exactly mean is a collective term for monads, operads, clubs, Lawvere theories, and in general all categorical constructions that may have “algebras”, or which describe formal operations in some (possibly generalized) way. In my mind they all encode some sort of “theory”, but the word “theory” seems to be reserved for “algebraic theory”/”Lawvere theory”.
Which term would you use instead?
(I hope it’s clear enough what I’m asking.)
]]>In overcategory, it is shown that the forgetful functor reflects limits, and it is mentioned that this is a consequence of the undercategory being the category of algebras for the monad What about the comma category , where and ? On the one hand, it seems like the diagrammatic proof still goes through and the forgetful functor which takes reflects limits. On the other hand, I cannot see a monad (or even just an endofunctor) for which comprises the algebras. Seems like it wants to be the functor , but this is not an endofunctor. Is there a better way to understand how behaves with respect to limits?
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