## Network Theory in Turin

23 May, 2015

Here are the slides of the talk I’m giving on Monday to kick off the Categorical Foundations of Network Theory workshop in Turin:

This is a long talk, starting with the reasons I care about this subject, and working into the details of one particular project: the categorical foundations of networks as applied to electrical engineering and control theory. There are lots of links in blue; click on them for more details!

## Information and Entropy in Biological Systems (Part 4)

21 May, 2015

I kicked off the workshop on Information and Entropy in Biological Systems with a broad overview of the many ways information theory and entropy get used in biology:

• John Baez, Information and entropy in biological systems.

Abstract. Information and entropy are being used in biology in many different ways: for example, to study biological communication systems, the ‘action-perception loop’, the thermodynamic foundations of biology, the structure of ecosystems, measures of biodiversity, and evolution. Can we unify these? To do this, we must learn to talk to each other. This will be easier if we share some basic concepts which I’ll sketch here.

The talk is full of links, in blue. If you click on these you can get more details. You can also watch a video of my talk:

## Information and Entropy in Biological Systems (Part 3)

20 May, 2015

We had a great workshop on information and entropy in biological systems, and now you can see what it was like. I think I’ll post these talks one a time, or maybe a few at a time, because they’d be overwhelming taken all at once.

So, let’s dive into Chris Lee’s exciting ideas about organisms as ‘information evolving machines’ that may provide ‘disinformation’ to their competitors. Near the end of his talk, he discusses some new results on an ever-popular topic: the Prisoner’s Dilemma. You may know about this classic book:

• Robert Axelrod, The Evolution of Cooperation, Basic Books, New York, 1984. Some passages available free online.

If you don’t, read it now! He showed that the simple ‘tit for tat’ strategy did very well in some experiments where the game was played repeatedly and strategies who did well got to ‘reproduce’ themselves. This result was very exciting, so a lot of people have done research on it. More recently a paper on this subject by William Press and Freeman Dyson received a lot of hype. I think this is a good place to learn about that:

• Mike Shulman, Zero determinant strategies in the iterated Prisoner’s Dilemma, The n-Category Café, 19 July 2012.

Chris Lee’s new work on the Prisoner’s Dilemma is here, cowritten with two other people who attended the workshop:

The art of war: beyond memory-one strategies in population games, PLOS One, 24 March 2015.

Abstract. We show that the history of play in a population game contains exploitable information that can be successfully used by sophisticated strategies to defeat memory-one opponents, including zero determinant strategies. The history allows a player to label opponents by their strategies, enabling a player to determine the population distribution and to act differentially based on the opponent’s strategy in each pairwise interaction. For the Prisoner’s Dilemma, these advantages lead to the natural formation of cooperative coalitions among similarly behaving players and eventually to unilateral defection against opposing player types. We show analytically and empirically that optimal play in population games depends strongly on the population distribution. For example, the optimal strategy for a minority player type against a resident tit-for-tat (TFT) population is ‘always cooperate’ (ALLC), while for a majority player type the optimal strategy versus TFT players is ‘always defect’ (ALLD). Such behaviors are not accessible to memory-one strategies. Drawing inspiration from Sun Tzu’s the Art of War, we implemented a non-memory-one strategy for population games based on techniques from machine learning and statistical inference that can exploit the history of play in this manner. Via simulation we find that this strategy is essentially uninvadable and can successfully invade (significantly more likely than a neutral mutant) essentially all known memory-one strategies for the Prisoner’s Dilemma, including ALLC (always cooperate), ALLD (always defect), tit-for-tat (TFT), win-stay-lose-shift (WSLS), and zero determinant (ZD) strategies, including extortionate and generous strategies.

