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)

]]>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.

On the other hand, you could define a simplicial complex to be 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.

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?

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.

]]>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?

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.

]]>My second comment is that when you are dealing with networks of Hamiltonian (a.k.a. conservative) systems, simplicial complexes come up naturally. Lee and I played around with them a bit but got nowhere. It may matter that they form a quasitopos. Perhaps we can talk about it next May.

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