Applied Category Theory Seminar

 

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:

• The mathematics of planet Earth.

• Props in network theory.

January 15, 2019: Jonathan Lorand
Classification 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.

Reading material:

• 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:

• David I. Spivak, The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits.

• 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.

Talk slides: Social contagion modeled on random networks.

Reading material:

• 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.

February 5, 2019: Jade Master
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.

Reading material:

• 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

Talk slides: The pi calculus: towards global computing.

Reading material:

• 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.

Reading material:

• Rémy Tuyéras, Category theory for genetics.

ACT 2019 – First Announcement

 

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?

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)

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!

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Some posts in this series:

• Part 1. CASCADE: the Complex Adaptive System Composition and Design Environment.

• 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.

• Part 9 – Network models from Petri nets with catalysts.

Open Petri Nets (Part 3)

I’ve been talking about my new paper with Jade Master:

• 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 and and get a morphism In a monoidal category you can also tensor morphisms and and get a morphism This of course relies on your ability to tensor objects. In a symmetric monoidal category you also have 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 so their associator:

their left and right unitor:

and even—disturbingly—their braiding:

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

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 and a category of ‘commutative monoidal categories’, which we call

An object of is a Petri net: that is, a set of places, a set of transitions, and source and target functions

where is the underlying set of the free commutative monoid on

More concretely, is the set of formal finite linear combinations of elements of with natural number coefficients. The set naturally includes in , and for any function

there is a unique monoid homomorphism

extending

A Petri net morphism from a Petri net

to a Petri net

is a pair of functions

making the two obvious diagrams commute:

There is a category 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 Explicitly, it’s a strict monoidal category such that for all objects and we have

and for all morphisms and

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

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

There’s a functor

sending any commutative monoidal category to its underlying Petri net. This Petri net has the set of objects as its set of places and the set of morphisms as its set of transitions, and

as its source and target maps.

Proposition. The functor has a left adjoint

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 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!

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• 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.

Open Petri Nets (Part 2)

I’d like to continue talking about this paper:

• John Baez and Jade Master, Open Petri nets.

Last time I explained, in a sketchy way, the double category of open Petri nets. This time I’d like to describe a ‘semantics’ for open Petri nets.

In his famous thesis Functorial Semantics of Algebraic Theories, Lawvere introduced the idea that semantics, as a map from expressions to their meanings, should be a functor between categories. This has been generalized in many directions, and the same idea works for double categories. So, we describe our semantics for open Petri nets as a map

from our double category of open Petri nets to a double category of relations. This map sends any open Petri net to its ‘reachability relation’.

In Petri net theory, a marking of a set is a finite multisubset of We can think of this as a way of placing finitely ‘tokens’—little black dots—on the elements of A Petri net lets us start with some marking of its places and then repeatedly change the marking by moving tokens around, using the transitions. This is how Petri nets describe processes!

For example, here’s a Petri net from chemistry:

Here’s a marking of its places:

But using the transitions, we can repeatedly change the marking. We started with one atom of carbon, one molecule of oxygen, one molecule of sodium hydroxide and one molecule of hydrochloric acid. But they can turn into one molecule of carbon dioxide, one molecule of sodium hydroxide and one molecule of hydrochloric acid:

These can then turn into one molecule of sodium bicarbonate and one molecule of hydrochloric acid:

Then these can turn into one molecule of carbon dioxide, one molecule of water and one molecule of sodium chloride:

People say one marking is reachable from another if you can get it using a finite sequence of transitions in this manner. (Our paper explains this well-known notion more formally.) In this example every marking has 0 or 1 tokens in each place. But that’s not typical: in general we could have any natural number of tokens in each place, so long as the total number of tokens is finite.

Our paper adapts the concept of reachability to open Petri nets. Let denote the set of markings of the set Given an open Petri net there is a reachability relation

This relation holds when we can take a given marking of feed those tokens into the Petri net move them around using transitions in and have them pop out and give a certain marking of leaving no tokens behind.

