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Stream: community: our work

Topic: Joe Moeller

Joe Moeller (Mar 29 2024 at 18:53):

I've accepted a postdoc position at CalTech. I'll be working with Aaron Ames on using category theory to gain a deeper understanding of stability and control of dynamical systems.

Evan Patterson (Mar 29 2024 at 18:55):

Congrats Joe!

Joe Moeller (Mar 29 2024 at 19:06):

Thanks! I'm super excited to work on really interesting ideas and at such a cool place. Cherry on top is that I'm back home in SoCal.

Matteo Capucci (he/him) (Mar 29 2024 at 19:38):

That's cool to hear! I'm curious what the approach is.

Joe Moeller (Mar 29 2024 at 20:55):

I think we should have a paper out within a reasonable amount of time. I'll sorta randomly throw "end of summer" out there. So the approach will be revealed then!

Niles Johnson (Mar 30 2024 at 12:32):

Congrats!! Sounds interesting!

Joe Moeller (Oct 11 2024 at 00:29):

John and Todd and I finally finished the sequel to our Schur functors paper:

2-rig extensions and the splitting principle
John Baez, Joe Moeller, Todd Trimble

Classically, the splitting principle says how to pull back a vector bundle in such a way that it splits into line bundles and the pullback map induces an injection on K-theory. Here we categorify the splitting principle and generalize it to the context of 2-rigs. A 2-rig is a kind of categorified "ring without negatives", such as a category of vector bundles with ⊕ as addition and ⊗ as multiplication. Technically, we define a 2-rig to be a Cauchy complete k-linear symmetric monoidal category where k has characteristic zero. We conjecture that for any suitably finite-dimensional object r of a 2-rig R, there is a 2-rig map E:R→R′ such that E(r) splits as a direct sum of finitely many "subline objects" and E has various good properties: it is faithful, conservative, essentially injective, and the induced map of Grothendieck rings K(E):K(R)→K(R′) is injective. We prove this conjecture for the free 2-rig on one object, namely the category of Schur functors, whose Grothendieck ring is the free λ-ring on one generator, also known as the ring of symmetric functions. We use this task as an excuse to develop the representation theory of affine categories - that is, categories enriched in affine schemes - using the theory of 2-rigs.

Joe Moeller (Oct 11 2024 at 00:31):

My pseudo-promised deadline of "end of summer" for my first paper with Aaron Ames has probably passed, though you couldn't tell from the weather. We are almost done. The basic theory is complete, we're just working on some more examples to include.

John Baez (Oct 11 2024 at 01:03):

Please finish soon so the weather here cools down! :sunglasses:

Joe Moeller (Feb 25 2025 at 03:13):

Ok, here’s part one!

https://arxiv.org/abs/2502.15276

Categorical Lyapunov Theory I: Stability of Flows

Aaron D. Ames, Joe Moeller, Paulo Tabuada

Lyapunov's theorem provides a fundamental characterization of the stability of dynamical systems. This paper presents a categorical framework for Lyapunov theory, generalizing stability analysis with Lyapunov functions categorically. Core to our approach is the set of axioms underlying a setting for stability, which give the necessary ingredients for “doing Lyapunov theory'' in a category of interest. With these minimal assumptions, we define the stability of equilibria, formulate Lyapunov morphisms, and demonstrate that the existence of Lyapunov morphisms is necessary and sufficient for establishing the stability of flows. To illustrate these constructions, we show how classical notions of stability, e.g., for continuous and discrete time dynamical systems, are captured by this categorical framework for Lyapunov theory. Finally, to demonstrate the extensibility of our framework, we illustrate how enriched categories, e.g., Lawvere metric spaces, yield settings for stability enabling one to “do Lyapunov theory'' in enriched categories.