And now for the talk! Click on the talk title here for Chris Lee’s slides, or go down and watch the video:

Abstract. Information theory is an intuitively attractive way of thinking about biological evolution, because it seems to capture a core aspect of biology—life as a solution to “information problems”—in a fundamental way. However, there are non-trivial questions about how to apply that idea, and whether it has actual predictive value. For example, should we think of biological systems as being actually driven by an information metric? One idea that can draw useful links between information theory, evolution and statistical inference is the definition of an information evolving machine (IEM) as a system whose elements represent distinct predictions, and whose weights represent an information (prediction power) metric, typically as a function of sampling some iterative observation process. I first show how this idea provides useful results for describing a statistical inference process, including its maximum entropy bound for optimal inference, and how its sampling-based metrics (“empirical information”, Ie, for prediction power; and “potential information”, Ip, for latent prediction power) relate to classical definitions such as mutual information and relative entropy. These results suggest classification of IEMs into several distinct types:

1. Ie machine: e.g. a population of competing genotypes evolving under selection and mutation is an IEM that computes an Ie equivalent to fitness, and whose gradient (Ip) acts strictly locally, on mutations that it actually samples. Its transition rates between steady states will decrease exponentially as a function of evolutionary distance.

2. “Ip tunneling” machine: a statistical inference process summing over a population of models to compute both Ie, Ip can directly detect “latent” information in the observations (not captured by its model), which it can follow to “tunnel” rapidly to a new steady state.

3. disinformation machine (multiscale IEM): an ecosystem of species is an IEM whose elements (species) are themselves IEMs that can interact. When an attacker IEM can reduce a target IEM’s prediction power (Ie) by sending it a misleading signal, this “disinformation dynamic” can alter the evolutionary landscape in interesting ways, by opening up paths for rapid co-evolution to distant steady-states. This is especially true when the disinformation attack targets a feature of high fitness value, yielding a combination of strong negative selection for retention of the target feature, plus strong positive selection for escaping the disinformation attack. I will illustrate with examples from statistical inference and evolutionary game theory. These concepts, though basic, may provide useful connections between diverse themes in the workshop.

## PROPs for Linear Systems

18 May, 2015

Eric Drexler likes to say: engineering is dual to science, because science tries to understand what the world does, while engineering is about getting the world to do what you want. I think we need a slightly less ‘coercive’, more ‘cooperative’ approach to the world in order to develop ‘ecotechnology’, but it’s still a useful distinction.

For example, classical mechanics is the study of what things do when they follow Newton’s laws. Control theory is the study of what you can get them to do.

Say you have an upside-down pendulum on a cart. Classical mechanics says what it will do. But control theory says: if you watch the pendulum and use what you see to move the cart back and forth correctly, you can make sure the pendulum doesn’t fall over!

Control theorists do their work with the help of ‘signal-flow diagrams’. For example, here is the signal-flow diagram for an inverted pendulum on a cart:

When I take a look at a diagram like this, I say to myself: that’s a string diagram for a morphism in a monoidal category! And it’s true. Jason Erbele wrote a paper explaining this. Independently, Bonchi, Sobociński and Zanasi did some closely related work:

• John Baez and Jason Erbele, Categories in control.

• Filippo Bonchi, Paweł Sobociński and Fabio Zanasi, Interacting Hopf algebras.

• Filippo Bonchi, Paweł Sobociński and Fabio Zanasi, A categorical semantics of signal flow graphs.

I’ll explain some of the ideas at the Turin meeting on the categorical foundations of network theory. But I also want to talk about this new paper that Simon Wadsley of Cambridge University wrote with my student Nick Woods:

• Simon Wadsley and Nick Woods, PROPs for linear systems.

This makes the picture neater and more general!

You see, Jason and I used signal flow diagrams to give a new description of the category of finite-dimensional vector spaces and linear maps. This category plays a big role in the control theory of linear systems. Bonchi, Sobociński and Zanasi gave a closely related description of an equivalent category, $\mathrm{Mat}(k),$ where:

• objects are natural numbers, and

• a morphism $f : m \to n$ is an $n \times m$ matrix with entries in the field $k,$

and composition is given by matrix multiplication.

But Wadsley and Woods generalized all this work to cover $\mathrm{Mat}(R)$ whenever $R$ is a commutative rig. A rig is a ‘ring without negatives’—like the natural numbers. We can multiply matrices valued in any rig, and this includes some very useful examples… as I’ll explain later.

Theorem. Whenever $R$ is a commutative rig, $\mathrm{Mat}(R)$ is the PROP for bicommutative bimonoids over $R.$

This result is quick to state, but it takes a bit of explaining! So, let me start by bringing in some definitions.

