The n-Category Café

Re: Open Systems in Classical Mechanics

And by replacing real numbers with quasiprobability distributions you get open systems in quantum mechanics.

Re: Open Systems in Classical Mechanics

This switch from cospans to spans feels to me like the duality between algebra and geometry, or more generally the dualization that happens when taking “functions on blah”. That is, if $A\to C \leftarrow B$ is a cospan and $R$ is an object, then $R^A \leftarrow R^C \to R^B$ is a span. Is there some sense in which your categories of spans are obtained by dualizing a category of cospans?

Re: Open Systems in Classical Mechanics

Cool! This is similar to the symplectic (bi-)categories studied by Alan Weinstein, but the composition is different.

Do the two projection maps in a span always have fibres that are orthogonal with respect to the symplectic pairing?

Re: Open Systems in Classical Mechanics

On the Category Theory Community Server, Morgan Rogers and I are having a nice conversation about this paper. He wrote:

It’s been a while since I did any hands-on physics, and I’m a little lazy to take a full dive into your paper, so forgive me if my question seems silly. When you say “We can equip all the symplectic manifolds in this story with Hamiltonians”, and then say it’s a big deal that you’re doing this for the feet of the spans as well as the apex, why is that? When equipping Hamiltonians, don’t you do so on the underlying category before taking spans? That is, if one is composing spans with Hamiltonians, shouldn’t the Hamiltonians of the intermediate system match up for that to make sense, and shouldn’t the Hamiltonian on the apex be constrained to be compatible with the Hamiltonians on the feet?

Another way of saying what I’m trying to ask is, why isn’t “HamSy” an exactly analogous construction to the category of spans of Poisson maps, but on a slightly richer category? I gather that taking pullbacks is not straightforward in the richer setting; is that the root of it?

I wrote:

When you say “We can equip all the symplectic manifolds in this story with Hamiltonians”, and then sat it’s a big deal that you’re doing this for the feet of the spans as well as the apex, why is that?

Just for people who didn’t read my blog article: I didn’t exactly say it was a “big deal”. It’s only “big” in a rather technical sense, namely that this prevents our theory of open classical systems from fitting into the “decorated cospan” or “structured cospan” frameworks that some of us have been working on for a while. Nobody who wasn’t closely following these technical developments would care much.

When equipping manifolds with Hamiltonians, don’t you do so on the underlying category before taking spans?

No, because we don’t want to require that the legs of the spans are morphisms “preserving” the Hamiltonians, and perhaps more importantly when we compose two spans to get the Hamiltonian on the new apex we add the Hamiltonians on the apices of the spans we’re composing (after pulling them back) and subtract the Hamiltonian on the common foot (after pulling it back).

This tricky-sounding rule says that when you glue together two physical systems by identifying a piece of one with an isomorphic piece of the other, the energy of the resulting system is the sum of the energies of the two systems you glued together, minus the energy in the piece you’ve identified. The subtraction is necessary to avoid “double-counting”.

Another way of saying what I’m trying to ask is, why isn’t “HamSy” an exactly analogous construction to the category of spans of Poisson maps, but on a slightly richer category?

I think not; I think $\mathsf{HamSy}$ is not a decorated or structured cospan category, thanks to the sneaky rule for getting the Hamiltonian of a composite system.

I should try to prove it’s not.

I gather that taking pullbacks is not straightforward in the richer setting; is that the root of it?

That’s a somewhat different issue that we also have to deal with.

He wrote:

When equipping manifolds with Hamiltonians, don’t you do so on the underlying category before taking spans?

No, because we don’t want to require that the legs of the spans are morphisms “preserving” the Hamiltonians, and perhaps more importantly when we compose two spans to get the Hamiltonian on the new apex we add the Hamiltonians on the apices of the spans we’re composing (after pulling them back) and subtract the Hamiltonian on the common foot (after pulling it back).

Aren’t there constraints on the Hamiltonians of subsystems of a common system? If not, why not, and if so, shouldn’t there be morphisms (possibly in the opposite direction to the Poisson maps) expressing this? The formula you describe does sound a lot like the formula you would get for a pushout, after all.

I wrote:

Aren’t there constraints on the Hamiltonians of subsystems of a common system?

Not sure what you mean. When we compose open systems, which are cospans with extra structure, we demand that the Hamiltonians on the feet match. But we don’t impose any relation between the Hamiltonian on the feet of a span and the Hamiltonian on its apex.

If not, why not?

Because there’s no way to determine the energy of a subsystem just from the energy of the whole system, or vice versa; in general they obey no relation whatsoever.

He wrote:

In the example of particles on ends of a spring you give in the blog post, the kinetic energy of either particle surely makes an additive contribution to the energy of the system, doesn’t it? Although the calculation for potentials seems more difficult…

As another example, if I have (a classical model of) two atoms in a molecule (such as the two hydrogens in a water molecule), their individual energies are not determined by the energy of the molecule, nor do their energies determine the energy of the entire system since there is another big piece attached, but there must surely be constraints in each direction?

I wrote:

In the example of particles on ends of a spring you give in the blog post, the kinetic energy of either particle surely makes an additive contribution to the energy of the system, doesn’t it?