Joe Moeller (Feb 25 2025 at 03:15):

What ended up happening is we kept adding and adding to the paper until it became a monster. This part one is ironically what we had done at the end of summer :upside_down:

Joe Moeller (Feb 25 2025 at 19:39):

I'm also speaking about this project in the Topos Institute's online colloquium on Thursday:
https://topos.institute/events/topos-colloquium/

Joe Moeller (Mar 05 2025 at 22:06):

Here's my talk at the Topos Institute's online colloquium:
https://www.youtube.com/live/dI033-TjymA?si=9C153AvSuKTYiwzy

Joe Moeller (May 30 2025 at 22:49):

Here's our second paper on Lyapunov theory:

Categorical Lyapunov Theory II: Stability of Systems
https://arxiv.org/abs/2505.22968
again with Aaron Ames, but this time with Sébastien Mattenet, grad student at UC Louvain.

Lyapunov's theorem provides a foundational characterization of stable equilibrium points in dynamical systems. In this paper, we develop a framework for stability for F-coalgebras. We give two definitions for a categorical setting in which we can study the stability of a coalgebra for an endofunctor F. One is minimal and better suited for concrete settings, while the other is more intricate and provides a richer theory. We prove a Lyapunov theorem for both notions of setting for stability, and a converse Lyapunov theorem for the second.

Joe Moeller (May 30 2025 at 22:51):

Each paper has it's own mini story, but the two combined tell the complete story of our categorical approach to Lyapunov theory.

Joe Moeller (May 30 2025 at 23:01):

An equilibrium point in a dynamical system is stable if every trajectory approaches it over time (or more generally, if it does not get too far away over time). That's easy enough to say, but actually checking this property directly means solving the system, which is generally very difficult and not practical if you are, for instance, telling a computer to do it over and over in real time on a walking robot that you don't want to fall over.

Lyapunov realized you can certify the stability of an equilibrium point if you can find a function on your space which satisfies two properties:

1. at any point, the function is decreasing in the direction that the vector field is pointing
2. it's "positive definite" relative to the eq point, i.e. it's 0 at the point and positive elsewhere.

Joe Moeller (May 30 2025 at 23:04):

This has a nice visual intuition where the function (called a Lyapunov function) is basically this paraboloid, and the dynamics of the system is essentially a ball rolling down. Stability is then just the noticing that the ball is going towards the low point in the long run, even if it's actual distance from it fluctuates in some particular interval of time.

Gijs_Hilhorst - Lyapunovfn.png

Joe Moeller (May 30 2025 at 23:19):

In our first paper, we study systems as actions of a monoid $T$ (for time) on some other object in a category. From this, we prove that the existence of a Lyapunov morphism (relative to the action and an equilibrium) implies stability of the equilibrium. This perspective essentially assumes that we've solved the system, which I've already pointed out is not realistic.

Our second paper treats systems as coalgebras of an endofunctor F. This perspective generalizes vector fields which are coalgebras of the tangent bundle functor on the category of smooth manifolds (satisfying an extra property). In this setup, we prove that a Lyapunov morphism (now relative to a coalgebra and an equilibrium) is a Lyapunov morphism for any monoid action which is a "solution" of it.

Taken together, these tell us that a Lyapunov morphism for a coalgebra implies the stability of the equilibrium point, i.e. Lyapunov's theorem, but at a super generalized level.

John Baez (May 31 2025 at 08:35):

This sounds very nice! You explained it very well here. Do you have some example applications of your super generalized Lyapunov theory?

Matteo Capucci (he/him) (Jun 03 2025 at 17:27):

Very nice Joe!

I guess from a very general point of view, one can interpret Lyapunov's insight as the fact that if you need to show that some state $s^*$ of a system $S$ is a stable fixpoint, it suffices to exhibit a map of systems $\phi:L \to S$ where $L$ is a system known to have a stable fixpoint $l^*$, and show that $\phi(l^*)=s^*$. Lyapunov's actual insight was that, for differential systems, a nice class of systems with known stable fixpoints are the gradient flows of smooth functions with a local minimum (at $0$).

Remarkably the way I told this story above isn't using at all the fact we are talking about attracting stable fixpoints. The above technique works for any kind of behaviour* of any kind of system. So perhaps I generalized too far?

*as long as it's functorially determined

Matteo Capucci (he/him) (Jun 03 2025 at 17:30):

Matteo Capucci (he/him) said:

So perhaps I generalized too far?