### Bicommutative bimonoids

We will work in any symmetric monoidal category, and draw morphisms as string diagrams.

A commutative monoid is an object equipped with a multiplication:

and a unit:

obeying these laws:

For example, suppose $\mathrm{FinVect}_k$ is the symmetric monoidal category of finite-dimensional vector spaces over a field $k$, with direct sum as its tensor product. Then any object $V \in \mathrm{FinVect}_k$ is a commutative monoid where the multiplication is addition:

$(x,y) \mapsto x + y$

and the unit is zero: that is, the unique map from the zero-dimensional vector space to $V.$

Turning all this upside down, cocommutative comonoid has a comultiplication:

and a counit:

obeying these laws:

For example, consider our vector space $V \in \mathrm{FinVect}_k$ again. It’s a commutative comonoid where the comultiplication is duplication:

$x \mapsto (x,x)$

and the counit is deletion: that is, the unique map from $V$ to the zero-dimensional vector space.

Given an object that’s both a commutative monoid and a cocommutative comonoid, we say it’s a bicommutative bimonoid if these extra axioms hold:

You can check that these are true for our running example of a finite-dimensional vector space $V.$ The most exciting one is the top one, which says that adding two vectors and then duplicating the result is the same as duplicating each one, then adding them appropriately.

Our example has some other properties, too! Each element $c \in k$ defines a morphism from $V$ to itself, namely scalar multiplication by $c:$

$x \mapsto c x$

We draw this as follows:

These morphisms are compatible with the ones so far:

Moreover, all the ‘rig operations’ in $k$—that is, addition, multiplication, 0 and 1, but not subtraction or division—can be recovered from what we have so far:

We summarize this by saying our vector space $V$ is a bicommutative bimonoid ‘over $k$‘.

More generally, suppose we have a bicommutative bimonoid $A$ in a symmetric monoidal category. Let $\mathrm{End}(A)$ be the set of bicommutative bimonoid homomorphisms from $A$ to itself. This is actually a rig: there’s a way to add these homomorphisms, and also a way to ‘multiply’ them (namely, compose them).

Suppose $R$ is any commutative rig. Then we say $A$ is a bicommutative bimonoid over $R$ if it’s equipped with a rig homomorphism

$\Phi : R \to \mathrm{End}(A)$

This is a way of summarizing the diagrams I just showed you! You see, each $c \in R$ gives a morphism from $A$ to itself, which we write as

The fact that this is a bicommutative bimonoid endomorphism says precisely this:

And the fact that $\Phi$ is a rig homomorphism says precisely this:

So sometimes the right word is worth a dozen pictures!

What Jason and I showed is that for any field $k,$ the $\mathrm{FinVect}_k$ is the free symmetric monoidal category on a bicommutative bimonoid over $k.$ This means that the above rules, which are rules for manipulating signal flow diagrams, completely characterize the world of linear algebra!

Bonchi, Sobociński and Zanasi used ‘PROPs’ to prove a similar result where the field is replaced by a sufficiently nice commutative ring. And Wadlsey and Woods used PROPS to generalize even further to the case of an arbitrary commutative rig!

But what are PROPs?

### PROPs

A PROP is a particularly tractable sort of symmetric monoidal category: a strict symmetric monoidal category where the objects are natural numbers and the tensor product of objects is given by ordinary addition. The symmetric monoidal category $\mathrm{FinVect}_k$ is equivalent to the PROP $\mathrm{Mat}(k),$ where a morphism $f : m \to n$ is an $n \times m$ matrix with entries in $k,$ composition of morphisms is given by matrix multiplication, and the tensor product of morphisms is the direct sum of matrices.

We can define a similar PROP $\mathrm{Mat}(R)$ whenever $R$ is a commutative rig, and Wadsley and Woods gave an elegant description of the ‘algebras’ of $\mathrm{Mat}(R)$. Suppose $C$ is a PROP and $D$ is a strict symmetric monoidal category. Then the category of algebras of $C$ in $D$ is the category of strict symmetric monoidal functors $F : C \to D$ and natural transformations between these.

