Last fall I read “A diagrammatic view of differential equations in physics” by Evan Patterson, Andrew Baas, Timothy Hosgood and James Fairbanks. They show that you can use diagrams to write down the sorts of equations on manifolds that physicists care about. In this post I’ll try to convey the main ideas of the paper to you so that: (1) I can get you excited about it and (2) I can refer back to these notes in the future.
So how does one think of an equation as a diagram?
The idea is pretty neat. First you’ve got to ask yourself: “what’s an equation?” Since an actually good answer to this questions can get rather philosophical and since I want to get to the point fast, I’ll wave my hands a bit: an equation is a way of relating “quantities” that are defined on some “space” of interest.
Then you’ve got to ask yourself: “what kind of mathematical structure can I use to impose relations on some quantities?”
If you’re thinking of graphs – the relational structure par excellence – then you’re not far off. They don’t really do the trick, but you’ll see in a second that diagrams (a.k.a. semantically-rich graphs) are exactly what we need.
(By the way, as an Italian who grew up in a culture where hand gestures can be very precise, I never really understood why “hand-waving” is considered another way of saying “sweeping details under the rug”.)
So having established the intutition behind where the diagrams come from, all that’s left is to figure out what we’ll mean by “space” and “physical quantities” defined on that space. This is summarized below.
| “Space” | A manifold \(M\) |
| Physical quantities defined in “space” | Sheaves on \(M\) |
| Equations | Diagrams relating physical quantities on the space (i.e. diagrams into the category of sheaves on \(M\)) |
I like to get the big picture first, but that’s just my general preference, so don’t worry if you’re lost a this point: I’ll unpack things below. Let’s start with an example (it’s the same example that’s given in the Patterson, Baas Hosgood & Fairbanks paper).
Example: the Discrete Heat Equation as a Diagrammatic Equation
I’ll be talking about the heat equation as an example, so, just as a reminder, let’s recall it: we say that a function \(u \colon (T \subseteq \mathbb{R}, U \subseteq \mathbb{R}^n) \to \mathbb{R}\) is a solution to the heat equation (where \(T\) is the time domain and \(U\) is the spacial domain) if \[\frac{\partial u}{\partial t} = \Delta u\] where \(\Delta u\) denotes the Laplacian \[\Delta u = \sum_{1}^n \frac{\partial^2 u}{\partial x_i^2}\] in terms of the Cartesian coordinates \(x_1, \dots, x_n\).
Now, since the goal is to explain how to think of equations as diagrams and since I’m trying to follow the Diagramamtic Equations paper closely here (so that you can seemelessly transition into reading it, if you’re interested), I’ll focus on the discrete heat equation.
In the discrete case, rather than thinking of space as \(\mathbb{R}^n\), we’ll think of it as a graph \(G\) with vertices \(V\) (for instance we could take \(G = \mathbb{Z}^n\)). Just as in the case above, we’ll want a solution (a “quantity”) to the heat equation on the spacial domain \(G\) to be a function of the form \(u \colon \mathbb{N} \times V \to \mathbb{R}\) (note that we’re thinking of everything, including time as being discrete).
The discrete derivative with respect to time of such a function \(u\) is then given as the operator \(\partial \colon \mathbb{R}^{\mathbb{N} \times V} \to \mathbb{R}^{\mathbb{N} \times V}\) which takes any function \(u\) to the function \[\partial \: u \colon (n, x) \mapsto u(n+1, x) – u(n,x).\] One can similarly define the discrete Laplacian as an operator \(\Delta \colon \mathbb{R}^{\mathbb{N} \times V} \to \mathbb{R}^{\mathbb{N} \times V} \). I won’t give the whole definition here (you can just check out page 6 of the paper) since the point I’m trying to get accross is that the heat equation can be written as the following diagram in \(\mathsf{Vect}_{\mathbb{R}}\) (the category of \(\mathbb{R}\)eal vector spaces and linear maps)
To undertand why this is an equation, notice that you can pick out a single vector \(u \in \mathbb{R}^{\mathbb{N} \times V}\) as a generalized element i.e. a linear map \(\mathbb{R} \to \mathbb{R}^{\mathbb{N} \times V}\) which takes the basis vector of \(\mathbb{R}\) to \(u\) (the vector we were trying to pick out). Then the commutativity of the following diagram is precisely the statement of the heat equation!