Perhaps I didn't, but I just moved the 'meat' of the problem to defining the right setting in which the concept of 'stable fixpoint' of a system makes sense (i.e. it is defined and it is functorial on the maps)... which is what you address in your papers.

Joe Moeller (Jun 03 2025 at 17:43):

John Baez said:

This sounds very nice! You explained it very well here. Do you have some example applications of your super generalized Lyapunov theory?

Sorry, travel + just forgetting prevented me from answering. Besides the classical continuous and discrete time settings, we also talk a bit about quantale-enriched categories (generalizing from Lawvere metric spaces) in the first paper, and labeled transition systems in the second paper. We have a few other examples that only get interesting in the asymptotic case, which we have worked out, but are writing a separate paper for.

Interestingly, we can write down examples that are non-deterministic/stochastic etc, but the definitions force us to only consider deterministic behaviors of those. My gut says something influencing this is the fact that our very first assumption is that we have a cartesian category, and maybe redoing things in a Markov category would result in something interesting. If anybody has thoughts, I'd love to talk.

A stone unturned is finding an example of a quantale Q for which the category of Q-enriched categories admits an endofunctor whose coalgebras are particularly interesting. As soon as we have such an example in hand, we have already proved a Lyapunov theorem for it. Again, if anybody has any ideas, I'd love to talk.

Joe Moeller (Jun 03 2025 at 18:04):

Matteo Capucci (he/him) said:

Very nice Joe!

I guess from a very general point of view, one can interpret Lyapunov's insight as the fact that if you need to show that some state $s^*$ of a system $S$ is a stable fixpoint, it suffices to exhibit a map of systems $\phi:L \to S$ where $L$ is a system known to have a stable fixpoint $l^*$, and show that $\phi(l^*)=s^*$. Lyapunov's actual insight was that, for differential systems, a nice class of systems with known stable fixpoints are the gradient flows of smooth functions with a local minimum (at $0$).

Unless I misunderstand what you mean, I think I'd say the first part is backwards. It's more like your certifying the stability of a system $S$ by mapping it into a simple stable system $V \colon S \to R$ (changing the letters just in case I am misunderstanding your point).

It is true that this does fit a more general theme of behavior classification through mapping into simple representatives of that behavior. There's a whole subfield of people studying "Lyapunov methods", essentially instantiating this for various notions of system and various notions of stability.

David Corfield (Jun 03 2025 at 19:18):

Joe Moeller said:

It's more like your certifying the stability of a system $S$ by mapping it into a simple stable system

Like Grodin and Harper certifying amortized cost by mapping into a simple coalgebra on a cost algebra here.

Joe Moeller (Jun 03 2025 at 19:54):

I looked at exactly this paper around a year ago, but I clearly missed something!

Matteo Capucci (he/him) (Jun 04 2025 at 06:36):

Joe Moeller said:

Matteo Capucci (he/him) said:

Very nice Joe!

I guess from a very general point of view, one can interpret Lyapunov's insight as the fact that if you need to show that some state $s^*$ of a system $S$ is a stable fixpoint, it suffices to exhibit a map of systems $\phi:L \to S$ where $L$ is a system known to have a stable fixpoint $l^*$, and show that $\phi(l^*)=s^*$. Lyapunov's actual insight was that, for differential systems, a nice class of systems with known stable fixpoints are the gradient flows of smooth functions with a local minimum (at $0$).

Unless I misunderstand what you mean, I think I'd say the first part is backwards. It's more like your certifying the stability of a system $S$ by mapping it into a simple stable system $V \colon S \to R$ (changing the letters just in case I am misunderstanding your point).

I think I'm saying something different, the map V is only used to construct the simple system L, namely by taking it's gradient flow. Then you map L to S.

Matteo Capucci (he/him) (Jun 04 2025 at 06:41):

What I mean by map matters. If I naively do that for 'differential Moore machines', clearly I don't get a map which exactly commutes with the vector field, ie it's not true that around the stable fixpoint, S is the gradient flow of V. Instead one has a weaker notion of map in which one only checks that the two vector fields are always 'oriented in the same direction', thus the first requirement of a Lyapunov function...