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

Applied Category Theory 2018 Schedule

13 April, 2018

Here’s the schedule of the ACT2018 workshop:

Click to enlarge!

They put me on last, either because my talk will be so boring that it’s okay everyone will have left, or because my talk will be so exciting that nobody will want to leave. I haven’t dared ask the organizers which one.

On the other hand, they’ve put me on first for the “school” which occurs one week before the workshop. Here’s the schedule for the ACT 2018 Adjoint School:

Applied Category Theory Course: Ordered Sets

7 April, 2018

My applied category theory course based on Fong and Spivak’s book Seven Sketches is going well. Over 250 people have registered for the course, which allows them to ask question and discuss things. But even if you don’t register you can read my “lectures”.

Here are all the lectures on Chapter 1, which is about adjoint functors between posets, and how they interact with meets and joins. We study the applications to logic – both classical logic based on subsets, and the nonstandard version of logic based on partitions. And we show how this math can be used to understand “generative effects”: situations where the whole is more than the sum of its parts!

Lecture 1 – Introduction
Lecture 2 – What is Applied Category Theory?
Lecture 3 – Chapter 1: Preorders
Lecture 4 – Chapter 1: Galois Connections
Lecture 5 – Chapter 1: Galois Connections
Lecture 6 – Chapter 1: Computing Adjoints
Lecture 7 – Chapter 1: Logic
Lecture 8 – Chapter 1: The Logic of Subsets
Lecture 9 – Chapter 1: Adjoints and the Logic of Subsets
Lecture 10 – Chapter 1: The Logic of Partitions
Lecture 11 – Chapter 1: The Poset of Partitions
Lecture 12 – Chapter 1: Generative Effects
Lecture 13 – Chapter 1: Pulling Back Partitions
Lecture 14 – Chapter 1: Adjoints, Joins and Meets
Lecture 15 – Chapter 1: Preserving Joins and Meets
Lecture 16 – Chapter 1: The Adjoint Functor Theorem for Posets
Lecture 17 – Chapter 1: The Grand Synthesis

If you want to discuss these things, please visit the Azimuth Forum and register! Use your full real name as your username, with no spaces, and use a real working email address. If you don’t, I won’t be able to register you. Your email address will be kept confidential.

I’m finding this course a great excuse to put my thoughts about category theory into a more organized form, and it’s displaced most of the time I used to spend on Google+ and Twitter. That’s what I wanted: the conversations in the course are more interesting!

Applied Category Theory Course

26 March, 2018

It just became a lot easier to learn about applied category theory, thanks to this free book:

• Brendan Fong and David Spivak, Seven Sketches in Compositionality: An Invitation to Applied Category Theory.

I’ve started an informal online course based on this book on the Azimuth Forum. I’m getting pretty sick of the superficial quality of my interactions on social media. This could be a way to do something more interesting.

The idea is that you can read chapters of this book, discuss them, try the exercises in the book, ask and answer questions, and maybe team up to create software that implements some of the ideas. I’ll try to keep things moving forward. For example, I’ll explain some stuff and try to help answer questions that people are stuck on. I may also give some talks or run discussions on Google Hangouts or similar software—but only when I have time: I’m more of a text-based guy. I may get really busy some times, and leave the rest of you alone for a while. But I like writing about math for at least 15 minutes a day, and more when I have time. Furthermore, I’m obsessed with applied category theory and plan to stay that way for at least a few more years.

If this sounds interesting, let me know here—and please visit the Azimuth Forum and register! Use your full real name as your username, with no spaces. I will add spaces and that will become your username. Use a real working email address. If you don’t, the registration process may not work.

Over 70 people have registered so far, so this process will take a while.

The main advantage of the Forum over this blog is that you can initiate new threads and edit your comments. Like here you can write equations in LaTeX. Like here, that ability is severely limited: for example you can’t define macros, and you can’t use TikZ. (Maybe someone could fix that.) But equations are better typeset over there—and more importantly, the ability to edit comments makes it a lot easier to correct errors in your LaTeX.

Please let me know what you think.

What follows is the preface to Fong and Spivak’s book, just so you can get an idea of what it’s like.


Category theory is becoming a central hub for all of pure mathematics. It is unmatched in its ability to organize and layer abstractions, to find commonalities between structures of all sorts, and to facilitate communication between different mathematical communities. But it has also been branching out into science, informatics, and industry. We believe that it has the potential to be a major cohesive force in the world, building rigorous bridges between disparate worlds, both theoretical and practical. The motto at MIT is mens et manus, Latin for mind and hand. We believe that category theory—and pure math in general—has stayed in the realm of mind for too long; it is ripe to be brought to hand.

