## Systems as Wiring Diagram Algebras

28 January, 2019

Check out the video of Christina Vasilakopoulou’s talk, the third in the Applied Category Theory Seminar here at U. C. Riverside! It was nicely edited by Paola Fernandez and uploaded by Joe Moeller.

Abstract. We will start by describing the monoidal category of labeled boxes and wiring diagrams and its induced operad. Various kinds of systems such as discrete and continuous dynamical systems have been expressed as algebras for that operad, namely lax monoidal functors into the category of categories. A major advantage of this approach is that systems can be composed to form a system of the same kind, completely determined by the specific way the composite systems are interconnected (‘wired’ together). We will then introduce a generalized system, called a machine, again as a wiring diagram algebra. On the one hand, this abstract concept is all-inclusive in the sense that discrete and continuous dynamical systems are sub-algebras; on the other hand, we can specify succinct categorical conditions for totality and/or determinism of systems that also adhere to the algebraic description.

• Patrick Schultz, David I. Spivak and Christina Vasilakopoulou, Dynamical systems and sheaves.

• Dmitry Vagner, David I. Spivak and Eugene Lerman, Algebras of open dynamical systems on the operad of wiring diagrams.

## The Mathematics of the 21st Century

13 January, 2019

Check out the video of my talk, the first in the Applied Category Theory Seminar here at U. C. Riverside. It was nicely edited by Paola Fernandez and uploaded by Joe Moeller.

Abstract. The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics—and be affected by it—just as dramatically as the agricultural revolution or industrial revolution. We should get ready!

The slides are rather hard to see in the video, but you can read them here while you watch the talk. Click on links in green for more information!

## Applied Category Theory Seminar

14 December, 2018

We’re going to have a seminar on applied category theory here at U. C. Riverside! My students have been thinking hard about category theory for a few years, but they’ve decided it’s time to get deeper into applications. Christian Williams, in particular, seems to have caught my zeal for trying to develop new math to help save the planet.

We’ll try to videotape the talks to make it easier for you to follow along. I’ll also start discussions here and/or on the Azimuth Forum. It’ll work best if you read the papers we’re talking about and then join these discussions. Ask questions, and answer any questions you can!

Here’s how the schedule of talks is shaping up so far. The talks are on Tuesdays 3:34–5:00 pm in Room 268 of Skye Hall, the mathematics building at U. C. Riverside.

January 8, 2019: John Baez
Mathematics in the 21st century

Abstract. The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics – and be affected by it – just as dramatically as the agricultural revolution or industrial revolution. We should get ready!

Talk slides: Mathematics in the 21st century.

Also try these slides and videos from related talks:

January 15, 2019: Jonathan Lorand
Problems in symplectic linear algebra

Abstract. In this talk we will look at various examples of classification problems in symplectic linear algebra: conjugacy classes in the symplectic group and its Lie algebra, linear lagrangian relations up to conjugation, tuples of (co)isotropic subspaces. I will explain how many such problems can be encoded using the theory of symplectic poset representations, and will discuss some general results of this theory. Finally, I will recast this discussion from a broader category-theoretic perspective.

Talk slides: Problems in symplectic linear algebra.

• Jonathan Lorand, Classifying linear canonical relations.

• Jonathan Lorand and Alan Weinstein, Decomposition of (co)isotropic relations.

January 22, 2019: Christina Vasilakopoulou
Systems as wiring diagram algebras

Abstract. We will start by describing the monoidal category of labeled boxes and wiring diagrams and its induced operad. Various kinds of systems such as discrete and continuous dynamical systems have been expressed as algebras for that operad, namely lax monoidal functors into the category of categories. A major advantage of this approach is that systems can be composed to form a system of the same kind, completely determined by the specific way the composite systems are interconnected (‘wired’ together). We will then introduce a generalized system, called a machine, again as a wiring diagram algebra. On the one hand, this abstract concept is all-inclusive in the sense that discrete and continuous dynamical systems are sub-algebras; on the other hand, we can specify succinct categorical conditions for totality and/or determinism of systems that also adhere to the algebraic description.