If for every choice of $D$ the category of algebras of $C$ in $D$ is equivalent to the category of algebraic structures of some kind in $D,$ we say $C$ is the PROP for structures of that kind. This explains the theorem Wadsley and Woods proved:

Theorem. Whenever $R$ is a commutative rig, $\mathrm{Mat}(R)$ is the PROP for bicommutative bimonoids over $R.$

The fact that an algebra of $\mathrm{Mat}(R)$ is a bicommutative bimonoid is equivalent to all this stuff:

The fact that $\Phi(c)$ is a bimonoid homomorphism for all $c \in R$ is equivalent to this stuff:

And the fact that $\Phi$ is a rig homomorphism is equivalent to this stuff:

This is a great result because it includes some nice new examples.

First, the commutative rig of natural numbers gives a PROP $\mathrm{Mat}.$ This is equivalent to the symmetric monoidal category $\mathrm{FinSpan},$ where morphisms are isomorphism classes of spans of finite sets, with disjoint union as the tensor product. Steve Lack had already shown that $\mathrm{FinSpan}$ is the PROP for bicommutative bimonoids. But this also follows from the result of Wadsley and Woods, since every bicommutative bimonoid $V$ is automatically equipped with a unique rig homomorphism

$\Phi : \mathbb{N} \to \mathrm{End}(V)$

Second, the commutative rig of booleans

$\mathbb{B} = \{F,T\}$

with ‘or’ as addition and ‘and’ as multiplication gives a PROP $\mathrm{Mat}(\mathbb{B}).$ This is equivalent to the symmetric monoidal category $\mathrm{FinRel}$ where morphisms are relations between finite sets, with disjoint union as the tensor product. Samuel Mimram had already shown that this is the PROP for special bicommutative bimonoids, meaning those where comultiplication followed by multiplication is the identity:

But again, this follows from the general result of Wadsley and Woods!

Finally, taking the commutative ring of integers $\mathbb{Z},$ Wadsley and Woods showed that $\mathrm{Mat}(\mathbb{Z})$ is the PROP for bicommutative Hopf monoids. The key here is that scalar multiplication by $-1$ obeys the axioms for an antipode—the extra morphism that makes a bimonoid into a Hopf monoid. Here are those axioms:

More generally, whenever $R$ is a commutative ring, the presence of $-1 \in R$ guarantees that a bimonoid over $R$ is automatically a Hopf monoid over $R.$ So, when $R$ is a commutative ring, Wadsley and Woods’ result implies that $\mathrm{Mat}(R)$ is the PROP for Hopf monoids over $R.$

Earlier, in their paper on ‘interacting Hopf algebras’, Bonchi, Sobociński and Zanasi had given an elegant and very different proof that $\mathrm{Mat}(R)$ is the PROP for Hopf monoids over $R$ whenever $R$ is a principal ideal domain. The advantage of their argument is that they build up the PROP for Hopf monoids over $R$ from smaller pieces, using some ideas developed by Steve Lack. But the new argument by Wadsley and Woods has its own charm.

In short, we’re getting the diagrammatics of linear algebra worked out very nicely, providing a solid mathematical foundation for signal flow diagrams in control theory!

## Carbon Emissions Stopped Growing?

15 May, 2015

In 2014, global carbon dioxide emissions from energy production stopped growing!

At least, that’s what preliminary data from the International Energy Agency say. It seems the big difference is China. The Chinese made more electricity from renewable sources, such as hydropower, solar and wind, and burned less coal.

In fact, a report by Greenpeace says that from April 2014 to April 2015, China’s carbon emissions dropped by an amount equal to the entire carbon emissions of the United Kingdom!

I want to check this, because it would be wonderful if true: a 5% drop. They say that if this trend continues, China will close out 2015 with the biggest reduction in CO2 emissions every recorded by a single country.

The International Energy Agency also credits Europe’s improved attempts to cut carbon emissions for the turnaround. In the US, carbon emissions has basically been dropping since 2006—with a big drop in 2009 due to the economic collapse, a partial bounce-back in 2010, but a general downward trend.

In the last 40 years, there have only been 3 times in which emissions stood still or fell compared to the previous year, all during global economic crises: the early 1980’s, 1992, and 2009. In 2014, however, the global economy expanded by 3%.