Purpose and audience

The purpose of this book is to offer a self-contained tour of applied category theory. It is an invitation to discover advanced topics in category theory through concrete real-world examples. Rather than try to give a comprehensive treatment of these topics—which include adjoint functors, enriched categories, proarrow equipments, toposes, and much more–we merely provide a taste. We want to give readers some insight into how it feels to work with these structures as well as some ideas about how they might show up in practice.

The audience for this book is quite diverse: anyone who finds the above description intriguing. This could include a motivated high school student who hasn’t seen calculus yet but has loved reading a weird book on mathematical logic they found at the library. Or a machine learning researcher who wants to understand what vector spaces, design theory, and dynamical systems could possibly have in common. Or a pure mathematician who wants to imagine what sorts of applications their work might have. Or a recently-retired programmer who’s always had an eerie feeling that category theory is what they’ve been looking for to tie it all together, but who’s found the usual books on the subject impenetrable.

For example, we find it something of a travesty that in 2018 there seems to be no introductory material available on monoidal categories. Even beautiful modern introductions to category theory, e.g. by Riehl or Leinster, do not include anything on this rather central topic. The basic idea is certainly not too abstract; modern human intuition seems to include a pre-theoretical understanding of monoidal categories that is just waiting to be formalized. Is there anyone who wouldn’t correctly understand the basic idea being communicated in the following diagram?

Many applied category theory topics seem to take monoidal categories as their jumping off point. So one aim of this book is to provide a reference—even if unconventional—for this important topic.

We hope this book inspires both new visions and new questions. We intend it to be self-contained in the sense that it is approachable with minimal prerequisites, but not in the sense that the complete story is told here. On the contrary, we hope that readers use this as an invitation to further reading, to orient themselves in what is becoming a large literature, and to discover new applications for themselves.

This book is, unashamedly, our take on the subject. While the abstract structures we explore are important to any category theorist, the specific topics have simply been chosen to our personal taste. Our examples are ones that we find simple but powerful, concrete but representative, entertaining but in a way that feels important and expansive at the same time. We hope our readers will enjoy themselves and learn a lot in the process.

How to read this book

The basic idea of category theory—which threads through every chapter—is that if one pays careful attention to structures and coherence, the resulting systems will be extremely reliable and interoperable. For example, a category involves several structures: a collection of objects, a collection of morphisms relating objects, and a formula for combining any chain of morphisms into a morphism. But these structures need to cohere or work together in a simple commonsense way: a chain of chains is a chain, so combining a chain of chains should be the same as combining the chain. That’s it!

We will see structures and coherence come up in pretty much every definition we give: “here are some things and here are how they fit together.” We ask the reader to be on the lookout for structures and coherence as they read the book, and to realize that as we layer abstraction on abstraction, it is the coherence that makes everything function like a well-oiled machine.

Each chapter in this book is motivated by a real-world topic, such as electrical circuits, control theory, cascade failures, information integration, and hybrid systems. These motivations lead us into and through various sorts of category-theoretic concepts.

We generally have one motivating idea and one category-theoretic purpose per chapter, and this forms the title of the chapter, e.g. Chapter 4 is “Collaborative design: profunctors, categorification, and monoidal categories.” In many math books, the difficulty is roughly a monotonically-increasing function of the page number. In this book, this occurs in each chapter, but not so much in the book as a whole. The chapters start out fairly easy and progress in difficulty.

The upshot is that if you find the end of a chapter very difficult, hope is certainly not lost: you can start on the next one and make good progress. This format lends itself to giving you a first taste now, but also leaving open the opportunity for you to come back at a later date and get more deeply into it. But by all means, if you have the gumption to work through each chapter to its end, we very much encourage that!

We include many exercises throughout the text. Usually these exercises are fairly straightforward; the only thing they demand is that the reader’s mind changes state from passive to active, rereads the previous paragraphs with intent, and puts the pieces together. A reader becomes a student when they work the exercises; until then they are more of a tourist, riding on a bus and listening off and on to the tour guide. Hey, there’s nothing wrong with that, but we do encourage you to get off the bus and make contact with the natives as often as you can.

