Decorated Cospans

Last time I talked about a new paper I wrote with Brendan Fong. It’s about electrical circuits made of ‘passive’ components, like resistors, inductors and capacitors. We showed these circuits are morphisms in a category. Moreover, there’s a functor sending each circuit to its ‘external behavior’: what it does, as seen by someone who can only measure voltages and currents at the terminals.

Our paper uses a formalism that Brendan developed here:

• Brendan Fong, Decorated cospans, Theory and Applications of Categories 30 (2015), 1096–1120.

The idea here is we may want to take something like a graph with edges labelled by positive numbers: and say that some of its nodes are ‘inputs’, while others are ‘outputs’: This lets us treat our labelled graph as a ‘morphism’ from the set $X$ to the set $Y.$

The point is that we can compose such morphisms. For example, suppose we have another one of these things, going from $Y$ to $Z$: Since the points of $Y$ are sitting in both things: we can glue them together and get a thing going from $X$ to $Z$: That’s how we compose these morphisms.

Note how we’re specifying some nodes of our original thing as inputs and outputs: We’re using maps from two sets $X$ and $Y$ to the set $N$ of nodes of our graph. And a bit surprisingly, we’re not demanding that these maps be one-to-one. That turns out to be useful—and in general, when doing math, it’s dumb to make your definitions forbid certain possibilities unless you really need to.

So, our thing is really a cospan of finite sets—that is, a diagram of finite sets and functions like this: together some extra structure on the set $N$. This extra structure is what Brendan calls a decoration, and it makes the cospan into a ‘decorated cospan’. In this particular example, a decoration on $N$ is a way of making it into the nodes of a graph with edges labelled by positive numbers. But one could consider many other kinds of decorations: this idea is very general.

To formalize the idea of ‘a kind of decoration’, Brendan uses a functor $F: \mathrm{FinSet} \to \mathrm{Set}$

sending each finite set $N$ to a set of $F(N).$ This set $F(N)$ is the set of decorations of the given kind that we can put on $N.$

So, for any such functor $F,$ a decorated cospan of finite sets is a cospan of finite sets: together with an element of $F(N).$

But in fact, Brendan goes further. He’s not content to use a functor $F: \mathrm{FinSet} \to \mathrm{Set}$

to decorate his cospans.

First, there’s no need to limit ourselves to cospans of finite sets: we can replace $\mathrm{FinSet}$ with some other category! If $C$ is any category with finite colimits, there’s a category $\mathrm{Cospan}(C)$ with:

• objects of $C$ as its objects,
• isomorphism classes of cospans between these as morphisms.

Second, there’s no need to limit ourselves to decorations that are elements of a set: we can replace $\mathrm{Set}$ with some other category! If $D$ is any symmetric monoidal category, we can define an element of an object $d \in D$ to be a morphism $f: I \to d$

where $I$ is the unit for the tensor product in $D.$

So, Brendan defines decorated cospans at this high level of generality, and shows that under some mild conditions we can compose them, just as in the pictures we saw earlier.

Here’s one of the theorems Brendan proves:

Theorem. Suppose $C$ is a category with finite colimits, and make $C$ into a symmetric monoidal category with its coproduct as the tensor product. Suppose $D$ is a symmetric monoidal category, and suppose $F: C \to D$ is a lax symmetric monoidal functor. Define an F-decorated cospan to be a cospan in $C$ together with an element of $F(N).$ Then there is a category with

• objects of $C$ as its objects,
• isomorphism classes of $F$-decorated cospans as its morphisms.

This is called the F-decorated cospan category, $FCospan.$ This category becomes symmetric monoidal in a natural way. It is then a dagger compact category.

(You may not know all this jargon, but ‘lax symmetric monoidal’, for example, talks about how we can take decorations on two things and get a decoration on their disjoint union, or ‘coproduct’. We need to be able to do this—as should be obvious from the pictures I drew. Also, a ‘dagger compact category’ is the kind of category whose morphisms can be drawn as networks.)

Brendan also explains how to get functors between decorated cospan categories. We need this in our paper on electrical circuits, because we consider several categories where morphisms is a circuit, or something that captures some aspect of a circuit. Most of these categories are decorated cospan categories. We want to get functors between them. And often we can just use Brendan’s general results to get the job done! No fuss, no muss: all the hard work has been done ahead of time.

I expect to use this technology a lot in my work on network theory.

10 Responses to Decorated Cospans

1. domenico says:

It is interesting the Decorated Cospan, it is similar to the field in physics: if there is a grid, in a N-dimensional space, and vectors along the field line, then Decorated Cospan seem that includes a physical field (within the limits of close points).

2. Brendan Fong says:

Thanks for the comment domenico! Could you please elaborate a bit though? It sounds very interesting, but I’m afraid I don’t quite get it. What the cospans correspond to in your example? And what would the decorations be?

• domenico says:

I thought a normal vector field, where the points and the vectors are the decorated cospans (the arrows of the cospans like real vectors), so that graph and flow curves can coincides with a limit of close distance of the vector field representation.

If it is so, then many theorems of the decorated cospans can be applied to the vector fields (with some limits), and vice versa.

3. I’ve got seven grad students working on this project—or actually eight, if you count Brendan Fong: I’ve been helping him on his dissertation, but he’s actually a student at Oxford.

Brendan was the first to join the project. I wanted him to work on electrical circuits, which are a nice familiar kind of network, a good starting point. But he went much deeper: he developed a general category­-theoretic framework for studying networks. We then applied it to electrical circuits, and other things as well.

4. I’ve got seven grad students working on this project—or actually eight, if you count Brendan Fong: I’ve been helping him on his dissertation, but he’s actually a student at Oxford.

Brendan was the first to join the project. I wanted him to work on electrical circuits, which are a nice familiar kind of network, a good starting point. But he went much deeper: he developed a general category­-theoretic framework for studying networks. We then applied it to electrical circuits, and other things as well.

5. At some point I gave Brendan Fong a project: describe the category whose morphisms are electrical circuits. He took up the challenge much more ambitiously than I’d ever expected, developing powerful general frameworks to solve not only this problem but also many others. He did this in a number of papers, most of which I’ve already discussed […]

6. But I want to say here what we do in our paper, because it’s pretty cool and it took a few years. To get things to work, we needed my student Brendan Fong to invent the right category-theoretic formalism: ‘decorated cospans’. But we also had to figure out the right way to think about open dynamical systems!

7. Two students in the Applied Category Theory 2018 school wrote a blog article about Brendan Fong’s theory of decorated cospans:

• Jonathan Lorand and Fabrizio Genovese, Hypergraph categories of cospans, The n-Category Café, 28 February 2018.

8. The monoidal Grothendieck construction plays an important role in our team’s work on network theory, in at least two ways. First, we use it to get a symmetric monoidal category, and then an operad, from any network model. Second, we use it to turn any decorated cospan category into a ‘structured cospan category’. I haven’t said anything about structured cospans yet, but they are an alternative approach to open systems, developed by my grad student Kenny Courser, that I’m very excited about. Stay tuned!

9. You may have heard of a similar idea: ‘decorated cospans’, invented by Brendan Fong. You may wonder what’s the difference!

Kenny’s talk explains the difference pretty well. Basically, decorated cospans that look isomorphic may not be technically isomorphic. There are just not enough isomorphisms between decorated cospans. As Kenny explains, structured cospans don’t suffer from this problem.

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