The Behavioral Approach to Systems Theory

19 June, 2018


Two more students in the Applied Category Theory 2018 school wrote a blog article about something they read:

• Eliana Lorch and Joshua Tan, The behavioral approach to systems theory, 15 June 2018.

Eliana Lorch is a mathematician based in San Francisco. Joshua Tan is a grad student in computer science at the University of Oxford and one of the organizers of Applied Category Theory 2018.

They wrote a great summary of this paper, which has been an inspiration to me and many others:

• Jan Willems, The behavioral approach to open and interconnected systems, IEEE Control Systems 27 (2007), 46–99.

They also list many papers influenced by it, and raise a couple of interesting problems with Willems’ idea, which can probably be handled by generalizing it.

Dynamical Systems and Their Steady States

17 June, 2018


As part of the Applied Category Theory 2018 school, Maru Sarazola wrote a blog article on open dynamical systems and their steady states. Check it out:

• Maru Sarazola, Dynamical systems and their steady states, 2 April 2018.

She compares two papers:

• David Spivak, The steady states of coupled dynamical systems compose according to matrix arithmetic.

• John Baez and Blake Pollard, A compositional framework for reaction networks, Reviews in Mathematical Physics 29 (2017), 1750028.
(Blog article here.)

It’s great, because I’d never really gotten around to understanding the precise relationship between these two approaches. I wish I knew the answers to the questions she raises at the end!

Applied Category Theory 2018/2019

15 June, 2018

A lot happened at Applied Category Theory 2018. Even as it’s still winding down, we’re already starting to plan a followup in 2019, to be held in Oxford. Here are some notes Joshua Tan sent out:

  1. Discussions: Minutes from the discussions can be found here.
  2. Photos: Ross Duncan took some very glamorous photos of the conference, which you can find here.

  3. Videos: Videos of talks are online here: courtesy of Jelle Herold and Fabrizio Genovese.

  4. Next year’s workshop: Bob Coecke will be organizing ACT 2019, to be hosted in Oxford sometime spring/summer. There will be a call for papers.

  5. Next year’s school: Daniel Cicala is helping organize next year’s ACT school. Please contact him at if you would like to get involved.

  6. Look forward to the official call for submissions, coming soon, for the first issue of Compositionality!

The minutes mentioned above contain interesting thoughts on these topics:

• Day 1: Causality
• Day 2: AI & Cognition
• Day 3: Dynamical Systems
• Day 4: Systems Biology
• Day 5: Closing

Tropical Algebra and Railway Optimization

24 May, 2018

Simon Willerton pointed out a wonderful workshop, which unfortunately neither he nor I can attend… nor Jamie Vicary, who is at Birmingham:

Tropical Mathematics & Optimisation for Railways, University of Birmingham, School of Engineering, Monday 18 June 2018.

If you can go, please do—and report back!

Tropical algebra involves the numbers (-\infty, \infty] made into a rig with minimization as the addition and addition as the multiplication. It’s called a rig because it’s a “ring without negatives”.

Tropical algebra is important in algebraic geometry, because if you take some polynomial equations and rewrite them replacing + with min and × with +, you get equations that describe shapes with flat pieces replacing curved surfaces, like this:

These simplified shapes are easier to deal with, but they shed light on the original curved ones! Click the picture for more on the subject from Johannes Rau.

Tropical algebra is also important for quantization, since classical mechanics chooses the path with minimum action while quantum mechanics sums over paths. But it’s also important for creating efficient railway time-tables, where you’re trying to minimize the total time it takes to get from one place to another. Finally these worlds are meeting!

Here’s the abstract, which shows that the reference to railway optimization is not just a joke:

Abstract. The main purpose of this workshop is to bring together specialists in tropical mathematics and mathematical optimisation applied in railway engineering and to foster further collaboration between them. It is inspired by some applications of tropical mathematics to the analysis of railway timetables. The most elementary of them is based on a controlled tropically linear dynamic system, which allows for a stability analysis of a regular timetable and can model the delay propagation. Tropical (max-plus) switching systems are one of the extensions of this elementary model. Tropical mathematics also provides appropriate mathematical language and tools for various other applications which willbe presented at the workshop.

The talks on mathematical optimisation in railway engineering will be given by Professor Clive Roberts and other prominent specialists working at the Birmingham Centre for Railway Research and Education (BCRRE). They will inform the workshop participants about the problems that are of actual interest for railways, and suggest efficient and practical methods of their solution.

For a glimpse of some of the category theory lurking in this subject, see:

• Simon Willerton, Project scheduling and copresheaves, The n-Category Café.


6 May, 2018

A new journal! We’ve been working on it for a long time, but we finished sorting out some details at ACT2018, and now we’re ready to tell the world!