Christina Vasilakopoulou’s talk was based on this paper:

• Patrick Schultz, David I. Spivak and Christina Vasilakopoulou, Dynamical systems and sheaves.

but she focused more on the algebraic description (and conditions for deterministic/total systems) rather than the sheaf theoretic aspect of the input types. This work builds on earlier papers such as these:

• Dmitry Vagner, David I. Spivak and Eugene Lerman, Algebras of open dynamical systems on the operad of wiring diagrams.

January 29, 2019: Daniel Cicala
Social contagion modeled on random networks

Abstract. A social contagion may manifest as a cultural trend, a spreading opinion or idea or belief. In this talk, we explore a simple model of social contagion on a random network. We also look at the effect that network connectivity, edge distribution, and heterogeneity has on the diffusion of a contagion.

• Mason A. Porter and James P. Gleeson, Dynamical systems on networks: a tutorial.

• Duncan J. Watts, A simple model of global cascades on random networks.

Backprop as functor: a compositional perspective on supervised learning

Abstract. Fong, Spivak and Tuyéras have found a categorical framework in which gradient descent algorithms can be constructed in a compositional way. To explain this, we first give a brief introduction to backprogation and gradient descent. We then describe their monoidal category Learn, where the morphisms are given by abstract learning algorithms. Finally, we show how gradient descent can be realized as a monoidal functor from Para, the category of Euclidean spaces with differentiable parameterized functions between them, to Learn.

• Brendan Fong, David I. Spivak and Rémy Tuyéras, Backprop as functor: a compositional perspective on supervised learning.

February 12, 2019: Christian Williams
The pi calculus: towards global computing

Abstract. Historically, code represents a sequence of instructions for a single machine. Each computer is its own world, and only interacts with others by sending and receiving data through external ports. As society becomes more interconnected, this paradigm becomes more inadequate – these virtually isolated nodes tend to form networks of great bottleneck and opacity. Communication is a fundamental and integral part of computing, and needs to be incorporated in the theory of computation.

To describe systems of interacting agents with dynamic interconnection, in 1980 Robin Milner invented the pi calculus: a formal language in which a term represents an open, evolving system of processes (or agents) which communicate over names (or channels). Because a computer is itself such a system, the pi calculus can be seen as a generalization of traditional computing languages; there is an embedding of lambda into pi – but there is an important change in focus: programming is less like controlling a machine and more like designing an ecosystem of autonomous organisms.

We review the basics of the pi calculus, and explore a variety of examples which demonstrate this new approach to programming. We will discuss some of the history of these ideas, called “process algebra”, and see exciting modern applications in blockchain and biology.

“… as we seriously address the problem of modelling mobile communicating systems we get a sense of completing a model which was previously incomplete; for we can now begin to describe what goes on outside a computer in the same terms as what goes on inside – i.e. in terms of interaction. Turning this observation inside-out, we may say that we inhabit a global computer, an informatic world which demands to be understood just as fundamentally as physicists understand the material world.” — Robin Milner

• Robin Milner, The polyadic pi calculus: a tutorial.

• Robin Milner, Communicating and Mobile Systems.

• Joachim Parrow, An introduction to the pi calculus.

February 19, 2019: Kenny Courser
Category theory for genetics

Abstract. Rémy Tuyéras has developed a categorical framework aimed at handling various commonly studied subjects from the theory of genetics. Some of these include alignment methods, CRISPR, homologous recombination, haplotypes, and genetic linkage. In this talk, I will introduce the foundations on which this framework is built and give a few examples related to DNA and RNA sequencing which are able to be described in this environment.

• Rémy Tuyéras, Category theory for genetics.

## ACT 2019 – First Announcement

2 October, 2018

animation by Marius Buliga

I’m helping organize ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019.