For example, consider this open Petri net

Here is a marking of

We can feed these tokens into and move them around using transitions in

They can then pop out into leaving none behind:

This gives a marking of that is ‘reachable’ from the original marking of

The main result of our paper is that the map sending an open Petri net to its reachability relation extends to a ‘lax double functor’

where is a double category having open Petri nets as horizontal 1-cells and is a double category having relations as horizontal 1-cells.

I can give you a bit more detail on those double categories, and also give you a clue about what ‘lax’ means, without it becoming too stressful.

Last time I said the double category has:

• sets as objects,

• functions as vertical 1-morphisms,

• open Petri nets as horizontal 1-cells—they look like this:

• morphisms between open Petri nets as 2-morphisms—an example would be the visually obvious map from this open Petri net:

to this one:

What about This double category has

• sets as objects,

• functions as vertical 1-morphisms,

• relations as horizontal 1-cells from to and

• maps between relations as 2-morphisms. Here a map between relations is a square

that obeys

So, the idea of the reachability semantics is that it maps:

• any set to the set consisting of all markings of that set.

• any function to the obvious function

(Yes, is a really a functor.)

• any open Petri net to its reachability relation

• any morphism between Petri nets to the obvious map between their reachability relations.

Especially if you draw some examples, all this seems quite reasonable and nice. But it’s important to note that is a lax double functor. This means that it does not send a composite open Petri net to the composite of the reachability relations for and So, we do not have

Instead, we just have

It’s easy to see why. Take to be this open Petri net:

and take to be this one:

Then their composite is this:

It’s easy to see that is a proper subset of In a token can move all the way from point 1 to point 5. But it does not do so by first moving through and then moving through It has to take a more complicated zig-zag path where it first leaves and enters then comes back into and then goes to

In our paper, Jade and I conjecture that we get

if we restrict the reachability semantics to a certain specific sub-double category of consisting of ‘one-way’ open Petri nets.

Finally, besides showing that

is a lax double functor, we also show that it’s symmetric monoidal. This means that the reachability semantics works as you’d expect when you run two open Petri nets ‘in parallel’.

In a way, the most important thing about our paper is that it illustrates some methods to study semantics for symmetric monoidal double categories. Kenny Courser and I will describe these methods more generally in our paper “Structured cospans.” They can be applied to timed Petri nets, colored Petri nets, and various other kinds of Petri nets. One can also develop a reachability semantics for open Petri nets that are glued together along transitions as well as places.

I hear that the company Statebox wants these and other generalizations. We aim to please—so we’d like to give it a try.

Next time I’ll wrap up this little series of posts by explaining how Petri nets give symmetric monoidal categories.

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• 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.

Open Petri Nets (Part 1)

Jade Master and I have just finished a paper on open Petri nets:

• John Baez and Jade Master, Open Petri nets.

Abstract. The reachability semantics for Petri nets can be studied using open Petri nets. For us an ‘open’ Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category, which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category Various choices of semantics for open Petri nets can be described using symmetric monoidal double functors out of Here we describe the reachability semantics, which assigns to each open Petri net the relation saying which markings of the outputs can be obtained from a given marking of the inputs via a sequence of transitions. We show this semantics gives a symmetric monoidal lax double functor from to the double category of relations. A key step in the proof is to treat Petri nets as presentations of symmetric monoidal categories; for this we use the work of Meseguer, Montanari, Sassone and others.

I’m excited about this, especially because our friends at Statebox are planning to use open Petri nets in their software. They’ve recently come out with a paper too:

• Fabrizio Romano Genovese and Jelle Herold, Executions in (semi-)integer Petri nets are compact closed categories.

Petri nets are widely used to model open systems in subjects ranging from computer science to chemistry. There are various kinds of Petri net, and various ways to make them ‘open’, and my paper with Jade only handles the simplest. But our techniques are flexible, so they can be generalized.

What’s an open Petri net? For us, it’s a thing like this:

The yellow circles are called ‘places’ (or in chemistry, ‘species’). The aqua rectangles are called ‘transitions’ (or in chemistry, ‘reactions’). There can in general be lots of places and lots of transitions. The bold arrows from places to transitions and from transitions to places complete the structure of a Petri net. There are also arbitrary functions from sets and into the set of places. This makes our Petri net into an ‘open’ Petri net.