So, the tide may be turning! But please remember: while carbon emissions may start dropping, they’re still huge. The amount of the CO2 in the air shot above 400 parts per million in March this year. As Erika Podest of NASA put it:

CO2 concentrations haven’t been this high in millions of years. Even more alarming is the rate of increase in the last five decades and the fact that CO2 stays in the atmosphere for hundreds or thousands of years. This milestone is a wake up call that our actions in response to climate change need to match the persistent rise in CO2. Climate change is a threat to life on Earth and we can no longer afford to be spectators.

Here is the announcement by the International Energy Agency:

Global energy-related emissions of carbon dioxide stalled in 2014, IEA, 13 March 2015.

Their full report on this subject will come out on 15 June 2015. Here is the report by Greenpeace EnergyDesk:

China coal use falls: CO2 reduction this year could equal UK total emissions over same period, Greenpeace EnergyDesk.

I trust them less than the IEA when it comes to using statistics correctly, but someone should be able to verify their claims if true.

## Resource Theories

12 May, 2015

by Brendan Fong

Hugo Nava-Kopp and I have a new paper on resource theories:

• Brendan Fong and Hugo Nava-Kopp, Additive monotones for resource theories of parallel-combinable processes with discarding.

A mathematical theory of resources is Tobias Fritz’s current big project. He’s already explained how ordered monoids can be viewed as theories of resource convertibility in a three part series on this blog.

Ordered monoids are great, and quite familiar in network theory: for example, a Petri net can be viewed as a presentation for an ordered commutative monoid. But this work started in symmetric monoidal categories, together with my (Oxford) supervisor Bob Coecke and Rob Spekkens.

The main idea is this: think of the objects of your symmetric monoidal category as resources, and your morphisms as ways to convert one resource into another. The monoidal product or ‘tensor product’ in your category allows you to talk about collections of your resources. So, for example, in the resource theory of chemical reactions, our objects are molecules like oxygen O2, hydrogen H2, and water H2O, and morphisms things like the electrolysis of water:

2H2O → O2 + 2H2

This is a categorification of the ordered commutative monoid of resource convertibility: we now keep track of how we convert resources into one another, instead of just whether we can convert them.

Categorically, I find the other direction easier to state: being a category, the resource theory is enriched over $\mathrm{Set}$, while a poset is enriched over the poset of truth values or ‘booleans’ $\mathbb{B} = \{0,1\}.$ If we ‘partially decategorify’ by changing the base of enrichment along the functor $\mathrm{Set} \to \mathbb{B}$ that maps the empty set to 0 and any nonempty set to 1, we obtain the ordered monoid corresponding to the resource theory.

But the research programme didn’t start at resource theories either. The starting point was ‘partitioned process theories’.

Here’s an example that guided the definitions. Suppose we have a bunch of labs with interacting quantum systems, separated in space. With enough cooperation and funding, they can do big joint operations on their systems, like create entangled pairs between two locations. For ‘free’, however, they’re limited to classical communication between the locations, although they can do the full range of quantum operations on their local system. So you’ve got a symmetric monoidal category with objects quantum systems and morphisms quantum operations, together with a wide (all-object-including) symmetric monoidal subcategory that contains the morphisms you can do with local quantum operations and classical communication (known as LOCC operations).

This general structure: a symmetric monoidal category (or SMC for short) with a wide symmetric monoidal subcategory, is called a partitioned process theory. We call the morphisms in the SMC processes, and those in the subSMC free processes.

There are a number of methods for building a resource theory (i.e. an SMC) from a partitioned process theory. The unifying idea though, is that your new SMC has the processes $f,g$ as objects, and morphisms $f \to g$ ways of using the free processes to build $g$ from $f.$

But we don’t have to go to fancy sounding quantum situations to find examples of partitioned process theories. Instead, just look at any SMC in which each object is equipped with an algebraic structure. Then the morphisms defining this structure can be taken as our ‘free’ processes.

For example, in a multigraph category every object has the structure of a ‘special commutative Frobenius algebra’. That’s a bit of a mouthful, but John defined it a while back, and examples include categories where morphisms are electrical circuits, and categories where morphisms are signal flow diagrams.