If you’re a grad student or postdoc interested in this subject, you should apply for the ‘school’ before January 30th. Later people can submit papers for the conference:

Dear all,

As part of a new growing community in Applied Category Theory, now with a dedicated journal Compositionality, a traveling workshop series SYCO, a forthcoming Cambridge U. Press book series Reasoning with Categories, and several one-off events including at NIST, we launch an annual conference+school series named Applied Category Theory, the coming one being at Oxford, July 15-19 for the conference, and July 22-26 for the school. The dates are chosen such that CT 2019 (Edinburgh) and the ACT 2019 conference (Oxford) will be back-to-back, for those wishing to participate in both.

There already was a successful invitation-only pilot, ACT 2018, last year at the Lorentz Centre in Leiden, also in the format of school+workshop.

For the conference, for those who are familiar with the successful QPL conference series, we will follow a very similar format for the ACT conference. This means that we will accept both new papers which then will be published in a proceedings volume (most likely a Compositionality special Proceedings issue), as well as shorter abstracts of papers published elsewhere. There will be a thorough selection process, as typical in computer science conferences. The idea is that all the best work in applied category theory will be presented at the conference, and that acceptance is something that means something, just like in CS conferences. This is particularly important for young people as it will help them with their careers.

Expect a call for submissions soon, and start preparing your papers now!

The school in ACT 2018 was unique in that small groups of students worked closely with an experienced researcher (these were John Baez, Aleks Kissinger, Martha Lewis and Pawel Sobociński), and each group ended up producing a paper. We will continue with this format or a closely related one, with Jules Hedges and Daniel Cicala as organisers this year. As there were 80 applications last year for 16 slots, we may want to try to find a way to involve more students.

We are fortunate to have a number of private sector companies closely associated in some way or another, who will also participate, with Cambridge Quantum Computing Inc. and StateBox having already made major financial/logistic contributions.

On behalf of the ACT Steering Committee,

John Baez, Bob Coecke, David Spivak, Christina Vasilakopoulou

## What is Applied Category Theory?

18 September, 2018

Tai-Danae Bradley has a new free “booklet” on applied category theory. It was inspired by the workshop Applied Category Theory 2018, which she attended, and I think it makes a great complement to Fong and Spivak’s book Seven Sketches and my online course based on that book:

• Tai-Danae Bradley, What is Applied Category Theory?

Abstract. This is a collection of introductory, expository notes on applied category theory, inspired by the 2018 Applied Category Theory Workshop, and in these notes we take a leisurely stroll through two themes (functorial semantics and compositionality), two constructions (monoidal categories and decorated cospans) and two examples (chemical reaction networks and natural language processing) within the field.

Check it out!

## Complex Adaptive System Design (Part 8)

22 August, 2018

John Foley, Joe Moeller and I have made some nice progress on compositional tasking for the Complex Adaptive System Composition and Design Environment project.

‘Compositional tasking’ means assigning tasks to networks agents in such a way that you can connect or even overlay such tasked networks and get larger ones. This lets you build up complex plans from smaller pieces.

In my last post in this series, I sketched an approach using ‘commitment networks’. A commitment network is a graph where nodes represent agents and edges represent commitments, like “A should move toward B either for 3 hours or until they meet, whichever comes first”. By overlaying such graphs we can build up commitment networks that describe complex plans of action. The rules for overlaying incorporate ‘automatic deconflicting’. In other words: don’t need to worry about agents being given conflicting duties as you stack up plans… because you’ve decided ahead of time what they should do in these situations.

I still like that approach, but we’ve been asked to develop some ideas more closely connected to traditional methods of tasking, like PERT charts, so now we’ve done that.

‘PERT’ stands for ‘program evaluation and review technique’. PERT charts were developed by the US Navy in 1957, but now they’re used all over industry to help plan and schedule large projects.

