Interesting stuff! So for general dynamical systems, you consider directed graphs with the subsystems as vertices, and a arrow from one subsystem to the other whenever the dynamics of the latter depends on the state of the former. For Hamiltonian systems, these relations necessarily always go both ways, and so it’s more natural to consider the simplicial complex of interactions instead. Right?

By the way, concerning the category of simplicial complexes: I’m currently studying The joy of cats (for unrelated reasons). They have some further results on , the category of simplicial complexes:

]]>28.21 Definition: A well-fibred topological construct for which is a quasitopos is called a

topological universe.[..]

28.23 Examples: [and some other cats] are topological universes.

I don’t see why orderings are needed for what we’re talking about here: networked dynamical systems.

What I am trying to say that in networks of Hamiltonian systems you would want to assign potentials to unordered collections of vertices. That means, as you said, that a network in this case is either a decorated symmetric complexes or a decorated symmetric set.

Now, here is an “obvious” interesting question: if these are the objects, what are the morphisms?

If your dynamical system is not Hamiltonian then you could have one part of the system drive another, and that looks directional to me. That’s why Golubitsky and Stewart and their collaborators have directed graphs… (OK, perhaps I should not put it quite so strongly, but it should be roughly correct)

]]>Eugene wrote:

When one says “simplicial complex”, are the vertices necessarily ordered?

I don’t know. Some people say a simplicial complex is a simplicial set with an extra property; with this sort of definition, the vertices in any simplex are ordered. This is what you’d want in algebraic topology: you want a simplex to determine a chain, so it had better have an orientation.

But it seems standard to say a simplicial complex is a set equipped with an arbitrary collection of finite subsets. That’s quite pretty, and now the vertices in any simplex (i.e., the points in one of the finite subsets) are unordered. This is the sort of simplicial complex I look at in my paper with Alex Hoffnung.

What do you call an

unorderedsimplicial complex?

How about an “unordered simplicial complex”?

I don’t know all the standard terminology here. But an “unordered simplicial set” is called a **symmetric set**: it’s a contravariant functor from the category of finite sets to . So, if a simplicial complex is a simplicial set with a special property, the analogous thing for symmetric sets deserves to be called a **symmetric complex**.

I don’t see why orderings are needed for what we’re talking about here: networked dynamical systems. Is that part of what you’re saying?

]]>thus unnecessarily limiting the generality of the idea you just mentioned?

I think I am confused and probably accidentally confused you.

When one says “simplicial complex”, are the vertices necessarily ordered? What do you call an * unordered* simplicial complex? Or, better yet, maybe for Hamiltonian systems one needs symmetric simplicial sets?

Yes, this is what I have in mind. I have no idea how it relates to electric circuits. I had a pretty decent physics education in high school, which let me coast for the first couple of years of college. But it was so long ago, I don’t remember any E&M any more.

The first thing Lee and I got stuck on is relating maps of simplicial complexes to maps between Hamiltonian systems.

Perhaps we should have tried Lagrangian correspondences, but we didn’t.

The second thing we got stuck on was port-Hamiltonian systems.

]]>Ah, yes. So then I can’t wait to hear his answer ;)

]]>That’s a nice guess, Tobias… but I’m a bit suspicious of whether this is what Eugene meant, because the same idea would work for arbitrary dynamical systems if we summed the vector fields generating time evolution, instead of the Hamiltonians. Since Eugene knows and loves networks of general dynamical systems, not just Hamiltonian systems, why would he have said

My second comment is that when you are dealing with networks of Hamiltonian (a.k.a. conservative) systems, simplicial complexes come up naturally.

thus unnecessarily limiting the generality of the idea you just mentioned? Except perhaps to deliberately confuse us.

]]>My second comment is that when you are dealing with networks of Hamiltonian (a.k.a. conservative) systems, simplicial complexes come up naturally.

Let me make a wild guess of how this goes!

Actual physical systems are often modelled as networks of interacting subsystems. If you draw each subsystem as a vertex and put a simplex for every collection of subsystems which interacts, then you get a simplicial complex! More precisely, the total Hamiltonian is a sum of interaction terms like this:

Here, is the simplicial complex describing which interactions there are, and ranges over all its simplices. If is a simplex, then any subset should be a simplex as well: if all the subsystems forming interact, the subsystems making up automatically interact as well, and this is why you get an (abstract) simplicial complex.

Alternatively, you can only sum over all maximal simplices and attribute each “smaller” interaction term to an arbitrary maximal simplex which contains it. (But for many Hamiltonians that would be quite an odd thing to do! Think e.g. of kinetic terms + pairwise interaction terms.)

In the case that the simplicial complex is a clique complex, this is closely related to Markov networks and the Hammersley-Clifford theorem.

So this is at least *one* way in which simplicial complexes come up in Hamiltonian systems — Eugene, is this what you have in mind? I wonder how it relates to electrical circuits?

Yay, finally someone says they like that paper! People who don’t like category theory are happy not knowing that diffeological spaces are concrete sheaves. People who do often want to drop the ‘concreteness’ condition and work simply with sheaves, thus obtaining a topos instead of a mere quasitopos. (This means generalizing diffeological spaces to things that don’t ‘have enough points’: e.g. you could get a thing with lots of smooth curves in it but only one point.) It’s nice to finally meet a reader who likes category theory but not too much.

My second comment is that when you are dealing with networks of Hamiltonian (a.k.a. conservative) systems, simplicial complexes come up naturally.

Sure, let’s talk about this! But a clue now would be nice, if it’s not top secret.

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