So these categories give partitioned process theories! This idea of partitioning the morphisms into ‘free’ ones and ‘costly’ ones is reminiscent of what I was saying earlier about the operad of wiring diagrams about it being useful to separate behavioural structure from interconnection structure.

This suggests that we can also view the free processes as generating some sort of operad, that describes the ways we allow ourselves to use free processes to turn processes into other processes. If we really want to roll a big machine out to play with this stuff, framed bicategories may also be interesting; Spivak is already using them to get at questions about operads. But that’s all conjecture, and a bit of a digression.

To get back to the point, this was all just to say that if you find yourself with a bunch of resistors, and you ask ‘what can I build?’, then you’re after the resource theory apparatus.

• Bob Coecke, Tobias Fritz and Rob W. Spekkens, A mathematical theory of resources.

## Cospans, Wiring Diagrams, and the Behavioral Approach

5 May, 2015

joint with Brendan Fong

We’re getting ready for the Turin workshop on the
Categorical Foundations of Network Theory. So, we’re trying to get our thoughts in order.

Last time we talked about understanding types of networks as categories of decorated cospans. Earlier, David Spivak told us about understanding networks as algebras of an operad. Both these frameworks work at capturing notions of modularity and interconnection. Are they then related? How?

In this post we want to discuss some similarities between decorated cospan categories and algebras for Spivak’s operad of wiring diagrams. The main idea is that the two approaches are ‘essentially’ equivalent, but that compared to decorated cospans, Spivak’s operad formalism puts greater emphasis on the distinction between the ‘duplication’ and ‘deletion’ morphisms and other morphisms in our category.

The precise details are still to be worked out—jump in and help us!

We begin with a bit about operads in general. Recall that an operad is similar to a category, except that instead of a set $\mathrm{hom}(x,y)$ of morphisms from any object $x$ to any object $y,$ you have a set $\mathrm{hom}(x_1,\dots,x_n;y)$ of operations from any finite list of objects $x_1,...,x_n$ to any object $y .$ If we have an operation $f \in \mathrm{hom}(x_1,\dots,x_n;y),$ we can call $x_1, \dots, x_n$ the inputs of $f$ and call $y$ the output of $f .$

We can compose operations in an operad. To understand how, it’s easiest to use pictures. We draw an operation in $\mathrm{hom}(x_1,\dots,x_n;y)$ as a little box with $n$ wires coming in and one wire coming out:

The input wires should be labelled with the objects $x_1, \dots, x_n$ and the output wire with the object $y,$ but I haven’t done this.

We are allowed to compose these operations as follows:

as long as the outputs of the operations $g_1,\dots,g_n$ match the inputs of the operation $f .$ The result is a new operation which we call $f \circ (g_1,\dots,g_n) .$

We demand that there be unary operations $1_x \in \mathrm{hom}(x;x)$ serving as identities for composition, and we impose an associative law that makes a composite of composites like this well-defined:

So far this is the definition of a operad without permutations. In a full-fledged permutative operad, we can also permute the inputs of an operation $f$ and get a new operation:

which we call $f \sigma$ if $\sigma$ is the the permutation of the inputs. We demand that $(f \sigma) \sigma' = f (\sigma \sigma') .$ And finally, we demand that permutations act in a way that is compatible with composition. For example:

Here we see that $(f \sigma) \circ (g_1, \dots, g_n)$ is equal to some obvious other thing.

Finally, there is a law saying

$f \circ (g_1 \sigma_1, \dots, g_n \sigma_n) = (f \circ (g_1 , \dots, g_n)) \sigma$

for some choice of $\sigma$ that you can cook up from the permuations $\sigma_i$ in an obvious way. We leave it as an exercise to work out the details. By the way, one well-known book on operads accidentally omits this law, so here’s a rather more lengthy exercise: read this book, see which theorems require this law, and correct their proofs!

