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Evan Patterson (Apr 18 2021 at 04:53):I understand that throughout his career Lawvere has worked on a categorical approach to continuum physics. I am curious to have a bird's eye view of this project. What have been its main themes and results? Who else has worked on it and what has it inspired/been inspired by? If I wanted to read about it, where should I start?
John Baez (Apr 18 2021 at 04:57):I see what you're doing here.
There's a kind of quantum jump from Lawvere's papers on this topic and Urs Schreiber's, it seems to me, with not a whole lot in between.
Evan Patterson (Apr 18 2021 at 04:58):To give you a sense of where I'm starting from, I didn't even realize that there was a connection between those two lines of work (besides having to do with physics!).
Fawzi Hreiki (Apr 18 2021 at 09:58):You may want to take a look at 'Categories in Continuum Physics' which has some nice short papers. I think the main contribution is mainly a meta-theoretical one which is just the introduction of the language of categories (and particularly SDG) into physics. Anders Kock has continued this line with some more substantial stuff relating to physics - I recall reading some papers by him on the heat equation and the wave equation in SDG.
Fawzi Hreiki (Apr 18 2021 at 10:00):I think the work of Urs Schreiber is maybe more related to the axiomatic cohesion and modality stuff but lifted to (infinity, 1)-categories
Fawzi Hreiki (Apr 18 2021 at 10:03):If you want a really easy paper on why categories should be useful in continuum physics, there's this paper.
Jade Master (Apr 18 2021 at 14:10):My guess is that a lot of Lawvere's ideas on continuum physics have never been followed up on. It seems to me that a lot of his ideas about this stuff he just mentioned once or twice in a lecture or a paper and he never followed up on them.
Jade Master (Apr 18 2021 at 14:34):For example Lawvere wrote this 9-page paper defining chaos using an adjunction https://conservancy.umn.edu/handle/11299/4775
Jade Master (Apr 18 2021 at 14:35):It's a really cool paper but I don't think anyone has worked on this idea since? I'd love for someone to tell me that I'm wrong.
John Baez (Apr 18 2021 at 15:18):Hmm, I'd never even heard of that paper, Jade!
John Baez (Apr 18 2021 at 15:21):Evan wrote:
To give you a sense of where I'm starting from, I didn't even realize that there was a connection between those two lines of work (besides having to do with physics!).
Briefly, Urs realized that Lawvere's thoughts on continuum physics, topos theory and adjoint modalities would become a lot more powerful if applied to Lurie's -toposes.
John Baez (Apr 18 2021 at 15:22):So, he went on a binge of trying to understand some of Lawvere's more difficult writings involving the philosopher Hegel and adjoint modalities. The linked nLab page will give you a flavor of the result.
John Baez (Apr 18 2021 at 15:27):Urs went on and used these ideas to study quantum field theory and string theory.
John Baez (Apr 18 2021 at 15:28):Crudely speaking, it's an approach that uses the ideas of synthetic differential geometry - using infinitesimals to think about calculus, but also differential forms.
Evan Patterson (Apr 18 2021 at 17:31):Thanks for the pointers and overview, everyone. I will check out some of these short papers.
John Baez (Apr 18 2021 at 19:13):By the way, I think I understand some of Lawvere's thoughts on continuum mechanics.
John Baez (Apr 18 2021 at 19:14):You might get started reading a bit of week200 of This Week's Finds, starting around where I say
Let me just briefly summarize, and then move on to his work on topos theory and mathematical physics.
John Baez (Apr 18 2021 at 19:26):One fun question, which I imagine you're wondering about, is whether his ideas are of any use in numerical methods for solving continuum mechanics problems (like fluid flow, etc.).
John Baez (Apr 18 2021 at 19:27):I don't see exactly how, but the connection between simplicial sets and differential graded algebras (like algebras of differential forms) might be one angle.
Tim Hosgood (Apr 18 2021 at 19:42):John Baez said:
I don't see exactly how, but the connection between simplicial sets and differential graded algebras (like algebras of differential forms) might be one angle.
this sounds like something that Paugum wrote about in his book on the mathematics behind quantum field theory, similar to the approach that e.g. Urs takes too. either way, it's something that I'm also very very interested in, so if anybody ever wants to talk about this then please do reach out and let me know! :smile:
Steve Huntsman (Apr 18 2021 at 20:35):IMO a good way to do physics would be to treat Glauber spins as open Markov processes and build up discrete spin models (how does the spin-coupling work?) then let the category theory take you into field theory. AFAIK nobody has tried this.
John Baez (Apr 18 2021 at 20:44):What are Glauber spins? I know about various spin systems like the Ising model, the Heisenberg model, etc...
Steve Huntsman (Apr 18 2021 at 23:57):A Glauber spin system is a continuous-time Markov process whose invariant distribution is the same as the Ising model—but as a Markov process it has dynamics—the spins flip. Glauber dynamics has taken on a much broader context in the arena of probabilistic graphical models since the original paper
Steve Huntsman (Apr 19 2021 at 00:28):https://en.wikipedia.org/wiki/Glauber_dynamics
Steve Huntsman (Apr 19 2021 at 01:38):...AKA Gibbs sampling
John Baez (Apr 19 2021 at 15:09):Okay, thanks. That's clear enough!
Spencer Breiner (Apr 19 2021 at 16:09):Fawzi Hreiki said:
If you want a really easy paper on why categories should be useful in continuum physics, there's this paper.
What a great resource! That link points to @Matt Earnshaw's (partial) github list/links to Lawvere's collected works .
Jade Master said:
For example Lawvere wrote this 9-page paper defining chaos using an adjunction https://conservancy.umn.edu/handle/11299/4775
But this paper isn't on it! Hopefully this will suffice to notify Matt that it's missing.
Matt Earnshaw (Apr 19 2021 at 16:48):thanks, the paper is not missing but the correct title is "Functorial remarks...". for some simple stuff on thermodynamics, Unity and Identity of Opposites in Calculus and Physics is cute though characteristically sketchy. it also has one of the all time best list of "key words"
Spencer Breiner (Apr 19 2021 at 17:03):So it is. I suppose it was foolish to expect the publisher to get the title right...
Spencer Breiner (Apr 19 2021 at 17:06):
Spencer Breiner (Apr 19 2021 at 17:06):... and over-simplistic Ctrl-F