It’s free to read, free to publish in, and it’s about building big things from smaller parts. Here’s the top of the journal’s home page right now:

Here’s the official announcement:

We are pleased to announce the launch of Compositionality, a new diamond open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Topics may concern foundational structures, an organizing principle, or a powerful tool. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition. To learn more about the scope and editorial policies of the journal, please visit our website at

Compositionality is the culmination of a long-running discussion by many members of the extended category theory community, and the editorial policies, look, and mission of the journal have yet to be finalized. We would love to get your feedback about our ideas on the forum we have established for this purpose:

Lastly, the journal is currently receiving applications to serve on the editorial board; submissions are due May 31 and will be evaluated by the members of our steering board: John Baez, Bob Coecke, Kathryn Hess, Steve Lack, and Valeria de Paiva.

We will announce a call for submissions in mid-June.

We’re looking forward to your ideas and submissions!

Best regards,

Brendan Fong, Nina Otter, and Joshua Tan

Symposium on Compositional Structures

4 May, 2018

As I type this, sitting in a lecture hall at the Lorentz Center, Jamie Vicary, University of Birmingham and University of Oxford, is announcing a new series of meetings:

Symposium on Compositional Structures.

The website, which will probably change, currently says this:

Symposium on Compositional Structures (SYCO)

The Symposium on Compositional Structures is a new interdisciplinary meeting aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language.

We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering discussion, disseminating new ideas, and spreading knowledge of open problems between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students. The meeting does not have proceedings.

While no list of topics could be exhaustive, SYCO welcomes submissions with a compositional focus related any the following areas, in particular from the perspective of category theory:

logical methods in computer science, including quantum and classical programming, concurrency, natural language processing and machine learning;

graphical calculi, including string diagrams, Petri nets and reaction networks;

languages and frameworks, including process algebras, proof nets, type theory and game semantics;

abstract algebra and pure category theory, including monoidal category theory, higher category theory, operads, polygraphs, and relationships to homotopy theory;

quantum algebra, including quantum computation and representation theory;

tools and techniques, including rewriting, formal proofs and proof assistants;

industrial applications, including case studies and real-world problem descriptions.


Meetings will involve both invited and contributed talks. The first meeting is planned for Autumn 2018, with more details to follow soon.


Some funding may be available to support travel and subsistence, especially for junior researchers who are speaking at the meeting.

Steering committee

The symposium is managed by the following people:

Ross Duncan, University of Strathclyde.
Chris Heunen, University of Edinburgh.
Aleks Kissinger, Radboud University Nijmegen.
Samuel Mimram, École Polytechnique.
Mehrnoosh Sadrzadeh, Queen Mary.
Pawel Sobocinski, University of Southampton.
Jamie Vicary, University of Birmingham and University of Oxford.

Applied Category Theory at NIST (Part 2)

18 April, 2018

Here are links to the slides and videos for most of the talks from this workshop:

Applied Category Theory: Bridging Theory & Practice, March 15–16, 2018, NIST, Gaithersburg, Maryland, USA. Organized by Spencer Breiner and Eswaran Subrahmanian.

They give a pretty good picture of what went on. Spencer Breiner put them up here; what follows is just a copy of what’s on his site.

Unfortunately, the end of Dusko Pavlovic’s talk, as well as Ryan Wisnesky’s and Steve Huntsman’s were lost due to a technical error. You can also find a Youtube playlist with all of the videos here.

Introduction to NIST:

Ram Sriram – NIST and Category Theory


Spencer Breiner – Introduction

Invited talks:

Bob Coecke – From quantum foundations to cognition via pictures


Dusko Pavlovic – Security Science in string diagrams (partial video)


John Baez – Compositional design and tasking of networks (part 1)


John Foley – Compositional design and tasking of networks (part 2)


David Spivak – A higher-order temporal logic for dynamical systems


Lightning Round Talks:

Ryan Wisnesky – Categorical databases (no video)

Steve Huntsman – Towards an operad of goals (no video)


Bill Regli – Disrupting interoperability (no slides)


Evan Patterson – Applied category theory in data science


Brendan Fong – data structures for network languages


Stephane Dugowson – A short introduction to a general theory of interactivity


Michael Robinson – Sheaf methods for inference


Cliff Joslyn – Seeking a categorical systems theory via the category of hypergraphs


Emilie Purvine – A category-theoretical investigation of the type hierarchy for heterogeneous sensor integration


Helle Hvid Hansen – Long-term values in Markov decision processes, corecursively


Alberto Speranzon – Localization and planning for autonomous systems via (co)homology computation


Josh Tan – Indicator frameworks (no slides)

Breakout round report