Kinetic energy is additive in a certain sense. But when you’re doing Hamiltonian mechanics with general symplectic manifolds, there’s no concept of “kinetic” versus “potential” energy: energy is just an arbitrary smooth function on your symplectic manifold of states.

In this situation there’s no general relation that holds between the energy of a whole system and the energies of two randomly chosen subsystems of that system, so we specify all three.

We also have a setup for open Lagrangian systems, which is formally analogous.

In the Lagrangian approach there’s a distinction between kinetic and potential energy built in - at least at the level of generality we work at (which is not maximally general). The configuration space (= space of possible “positions”) is a Riemannian manifold. A point in the tangent bundle specifies a position and velocity. The Riemannian metric lets us compute the kinetic energy from the velocity. The potential energy is an arbitrary smooth function on the Riemannian manifold. The Lagrangian is the kinetic energy minus the potential energy.

Now our open systems are spans of Riemannian manifolds where the apex and feet are equipped with potential functions. When we compose these spans we do the same trick of adding the potentials for the apices of the two spans and subtracting off the potential for their common foot — again to prevent “double-counting”.

Re: Open Systems in Classical Mechanics

This blog was posted at the perfect time for me (or, I am seeing it at the perfect time). Let me explain:

I have been working on compositional systems theory for a bit now. I had come up with a definition of “dynamical system doctrine” and a way of turning such a doctrine into a doubly indexed category (unital lax functor from a double category into the double category of categories, functors, and profunctors, or as I later realized, double copresheaf) of “systems”. By “way” here, I mean really cartesian 2-functor.

But I knew that this wasn’t the only way to think about composable systems. There are plenty others: in particular, there are the Baez-Fong-et-al style decorated/structured cospan systems, which include things like circuits and Petri nets. And there are the Willems-style variable sharing systems. All of them should give doubly indexed categories, because a doubly indexed category just says what it means to compose systems via an interconnection pattern, and how maps of systems can change their interface.

As it turned out, one of my technical pieces in my construction of the doubly indexed category of systems from my “doctrine” — the “vertical slice construction” that takes a double functor into a doubly indexed category, indexed by its codomain — could be repurposed for each of these two other ways of composing systems!

So this is what I’ve been thinking about recently: “paradigms of composition”. My current definition is that a paradigm of composition consists of a cartesian 2-category of “doctrines”, and a cartesian 2-functor which takes a doctrine to a doubly indexed category of “systems” in that doctrine, indexed by their “interface”. (Cartesian-ness is really there so that this construction preserves pseudo-monoids and then we don’t have to check coherences by hand…) Here are a few examples:

The port-plugging paradigm is defined like this: A port-plugging doctrine is a category $\mathcal{C}$ with finite colimits (“of systems”) with a chosen object $P$ (“a port”). The object $P$ induces a finite colimit preserving functor from finite sets to $\mathcal{C}$ which sends a finite set to the coproduct of $P$. We then define the doubly indexed category of systems in a doctrine by taking the double functor $0 : \ast \to \text{coSpan}(\mathcal{C})$ which picks out the initial object, taking the vertical slice construction of this to get a doubly indexed category indexed by $\text{coSpan}(\mathcal{C})$, and then restricting this along the double functor $\text{coSpan}(\text{Fin}) \to \text{coSpan}(\mathcal{C})$ induced by $P$. Unfolding this definition, we can see that an interface in a doctrine is a finite set of ports, and a system with that interface is a system (object of $\mathcal{C}$) with some “boundary” ports pointed to by that finite set. A interaction pattern between interfaces is a cospan of finite sets, and to have systems interact along this pattern we take the pushout in $\mathcal{C}$, gluing the ports together. Maps of systems are just the maps in $\mathcal{C}$, indexed by their action on ports. So this is a version of the “structured cospan” POV on systems. We can also do this with ports of various types which can only be glued if their types match up.

Another example is the input/output paradigm I talked about at ACT2020 and at some talks at MIT. An input/output doctrine is an indexed category whose base is a category of spaces and whose fibers are categories of bundles, equipped with a section that sends every space to its “tangent bundle”. Composition is done with generalized lenses. My talks about this can be found on my web-page.

Finally, the variable sharing paradigm is constructed similarly (but not perfectly dually) to the port-plugging paradigm. A doctrine in the variable sharing paradigm is a category $\mathcal{C}$ with finite limits. The doubly indexed category of systems is the vertical slice construction of $1 : \ast \to \text{Span}(\mathcal{C})$. A system with interface $I$ for this doctrine is then a map $S \to I$ exposing some variables of the system, and composition is given by pullback along a span which tells us how to share exposed variables.

This brings me back to this blog post. I knew of a “Willems-style” doctrine in this paradigm by taking the doctrine to be the category of Schultz-Spivak behavior types. But having read some of Urs’ writing on the nLab, I expected that Hamiltonian mechanics be a doctrine in the variable sharing paradigm. But I was having some trouble understanding what that really meant.

This blog post explains it to me wonderfully! The reason I was having trouble fitting it into this variable sharing paradigm is because its a bit more complicated than just spans in a category of symplectic manifolds and maps preserving the symplectic form. I wonder if there is a nice paradigm which includes the Hamiltonian and Lagrangian systems as you describe them — composing span-like — as doctrines. Or, perhaps, an expansion of the categories in the diffeological direction which lets them be variable sharing doctrines. I’m excited to read the paper!