ACT 2018 – Applied Category Theory Research School

I have some exciting news. We are recruiting participants for the inaugural Applied Category Theory (ACT 2018) research school, which will culminate with an intensive research week held at the Lorentz Center in Leiden, the Netherlands, in late April 2018.


The application deadline is soon: 1st November 2017.


If this sounds interesting, read on. Even if you think that this is not for you, maybe you know someone who would be interested: in that case, please pass this information on!



There are four mentors: John Baez, Aleks Kissinger, Martha Lewis and myself.


The high-level goal of the school is to bring early career researchers (i.e. late PhD through to postdoc stage) into the applied category theory community, and (hopefully!) do some exciting research together. In particular, we encourage applications from outside the category theory clique: broad computer science, physical science, engineering and even social science.


Each mentor chose a theme and a couple of papers for the reading list. The themes are:


  • John Baez – Semantics for open Petri nets and reaction networks
  • Aleks Kissinger – Unification of the logic of causality
  • Martha Lewis – Compositional approaches to linguistics and cognition
  • Pawel Sobocinski – Modelling of open and interconnected systems


I’ll explain what my theme is all about, but first I should say a bit about the innovative format of ACT 2018 developed by the excellent organisers, Brendan Fong and Nina Otter.


Preparations for the April research week will begin already in January 2018: the selected participants will video-conference with mentors, and go through a reading list to prepare them for an intense working week at the school itself; the hope is to get to the point where there is a real possibility of doing some original research, and maybe even obtaining publishable results!


My theme is “modelling of open and interconnected systems” and the two papers I chose are:

  • A Carboni and RFC Walters. Cartesian bicategories I. Journal of pure and applied algebra 49.1-2 (1987): 11-32.
  • JC Willems. The behavioral approach to open and interconnected systems. IEEE Control Systems 27.6 (2007): 46-99.


Below is a short advertising spiel that explains what I mean by the title and why I chose these two papers.


The data revolution of recent years means that scientific models (in physics, biology, medicine, chemistry, climate science, computer science, systems theory, …) are getting larger, more fine-grained and detailed and ever more complex. Algorithms that answer questions about model behaviour are often exponentional in the size of the model, meaning that–even with advances in hardware and software parallelisation–studying monolithic models usually does not scale. Moreover, monolithic models are poor engineering practice; for example, they are not portable in the sense that standard components cannot be reused from one application to another. These are just some of the reasons why a comprehensive mathematical theory of open and interconnected systems is urgently needed.


As Willems argued in The behavioral approach to open and interconnected systems, models for such systems must be fundamentally relational in order to be useful – the underlying physical equations that govern behaviour (e.g. Ohm’s law, the gas laws, …) are seldom causal: they merely govern the relationship between various system variables. This contrasts with the human custom of thinking causally about system behaviour. Willems argued that causality–while useful for human intuition–is physically dubious, not compositional and often complicates the mathematics.


To arrive at a comprehensive relational theory of open systems, category theory is unavoidable: one needs the generality to cover the wide variety of mathematical universes relevant in the various application areas, while identifying common structure. Carboni and Walters’ Cartesian Bicategories of relations offers a general and elegant framework for relational theories and has the potential to be relevant across multiple application areas.


I really hope that we manage to attract some engineers and system theorists. My research questions are pretty open-ended:


  1. Assemble a menagerie of relational theories from a variety of disciplines
  2. Evaluate cartesian bicategories of relations as a mathematical foundation of open and interconnected systems
  3. Study extensions (abelian bicategories, nonlinear variants)
I think it will be a lot of fun!


The official call for applications is below.
 We’re delighted to announce the Applied Category Theory 2018 Adjoint School, an initiative to bring early career researchers into the applied category theory community.


The Adjoint School comprises two phases: (1) an online reading seminar based on the recent Kan Extension Seminars, and (2) a four day research week at the Lorentz Center, Leiden, The Netherlands. Participants will also be invited to attend Applied Category Theory 2018, which will be held immediately following the research week, also at the Lorentz Center.

During the school, participants will work under the mentorship of four mentors, on one of the following research projects: 

  • John Baez: Semantics for open Petri nets and reaction networks
  • Aleks Kissinger: Unification of the logic of causality
  • Martha Lewis: Compositional approaches to linguistics and cognition
  • Pawel Sobocinski: Modelling of open and interconnected systems
The online seminar begins in early January 2018, and will run until the research week begins on April 23rd, 2018. Applied Category Theory 2018 will be held April 30th to May 4th.


Applications to participate in the school are now open. For more details, including how to apply, please see here: or contact the organisers. Applications close November 1st.


On behalf of the organisers,
Brendan Fong (

9 thoughts on “ACT 2018 – Applied Category Theory Research School

  1. Hi, Pawel, this program looks really cool, I attended the AMS MRC for HoTT ( this past summer which has some similarities and had a great experience. Not sure if I’ll make this one but definitely considering it.

    I have a question about cartesian bicategories. Specifically, what’s your view on the tradeoffs between using them vs proarrow equipments/framed bicategories? I’ve been working on some programming language semantics research where adjunctions in a bicategory-like structure play a central role and I’ve mostly used Mike Shulman’s papers on equipments as my reference, so I’m curious about the differences.


    • Hi Max,

      I’m not an expert on all the subtleties involved but the way it’s organised in my head is as follows:

      1. There is the obvious functor Set->Rel that sends a function to its graph. This raises the question of “how does one construct a category of relations?”. There are several nonequivalent ways of going about this; one way is to look at spans, another is to look at profunctors. Both have spawned a large body of literature. The work on equipments follows the profunctor approach, the idea, as I understand it is that the equipment gives you a notion of “internal profunctors”.

      2. Another question is to understand the *algebra* of relations. Here, there’s Freyd-Scedrov research thread on allegories, and cartesian bicategories. So I think of cartesian bicategories as a general notion of algebra of relations.


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