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

Cognition, Convexity, and Category Theory

15 June, 2018

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

• Tai-Danae Bradley and Brad Theilman, Cognition, convexity and category theory, The n-Category Café, 10 March 2018.

Tai-Danae Bradley is a mathematics PhD student at the CUNY Graduate Center and well-known math blogger. Brad Theilman is a grad student in neuroscience at the Gentner Lab at U. C. San Diego. I was happy to get to know both of them when the school met in Leiden.

In their blog article, they explain this paper:

• Joe Bolt, Bob Coecke, Fabrizio Genovese, Martha Lewis, Dan Marsden, and Robin Piedeleu, Interacting conceptual spaces I.

Fans of convex sets will enjoy this!

Applied Category Theory Course: Databases

6 June, 2018


In my online course we’re now into the third chapter of Fong and Spivak’s book Seven Sketches. Now we’re talking about databases!

To some extent this is just an excuse to (finally) introduce categories, functors, natural transformations, adjoint functors and Kan extensions. Great stuff, and databases are a great source of easy examples.

But it’s also true that Spivak helps run a company called Categorical Informatics that actually helps design databases using category theory! And his partner, Ryan Wisnesky, would be happy to talk to people about it. If you’re interested, click the link: he’s attending my course.

To read and join discussions on Chapter 3 go here:

Chapter 3

You can also do exercises and puzzles, and see other people’s answers to these.

Here are the lectures I’ve given so far:

Lecture 34 – Chapter 3: Categories
Lecture 35 – Chapter 3: Categories versus Preorders
Lecture 36 – Chapter 3: Categories from Graphs
Lecture 37 – Chapter 3: Presentations of Categories
Lecture 38 – Chapter 3: Functors
Lecture 39 – Chapter 3: Databases
Lecture 40 – Chapter 3: Relations
Lecture 41 – Chapter 3: Composing Functors
Lecture 42 – Chapter 3: Transforming Databases
Lecture 43 – Chapter 3: Natural Transformations
Lecture 44 – Chapter 3: Categories, Functors and Natural Transformations
Lecture 45 – Chapter 3: Composing Natural Transformations
Lecture 46 – Chapter 3: Isomorphisms
Lecture 47 – Chapter 3: Adjoint Functors
Lecture 48 – Chapter 3: Adjoint Functors

Workshop on Compositional Approaches

24 May, 2018

This looks great too:

Workshop on Compositional Approaches in Physics, Natural Language Processing, and Social Sciences, 2 September 2018, Nice, France.

Compositional Approaches for Physics, NLP, and Social Sciences (CAPNS 2018) will be colocated with QI 2018. The workshop is a continuation and extension of the Workshop on Semantic Spaces at the Intersection of NLP, Physics and Cognitive Science held in June 2016.

The ability to compose parts to form a more complex whole, and to analyze a whole as a combination of elements, is desirable across disciplines. In this workshop we bring together researchers applying compositional approaches to NLP, Physics, Cognitive Science, and Game Theory. The interplay between these disciplines will foster theoretically motivated approaches to understanding how meanings of words interact in sentences and discourse, how concepts develop, and how complex games can be analyzed. Commonalities between the compositional mechanisms employed may be extracted, and applications and phenomena traditionally thought of as ‘non-compositional’ will be examined.

Topics of interests include (but are not restricted to):
Applications of quantum logic in natural language processing and cognitive science
Compositionality in vector space models of meaning
Compositionality in conceptual spaces
Compositional approaches to game theory
Reasoning in vector spaces and conceptual spaces
Conceptual spaces in linguistics
Game-theoretic models of language and conceptual change
Diagrammatic reasoning for natural language processing, cognitive science, and game theory
Compositional explanations of so-called ‘non-compositional’ phenomena such as metaphor

June 30th: Paper submission
July 15th: Notification to contributors
September 2nd: Workshop date

Gerhard Jäger, Professor of General Linguistics, University of Tübingen
Paul Smolensky, Principal Researcher, Microsoft Research, and Krieger-Eisenhower Professor of Cognitive Science, Johns Hopkins University

We invite:
Original contributions (up to 12 pages) of previously unpublished work. Submission of substantial, albeit partial results of work in progress is welcomed.

Extended abstracts (3 pages) of previously published work that is recent and relevant to the workshop. These should include a link to a separately published paper or preprint.

Contributions should be submitted at:

Peter Bruza, Queensland University of Technology
Trevor Cohen, University of Texas
Fredrik Nordvall Forsberg, University of Strathclyde
Liane Gabora, University of British Columbia
Peter Gärdenfors, Lund University
Helle Hvid Hansen, TU Delft
Chris Heunen, University of Edinburgh
Peter Hines, University of York
Alexander Kurz, University of Leicester
Antonio Lieto, University of Turin
Glyn Morrill, Universitat Politècnica de Catalunya
Dusko Pavlovic, University of Hawaii
Taher Pilehvar, University of Cambridge
Emmanuel Pothos, City, University of London
Matthew Purver, Queen Mary University of London
Mehrnoosh Sadrzadeh, Queen Mary University of London
Marta Sznajder, Munich Center for Mathematical Philosophy
Pawel Sobocinski, University of Southampton
Dominic Widdows, Grab Technologies
Geraint Wiggins, Vrije Universiteit Brussel
Victor Winschel, OICOS GmbH
Philipp Zahn, University of St. Gallen
Frank Zenker, University of Konstanz

Bob Coecke, University of Oxford
Jules Hedges, University of Oxford
Dimitri Kartsaklis, University of Cambridge
Martha Lewis, ILLC, University of Amsterdam
Dan Marsden, University of Oxford

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.