Here’s simple example:

The nodes in this graph are different states, like “you have built the car but not yet put on the tires”. The edges are different tasks, like “put the tires on the car”. Each state is labelled with an arbitrary name: 10, 20, 30, 40 and 50. The tasks also have names: A, B, C, D, E, and F. More importantly, each task is labelled by the amount of time that task requires!

Your goal is to start at state 10 and move all the way to state 50. Since you’re bossing lots of people around, you can make them do tasks simultaneously. However, you can only reach a state after you have done all the tasks leading up to that state. For example, you can’t reach state 50 unless you have already done all of tasks C, E, and F. Some typical questions are:

• What’s the minimum amount of time it takes to get from state 10 to state 50?

• Which tasks could take longer, without changing the answer to the previous question? How much longer could each task take, without changing the answer? This amount of time is called the slack for that task.

There are known algorithms for solving such problems. These help big organizations plan complex projects. So, connecting compositional tasking to PERT charts seems like a good idea.

At first this seemed confusing because in our previous work the nodes represented agents, while in PERT charts the nodes represent states. Of course graphs can be used for many things, even in the same setup. But the trick was getting everything to fit together nicely.

Now I think we’re close.

John Foley has been working out some nice example problems where a collection of agents need to move along the edges of a graph from specified start locations to specified end locations, taking routes that minimize their total fuel usage. However, there are some constraints. Some edges can only be traversed by specified teams of agents: they can’t go alone. Also, no one agent is allowed to run out of fuel.

This is a nice problem because while it’s pretty simple and specific, it’s representative of a large class of problems where a collection of agents are trying to carry out tasks together. ‘Moving along the edge of a graph’ can stand for a task of any sort. The constraint that some edges can only be traversed by specified teams is then a way of saying that certain tasks can only be accomplished by teams.

Furthermore, there are nice software packages for optimization subject to constraints. For example, John likes one called Choco. So, we plan to use one of these as part of the project.

What makes this all compositional is that John has expressed this problem using our ‘network model’ formalism, which I began sketching in Part 6. This allows us to assemble tasks for larger collections of agents from tasks for smaller collections.

Here, however, an idea due to my student Joe Moeller turned out to be crucial.

In our first examples of network models, explained earlier in this series, we allowed a monoid of networks for any set of agents of different kinds. A monoid has a binary operation called ‘multiplication’, and the idea here was this could describe the operation of ‘overlaying’ networks: for example, laying one set of communication channels, or committments, on top of another.

However, Joe knew full well that a monoid is a category with one object, so he pushed for a generalization that allowed not just a monoid but a category of networks for any set of agents of different kinds. I didn’t know what this was good for, but I figured: what the heck, let’s do it. It was a mathematically natural move, and it didn’t make anything harder—in fact it clarified some of our constructions, which is why Joe wanted to do it.

Now that generalization is proving to be crucial! We can take our category of networks to have states as objects and tasks (ways of moving between states) as morphisms! So, instead of ‘overlaying networks’, the basic operation is now composing tasks.

So, we now have a framework where if you specify a collection of agents of different kinds, we can give you the category whose morphisms are tasks those agents can engage in.

An example is John’s setup where the agents are moving around on a graph.

But this framework also handles PERT charts! While the folks who invented PERT charts didn’t think of them this way, one can think of them as describing categories of a certain specific sort, with states as objects and tasks as morphisms.

So, we now have a compositional framework for PERT charts.

I would like to dive deeper into the details, but this is probably enough for one post. I will say, though, that we use some math I’ve just developed with my grad student Jade Master, explained here:

Open Petri nets (part 3), Azimuth, 19 August 2018.

The key is the relation between Petri nets and PERT charts. I’ll have more to say about that soon, I hope!

Some posts in this series:

Part 2. Metron’s software for system design.

Part 3. Operads: the basic idea.

Part 4. Network operads: an easy example.

Part 5. Algebras of network operads: some easy examples.

Part 6. Network models.

Part 7. Step-by-step compositional design and tasking using commitment networks.

Part 8. Compositional tasking using category-valued network models.