We can think of open Petri nets as morphisms between finite sets. There’s a way to compose them! Suppose we have an open Petri net from to where now I’ve given names to the points in these sets:

We write this as for short, where the funky arrow reminds us this isn’t a function between sets. Given another open Petri net for example this:

the first step in composing and is to put the pictures together:

At this point, if we ignore the sets we have a new Petri net whose set of places is the disjoint union of those for and

The second step is to identify a place of with a place of whenever both are images of the same point in . We can then stop drawing everything involving and get an open Petri net which looks like this:

Formalizing this simple construction leads us into a bit of higher category theory. The process of taking the disjoint union of two sets of places and then quotienting by an equivalence relation is a pushout. Pushouts are defined only up to canonical isomorphism: for example, the place labeled in the last diagram above could equally well have been labeled or This is why to get a category, with composition strictly associative, we need to use isomorphism classes of open Petri nets as morphisms. But there are advantages to avoiding this and working with open Petri nets themselves. Basically, it’s better to work with things than mere isomorphism classes of things! If we do this, we obtain not a category but a bicategory with open Petri nets as morphisms.

However, this bicategory is equipped with more structure. Besides composing open Petri nets, we can also ‘tensor’ them via disjoint union: this describes Petri nets being run in parallel rather than in series. The result is a symmetric monoidal bicategory. Unfortunately, the axioms for a symmetric monoidal bicategory are cumbersome to check directly. Double categories turn out to be more convenient.

Double categories were introduced in the 1960s by Charles Ehresmann. More recently they have found their way into applied mathematics. They been used to study various things, including open dynamical systems:

• Eugene Lerman and David Spivak, An algebra of open continuous time dynamical systems and networks.

open electrical circuits and chemical reaction networks:

• Kenny Courser, A bicategory of decorated cospans, Theory and Applications of Categories 32 (2017), 995–1027.

open discrete-time Markov chains:

• Florence Clerc, Harrison Humphrey and P. Panangaden, Bicategories of Markov processes, in Models, Algorithms, Logics and Tools, Lecture Notes in Computer Science 10460, Springer, Berlin, 2017, pp. 112–124.

and coarse-graining for open continuous-time Markov chains:

• John Baez and Kenny Courser, Coarse-graining open Markov processes. (Blog article here.)

As noted by Shulman, the easiest way to get a symmetric monoidal bicategory is often to first construct a symmetric monoidal double category:

• Mike Shulman, Constructing symmetric monoidal bicategories.

The theory of ‘structured cospans’ gives a systematic way to build symmetric monoidal double categories—Kenny Courser and I are writing a paper on this—and Jade and I use this to construct the symmetric monoidal double category of open Petri nets.

A 2-morphism in a double category can be drawn as a square like this:

We call and ‘objects’, and ‘vertical 1-morphisms’, and ‘horizontal 1-cells’, and a ‘2-morphism’. We can compose vertical 1-morphisms to get new vertical 1-morphisms and compose horizontal 1-cells to get new horizontal 1-cells. We can compose the 2-morphisms in two ways: horizontally and vertically. (This is just a quick sketch of the ideas, not the full definition.)

In our paper, Jade and I start by constructing a symmetric monoidal double category with:

• sets as objects,

• functions as vertical 1-morphisms,

• open Petri nets as horizontal 1-cells,

• morphisms between open Petri nets as 2-morphisms.

(Since composition of horizontal 1-cells is associative only up to an invertible 2-morphism, this is technically a pseudo double category.)

What are the morphisms between open Petri nets like? A simple example may be help give a feel for this. There is a morphism from this open Petri net:

to this one:

mapping both primed and unprimed symbols to unprimed ones. This describes a process of ‘simplifying’ an open Petri net. There are also morphisms that include simple open Petri nets in more complicated ones, etc.

This is just the start. Our real goal is to study the semantics of open Petri nets: that is, how they actually describe processes! And for that, we need to think about the free symmetric monoidal category on a Petri net. You can read more about those things in Part 2 and Part 3 of this series.

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• 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.