Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: applied category theory

Topic: Dynamics

Jules Hedges (Apr 03 2020 at 11:10):

I believe the category they're talking about over in #ACT@UCR seminar is the one defined in https://arxiv.org/abs/1704.02051, whose morphisms are algebraic vector fields

Jules Hedges (Apr 03 2020 at 11:12):

(I think there's also a syntactic version somewhere, where morphisms are syntactic systems of differential equations. It might be I imagined that, if it doesn't exist then I think it should)

Jules Hedges (Apr 03 2020 at 11:12):

Reminds me I should bring this up in the #economics stream

(=_=) (Apr 03 2020 at 12:23):

Jules Hedges said:

(I think there's also a syntactic version somewhere, where morphisms are syntactic systems of differential equations. It might be I imagined that, if it doesn't exist then I think it should)

Yeah, that sounds like it'd be interesting for @Archibald Juniper.

Daniel Geisler (Apr 03 2020 at 16:18):

Reviewing Dynam I get the feeling that it is not as general as a classical dynamical system. Does CT have a category appropriate for the following?

Arnold and Avez (1968), Perspectives of nonlinear dynamics, Vol 1, E. Atlee Jackson, p 51.
Let $M$ be a measure preserving manifold, $\mu$ a measure on $M$ defined by a continuous positive density, $f^t:M \to M$ a one -parameter group of measure-preserving diffeomorphisms. The collection $(M,\mu,f^t)$ is called a classical dynamical system.

John Baez (Apr 03 2020 at 18:37):

I discussed a category where open dynamical systems are morphisms here:

• John Baez and Blake Pollard, A compositional framework for reaction networks.

So if that is the "Dynam" you are talking about, read Section 6, "Open dynamical systems".

There are many variations on this theme waiting to be developed.

John Baez (Apr 03 2020 at 18:39):

There is an obvious category where an object is a manifold equipped with a measure-preserving one-parameter group of diffeomorphisms.

John Baez (Apr 03 2020 at 18:41):

However, it might best to start with a category $X$ where the objects are manifolds with measure and the morphisms are measure-preserving diffeomorphisms.

John Baez (Apr 03 2020 at 18:42):

Then we can talk about actions of the real line, or any other Lie group, on such objects.

John Baez (Apr 03 2020 at 18:43):

To get the action to be smooth as a function of time it's best to treat these categories as enriched in diffeological spaces.

John Baez (Apr 03 2020 at 18:43):

Then we can think of a smooth 1-parameter group of measure-preserving functions as an enriched functor $F : \mathbb{R} \to X$.

Jade Master (Apr 03 2020 at 18:45):

Yup. I describe a bit of this perspective here: https://jadeedenstarmaster.wordpress.com/2019/03/31/dynamical-systems-with-category-theory-yes/

John Baez (Apr 03 2020 at 18:51):

By the way, working with diffeological spaces is important if you want to make it really easy to have the space of smooth maps from one "manifold" to another be something like a "manifold" again. The category of manifolds doesn't have this property but the category of diffeological spaces does. We say it's cartesian closed. In fact it's even better: it's a quasitopos, as I showed with Alex Hoffnung:

• Convenient categories of smooth spaces.

Jules Hedges (Apr 03 2020 at 19:06):

I'm going to make a guess that these 2 different categories - one with dynamical systems as objects, one with open dynamical systems as morphisms - ought to fuse together to make something nice, like a monoidal bicategory

Jade Master (Apr 03 2020 at 19:19):

Well if you think of a dynamical system as a manifold with a vector field then I'm pretty sure the theory of structured cospans gives you something like what you are looking for.

Jules Hedges (Apr 03 2020 at 19:21):

Neat. I wonder whether you can say anything interesting in that language, maybe with surface diagrams

Jules Hedges (Apr 03 2020 at 19:22):

If you have a pair of string diagrams for open dynamical systems, a morphism between them would be drawn as a cobordism between the string diagrams

James Fairbanks (Apr 07 2020 at 01:44):

I'd be really interested in implemented structured cospans for dynamical systems if the details work out

John Baez (Apr 07 2020 at 02:02):

Never heard of 'em being "implemented" yet. It's bound to happen... when it does, someone please let me know!

Jules Hedges (Apr 07 2020 at 09:34):

I speculated about a general implementation of decorated cospans, at least on $\mathbf{FinSet}$, I think that would be massively valuable... not sure whether structured cospans would be easier or harder... but in any case, Catlab is definitely the place it should be done

Evan Patterson (Apr 07 2020 at 17:20):

@James Fairbanks and Micah Halter implemented a case of decorated cospans for SemanticModels.jl. We're planning to move a generalized version of this into Catlab, but it hasn't happened yet. Hopefully pretty soon.

John Baez (Apr 08 2020 at 06:24):

There are cosets, posets, tosets, prosets, and (darn, it doesn't rhyme) pomsets.

Todd Trimble (Apr 08 2020 at 15:08):

Also losets (linearly ordered), wosets (well-ordered).

sarahzrf (Apr 08 2020 at 15:09):

woset

sarahzrf (Apr 08 2020 at 15:09):

amazing

Todd Trimble (Apr 08 2020 at 15:11):

I've not seen those outside the nLab. Could be coinages of Toby Bartels.

sarahzrf (Apr 08 2020 at 15:12):

well, i like proset :)

Reid Barton (Apr 08 2020 at 15:17):

I'm not proud to admit I parsed "coinages" as "co-inages" at first. I mean it's only one letter off from "coimages".

Reid Barton (Apr 08 2020 at 15:17):

I also regularly have trouble with the word "cosplay".

Matteo Capucci (he/him) (Apr 08 2020 at 15:59):

John Baez said:

There are cosets, posets, tosets, prosets, and (darn, it doesn't rhyme) pomsets.

What's are prosets and pomsets?

Jules Hedges (Apr 08 2020 at 16:37):

As the inventor of a thing called "coplay", my brain also has trouble with cosplay. And so does my autocorrect

Todd Trimble (Apr 08 2020 at 18:16):

Prosets are preordered sets. Don't know about pomsets (typo for homsets?).

Oscar Cunningham (Apr 08 2020 at 18:18):

Homsets in a $(1,2)$-category?

sarahzrf (Apr 08 2020 at 19:58):

surely that would be a promset

Gershom (Apr 08 2020 at 21:20):

pomsets are a model of concurrency given by vaughn pratt. its short for "partially ordered multiset" and they're pretty intersesting.