## Open Petri Nets (Part 3)

19 August, 2018

• John Baez and Jade Master, Open Petri nets.

In Part 1 we saw the double category of open Petri nets; in Part 2 we saw the reachability semantics for open Petri nets as a double functor. Now I’d like to wrap up by showing you the engine beneath the hood of our results.

I fell in love with Petri nets when I realized that they were really just presentations of free symmetric monoidal categories. If you like category theory, this turns Petri nets from something mysterious into something attractive.

In any category you can compose morphisms $f\colon X \to Y$ and $g\colon Y \to Z$ and get a morphism $gf \colon X \to Z.$ In a monoidal category you can also tensor morphisms $f \colon X \to X'$ and $g \colon Y \to Y'$ and get a morphism $f \otimes g \colon X \otimes X' \to Y \otimes Y'.$ This of course relies on your ability to tensor objects. In a symmetric monoidal category you also have $X \otimes Y \cong Y \otimes X.$ And of course, there is more to it than this. But this is enough to get started.

A Petri net has ‘places’ and also ‘transitions’ going between multisets of places:

From this data we can try to generate a symmetric monoidal category whose objects are built from the places and whose morphisms are built from the transitions. So, for example, the above Petri net would give a symmetric monoidal category with an object

2 susceptible + infected

and a morphism from this to the object

susceptible + 2 infected

(built using the transition infection), and a morphism
from this to the object

susceptible + infected + resistant

(built using the transition recovery) and so on. Here we are using $+$ to denote the tensor product in our symmetric monoidal category, as usual in chemistry.

When we do this sort of construction, the resulting symmetric monoidal category is ‘free’. That is, we are not imposing any really interesting equations: the objects are freely generated by the places in our Petri net by tensoring, and the morphisms are freely generated by the transitions by tensoring and composition.

That’s the basic idea. The problem is making this idea precise!

Many people have tried in many different ways. I like this approach the best:

• José Meseguer and Ugo Montanari, Petri nets are monoids, Information and Computation 88 (1990), 105–155.

but I think it can be simplified a bit, so let me describe what Jade and I did in our paper.

The problem is that there are different notions of symmetric monoidal category, and also different notions of morphism between Petri nets. We take the maximally strict approach, and work with ‘commutative’ monoidal categories. These are just commutative monoid objects in $\mathrm{Cat},$ so their associator:

$\alpha_{A,B,C} \colon (A + B) + C \stackrel{\sim}{\longrightarrow} A + (B + C)$

their left and right unitor:

$\lambda_A \colon 0 + A \stackrel{\sim}{\longrightarrow} A$

$\rho_A \colon A + 0 \stackrel{\sim}{\longrightarrow} A$

and even—disturbingly—their braiding:

$\sigma_{A,B} \colon A + B \stackrel{\sim}{\longrightarrow} B + A$

are all identity morphisms.

The last would ordinarily be seen as ‘going too far’, since while every symmetric monoidal category is equivalent to one with trivial associator and unitors, this ceases to be true if we also require the braiding to be trivial. However, it seems that Petri nets most naturally serve to present symmetric monoidal categories of this very strict sort. There just isn’t enough information in a Petri net to make it worthwhile giving them a nontrivial braiding

$\sigma_{A,B} \colon A + B \stackrel{\sim}{\longrightarrow} B+A$

It took me a while to accept this, but now it seem obvious. If you want a nontrivial braiding, you should be using something a bit fancier than a Petri net.