Operads are similar to symmetric monoidal categories. The idea is that in a symmetric monoidal category you can just form the tensor product $x_1 \otimes \dots \otimes x_n$ and talk about the set of morphisms $x_1 \otimes \cdots \otimes \cdots x_n \to y .$ Indeed any symmetric monoidal category gives an operad in this way: just define $\mathrm{hom}(x_1,...,x_n;y)$ to be $\mathrm{hom}(x_1 \otimes \cdots \otimes x_n, y) .$ If we do this with Set, which is a symmetric monoidal category using the usual cartesian product of sets, we get an operad called $\mathrm{Set}.$

An algebra for an operad $O$ is an operad homomorphism $O \to \mathrm{Set}.$ We haven’t said what an operad homomorphism is, but you can probably figure it out yourself. The point is this: an algebra for $O$ turns the abstract operations in $O$ into actual operations on sets!

Finally, we should warn you that operads come in several flavors, and we’ve been talking about ‘typed permutative operads’. ‘Typed’ means that there’s more than one object; ‘permutative’ means that we have the ability to permute the input wires. When people say ‘operad’, they often mean an untyped permutative operad. For that, just specialize down to the case where there’s only one object $x.$

You can see a fully precise definition of untyped permutative operads here:

along with the definition of an untyped operad without permutations.

### The operad of wiring diagrams

Spivak’s favorite operad is the operad of wiring diagrams. The operad of wiring diagrams $WD$ is an operad version of $\mathrm{Cospan}(\mathrm{FinSet}),$ constructed in the vein suggested above: the objects are finite sets, and an operation from a list of sets $X_1,...,X_n$ to a set $Y$ is a cospan

$X_1+ \cdots +X_n \rightarrow S \leftarrow Y$

Spivak draws such a thing as a big circle with $n$ small circles cut out from the interior:

The outside of the big circle has a set $Y$ of terminals marked on it, and each small circle has a set $X_i$ of terminals marked on it. Then in the interior of this shape there are wires connecting these terminals. This what he calls a wiring diagrams.

You compose these wiring diagrams by pasting other wiring diagrams into each of the small circles.

The relationship with our Frobenius monoid diagrams is pretty simple: we draw our ‘wiring diagrams’ $X \to Y$ in a square, with the $X$ terminals on the left and $Y$ terminals on the right. To get a Spivak-approved wiring diagram, glue the top and bottom edges of this square together, then flatten the cylinder you get down into an annulus, with the $X$-side on the inside and $Y$-side on the outside. If $X = X_1+X_2$ you can imagine gluing opposite edges of the inside circle together to divide it into two small circles accordingly, and so on.

### Relational algebras of type A

Algebras for wiring diagrams tell you what components you have available to wire together with your diagrams. An algebra for the operad of wiring diagrams is an operad homomorphism

$WD \to \mathrm{Set}$

What does this look like? Just like a functor for categories, it assigns to each natural number a set, and each wiring diagram a function.

In work related to decorated cospans (such as our paper on circuits or John and Jason’s work on signal flow diagrams), our semantics usually is constructed from a field of values—not a physicist’s ‘field’, bt an algebraist’s sort of ‘field’, where you can add, multiply, subtract and divide. For example, we like being able to assign a real number like a velocity, or potential, or current to a variable. This gives us vector spaces and a bunch of nice linear-algebraic structures.

Spivak works more generally: he’s interested in the structure when you just have a set of values. While this means we can’t do some of the nice things we could do with a field, it also means this framework can do things like talk about logic gates, where the variables are boolean ones, or number theoretic questions, where you’re interested in the natural numbers.

So to discuss semantics we pick a set $A$ of values, such as the real numbers or natural numbers or booleans or colors. We imagine then associating elements of this set to each wire in a wiring diagram. More technically, the algebra

$\mathrm{Rel}A: WD \to \mathrm{Set}$

then maps each finite set $X$ to the power set $\mathcal{P}(A^X)$ of the set $A^X$ of functions $X \to A .$

On the morphisms (the wiring diagrams themselves), this functor behaves as follows. Note that a function ($X \to A$) can be thought of as an ‘X-vector’ $(a_1,...,a_x)$ of ‘A-coordinates’. A wiring diagram $X \to Y$ is just a cospan

$X \to N \leftarrow Y$

in $\mathrm{FinSet},$ so it can be thought of as some compares

$X \to N$

followed by some copies

$N \to Y$

Thus, given a wiring diagram $X \to Y,$ we can consider a partial function that maps an $X$-vector to the $Y$-vector by doing these compares, and if it passes them does the copies and returns the resulting $Y$-vector, but if not returns ‘undefined’. We can then define a map $\mathcal{P}(A^X) \to \mathcal{P}(A^Y)$ which takes a set of $X$-vectors to its image under this partial function.