Summarising what we’ve learnt so far, we have the following table.
| Traditional view | Diagrammatic View |
| An equation \(e\) | A diagram \(d_e\) whose morphisms are the operators of \(e\). |
| A solution \(u\) to \(e\) | A cone over the diagram \(d_e\). |
Diagrammatic Equations
If you get the general idea by now (please ask questions in the comments, if not!) then it’s time to do two things: (1) think of the continuous case and (2) figure out what kinds of properties we want the “operators” to have (and what category we ought to live in). I’ll give a very concise summary below of how to think of all of this. For that, though, you’ll need to know about sheaves.
Sheaves. I already explained sheaves in some previous posts (see §1, §2 and §3 if you want to learn a bunch about them) but, for what I’ll be speaking about in this post, we can stick with the more down-to-Earth notion of a sheaf on a topological space.
Df. Let \(X\) be a topological space and \(\mathsf{S}\) be a category. An \(S\)-sheaf on \(X\) is a contravariant functor \[\mathcal{F} \colon (\mathcal{O}X)^{op} \to \mathsf{S} \] from the poset of open sets of \(X\) to \(\mathsf{S}\) satisfying the sheaf condition; i.e. the requirement that, if you give me a collection \((U_i)_{i \in I}\) of opens in \(X\) whose union is an open \(U\), then \(\mathcal{F}U\) is the limit of the \(FU_i\) over the diagram induced by \(I\) (in other words \(\mathcal{F}\) takes colimits to limits).
I’m telling you about sheaves becasue we’ll use them to represent the physical quantities involved in our diagrammatic equations: these quantities will be described as sheaves on a fixed space-time domain represented by a manifold \(M\) with corners.
For this post, I’ll only care about \(\mathsf{Vect}_{\mathbb{R}}\)-valued sheaves such as the sheaf \(\mathfrak{X}_M\) of smooth vector fields on a manifold \(M\). Such sheaves (as do all sheaves for that matter) assemble into a category \(\mathsf{Sh}(M, \mathsf{Vect}_{\mathbb{R}})\) of \(\mathsf{Vect}_{\mathbb{R}}\)-valued sheaves on \(M\) (the morphisms are natural transformations). Following the notation in the Diagrammatic Equations paper, I’ll abbrevite \(\mathsf{Sh}(M, \mathsf{Vect}_{\mathbb{R}})\) as \(\mathsf{Sh}_\mathbb{R} M\).
Recall that in our earlier Heat Equation example the solution to the heat equation on \(\mathbb{R}^n\) was a function \[u \colon \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}\] assigning quantities (heat, for instance) to space-time. The idea in Patterson, Baas Hosgood & Fairbanks’ paper is to replace \(\mathbb{R}^{T \times \mathbb{R}^n}\) by a sheaf on a fixed manifold \(M\) representing space-time. Indeed here’s how they put it:
“[p]hysical quantities are taken to be sections of sheaves on the spatial domain M or the space-time domain \(M \times [0, \infty)\). Loosely speaking, a sheaf on a topological space is a coherent family of quantities defined on open subsets that can be glued together from smaller subsets whenever they agree on the overlaps.“
We’ve already seen such a sheaf \(\mathfrak{X}_M\) (first on the list below) and the following screenshot has a few more physically-relevant examples they give in the paper.
Finally, with this view in mind, we can speak of a diagram \[e \colon J \to \mathsf{Sh}_{\mathbb{R}} M \] as a diagrammatic equation \(e\) on a manifold \(M\) whose solutions (as in the example above) are cones over \(e\).
Even though I’m only thinking of \(\mathsf{Vect}_{\mathbb{R}}\)-valued sheaves (note that things are presented in greater generality in the original paper), I still find this “definition” of a diagramamtic equation remarkably satisfying.
Until next time,
Ben