Thus, we construct adjoint functors between a category of Petri nets, which we call $\textrm{Petri},$ and a category of ‘commutative monoidal categories’, which we call $\textrm{CMC}.$

An object of $\textrm{Petri}$ is a Petri net: that is, a set $S$ of places, a set $T$ of transitions, and source and target functions

$s, t \colon T \to \mathbb{N}[S]$

where $\mathbb{N}[S]$ is the underlying set of the free commutative monoid on $S.$

More concretely, $\mathbb{N}[S]$ is the set of formal finite linear combinations of elements of $S$ with natural number coefficients. The set $S$ naturally includes in $\mathbb{N}[S]$, and for any function

$f \colon S \to S'$

there is a unique monoid homomorphism

$\mathbb{N}[f] \colon \mathbb{N}[S] \to \mathbb{N}[S']$

extending $f.$

A Petri net morphism from a Petri net

$s, t \colon T \to \mathbb{N}[S]$

to a Petri net

$s', t' \colon T' \to \mathbb{N}[S']$

is a pair of functions

$f \colon T \to T'$

$g \colon S \to S'$

making the two obvious diagrams commute:

There is a category $\textrm{Petri}$ with Petri nets as objects and Petri net morphisms as morphisms.

On the other hand, a commutative monoidal category is a commutative monoid object in $\mathrm{Cat}.$ Explicitly, it’s a strict monoidal category $(C,+,0)$ such that for all objects $A$ and $B$ we have

$A + B = B + A$

and for all morphisms $f$ and $g$

$f + g = g + f$

Note that a commutative monoidal category is the same as a strict symmetric monoidal category where the symmetry isomorphisms

$\sigma_{A,B} \colon A + B \stackrel{\sim}{\longrightarrow} B+A$

are all identity morphisms. Every strict monoidal functor between commutative monoidal categories is automatically a strict symmetric monoidal functor. So, we let $\mathrm{CMC}$ be the category whose objects are commutative monoidal categories and whose morphisms are strict monoidal functors.

There’s a functor

$U \colon \mathrm{CMC} \to \mathrm{Petri}$

sending any commutative monoidal category $C$ to its underlying Petri net. This Petri net has the set of objects $\mathrm{Ob}(C)$ as its set of places and the set of morphisms $\mathrm{Mor}(C)$ as its set of transitions, and

$s, t \colon \mathrm{Mor}(C) \to \mathrm{Ob}(C) \hookrightarrow \mathbb{N}[\mathrm{Ob}(C)]$

as its source and target maps.

Proposition. The functor $U \colon \mathrm{CMC} \to \mathrm{Petri}$ has a left adjoint

$F \colon \mathrm{Petri} \to \mathrm{CMC}$

This is Proposition 10 in our paper, and we give an explicit construction of this left adjoint.

So that’s our conception of the free commutative monoidal category on a Petri net. It’s pretty simple. How could anyone have done anything else?

Montanari and Meseguer do almost the same thing, but our category of Petri nets is a subcategory of theirs: our morphisms of Petri nets send places to places, while they allow more general maps that send a place to a formal linear combination of places. On the other hand, they consider a full subcategory of our $\mathrm{CMC}$ containing only commutative monoidal categories whose objects form a free commutative monoid.

Other papers do a variety of more complicated things. I don’t have the energy to explain them all, but you can see some here:

• Pierpaolo Degano, José Meseguer and Ugo Montanari, Axiomatizing net computations and processes, in Logic in Computer Science 1989, IEEE, New Jersey, 1989, pp. 175–185.

• Vladimiro Sassone, Strong concatenable processes: an approach to the category of Petri net computations, BRICS Report Series, Dept. of Computer Science, U. Aarhus, 1994.

• Vladimiro Sassone, On the category of Petri net computations, in Colloquium on Trees in Algebra and Programming, Springer, Berlin, 1995.

• Vladimiro Sassone, An axiomatization of the algebra of Petri net concatenable processes, in Theoretical Computer Science 170 (1996), 277–296.

• Vladimiro Sassone and Pavel Sobociński, A congruence for Petri nets, Electronic Notes in Theoretical Computer Science 127 (2005), 107–120.

Getting the free commutative monoidal category on a Petri net right is key to developing the reachability semantics for open Petri nets in a nice way. But to see that, you’ll have to read our paper!

Part 1: the double category of open Petri nets.

Part 2: the reachability semantics for open Petri nets.

Part 3: the free symmetric monoidal category on a Petri net.