This semantics is called the relational $WD$-algebra of type A. We can think of it as being like the ‘light operations’ fragment of the signal flow calculus. By ‘light operations’, we mean the operations of duplication and deletion, which form a cocommutative comonoid:

and their time-reversed versions, ‘coduplication’ and ‘codeletion’, which form a commutatative monoid:

These fit together to form a Frobenius monoid, meaning that these equations hold:

And it’s actually extra-special, meaning that these equations hold:

Note that we can’t do the ‘dark operations’, because we only have a set $A$ of values, not a field, and the dark operations involve addition and zero!

### Operads and the behavioral approach

In formulating Frobenius monoids this way, Spivak achieves something that we’ve been working hard to find ways to achieve: a separation of the behavioral structure from the interconnection structure.

What do I mean by this? In his ‘behavioral approach‘, Willems makes the point that for all their elaborate and elegant formulation, in the end physical laws just come down to dividing the set of what might a priori be possible (the ‘universum’) into the set of things that actually are possible (the ‘behavior’), and the set of things that aren’t). Here the universum is the set $A^X$: a priori, on each of the wires in $X,$ we might associate any value of $A .$ For example, to the two wires at the ends of a resistor, we might a priori associate any pair of currents. But physical law, here Kirchhoff’s current law, says otherwise: the currents must be equal and opposite. So the ‘behavior’ is the subset $(i,-i)$ of the universum $\mathbb{R}^2.$

So you can say that to each object $X$ in the operad of wiring diagrams the relational algebra of type $A$ associates the set $\mathcal{P}(A^X)$ of possible behaviors—the universum is $A^X .$ ($\mathcal{P}(A^X)$ forms some sort of meta-universum, where you can discuss physical laws about physical laws, commonly called ‘principles’.)

The second key aspect of the behavioral approach is that the behaviors of larger systems can be constructed from the behaviors of its subsystems, if we understand the so-called ‘interconnection structure’ well enough. This is a key principle in engineering: we build big, complicated systems from a much smaller set of components, whether it be electronics from resistors and inductors, or mechanical devices from wheels and rods and ropes, or houses from Lego bricks. The various interconnection structures here are the wiring diagrams, and our relational algebras say they act by what Willems calls ‘variable sharing’.

This division between behavior and interconnection motivates the decorated cospan construction (where the decorations are the ‘components’, the cospans the ‘interconnection’) and also the multigraph categories discussed by Aleks Kissinger (where morphisms are the ‘components’, and the Frobenius monoid operations are the ‘interconnection’):

• Aleks Kissinger, Finite matrices are complete for (dagger-)multigraph categories.

So it’s good to have this additional way of thinking about things in our repertoire: operads describe ‘interconnection’, their algebras ‘behaviors’.

The separation Spivak achieves, however, seems to me to come at the cost of neat ways to talk about individual components, and perhaps this can be seen as the essential difference between the two approaches. By including our components as morphisms, we can talk more carefully about them and additional structure individual components have. On the other hand, by lumping all the components into the objects, Spivak can talk more carefully about how the interconnection structure acts on all behaviors at once.

### Other operads of wiring diagrams

One advantage of the operad approach is that you can easily tweak your operad to talk about different sorts of network structure. Sometimes you can make similar adjustments with decorated cospans too, such as working over the category of typed finite sets, rather than just finite sets, to discuss networks in which wires have types, and only wires of the same types can be connected together. A physical example is a model of a hydroelectric power plant, where you don’t want to connect a water pipe with an electrical cable! This is also a common technique in computer science, where you don’t want to try to multiply two strings of text, or try to interpret a telephone number as a truth value.

But some modifications are harder to do with decorated cospans. In some other papers, Spivak employs a more restricted operad of wiring diagrams, in which joining wires and terminating wires is not allowed, among other things. He uses this to formalise graphical languages for certain types of discrete-time processes, open dynamical systems, including mode-dependent ones.