Applied Category Theory at NIST (Part 1)

17 February, 2018

I think it’s really cool how applied category theory is catching on. My former student Blake Pollard is working at the National Institute of Standards and Technology on applications of category theory to electrical engineering. He’s working with Spencer Breiner… and now Breiner is helping run a workshop on this stuff:

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

It’s by invitation only, but I can’t resist mentioning its existence. Here’s the idea:

What: The Information Technology Laboratory at NIST is pleased to announce a workshop on Applied Category Theory to be held at NIST’s Gaithersburg, Maryland campus on March 15 & 16, 2018. The meeting will focus on practical avenues for introducing methods from category theory into real-world applications, with an emphasis on examples rather than theorems.

Who: The workshop aims to bring together two distinct groups. First, category theorists interested in pursuing applications outside of the usual mathematical fields. Second, domain experts and research managers from industry, government, science and engineering who have in mind potential domain applications for categorical methods.

Intended Outcomes: A proposed landscape of potential CT applications and the infrastructure needed to realize them, together with a 5-10 year roadmap for developing the field of applied category theory. This should include perspectives from industry, academia and government as well as major milestones, potential funding sources, avenues for technology transfer and necessary improvements in tool support and methodology. Exploratory collaborations between category theorists and domain experts. We will ask that each group come prepared to meet the other side. Mathematicians should be prepared with concrete examples that demonstrate practical applications of CT in an intuitive way. Domain experts should bring to the table specific problems to which they can devote time and/or funding as well as some reasons about why they think CT might be relevant to this application.

Invited Speakers:
John Baez (University of California at Riverside) and John Foley (Metron Scientific Solutions).
Bob Coecke (University of Oxford).
Dusko Pavlovic (University of Hawaii).

Some other likely participants include Chris Boner (Metron), Arquimedes Canedo (Siemens at Princeton), Stephane Dugowson (Supméca), William Edmonson (North Carolina A&T), Brendan Fong (MIT), Mark Fuge (University of Maryland), Jack Gray (Penumbra), Helle Hansen (Delft), Jelle Herold (Statebox), Marisa Hughes (Johns Hopkins), Steve Huntsman (BAE Systems), Patrick Johnson (Dassault Systèmes), Al Jones (NIST), Cliff Joslyn (Pacific Northwest National Laboratory), Richard Malek (NSF), Tom Mifflin (Metron), Ira Monarch (Carnegie Mellon), John Paschkewitz (DARPA), Evan Patterson (Stanford), Blake Pollard (NIST), Emilie Purvine (Pacific Northwest National Laboratory), Mark Raugas (Pacific Northwest National Laboratory), Bill Regli (University of Maryland), Michael Robinson (American U.) Alberto Speranzon (Honeywell Aerospace), David Spivak (MIT), Eswaran Subrahmanian (Carnegie Mellon), Jamie Vicary (Birmingham and Oxford), and Ryan Wisnesky (Categorical Informatics).

A bunch of us will stay on into the weekend and talk some more. I hope we make a lot of progress—and I plan to let you know how it goes!

I’ll be giving a joint talk with John Foley about our work using operads to design networks. This work is part of the Complex Adaptive System Composition and Design Environment project being done by Metron Scientific Solutions and managed by John Paschkewitz at DARPA.

Linguistics Using Category Theory

11 February, 2018


Now students in the Applied Category Theory 2018 school are reading about categories applied to linguistics. Read the blog article here for more:

• Jade Master and Cory Griffith, Linguistics using category theory, The n-Category Café, 6 February 2018.

This was written by my grad student Jade Master along with Cory Griffith, an undergrad at Stanford. Jade is currently working with me on the semantics of open Petri nets.

What’s the basic idea of this linguistics and category theory stuff? I don’t know much about this, but I can say a bit.

Since category theory is great for understanding the semantics of programming languages, it makes sense to try it for human languages, even though they’re much harder. The first serious attempt I know was by Jim Lambek, who introduced pregroup grammars in 1958:

• Joachim Lambek, The mathematics of sentence structure, Amer. Math. Monthly 65 (1958), 154–170.

In this article he hid the connection to category theory. But when you start diagramming sentences or phrases using his grammar, as below, you get planar string diagrams as shown above. So it’s not surprising—if you’re in the know—that he’s secretly using monoidal categories where every object has a right dual and, separately, a left dual.

This fact is just barely mentioned in the Wikipedia article:

Pregroup grammar.

but it’s explained in more detail here:

• A. Preller and J. Lambek, Free compact 2-categories, Mathematical Structures in Computer Science 17 (2005), 309-340.

This stuff is hugely fun, so I’m wondering why I never looked into it before! When I talked to Lambek, who is sadly no longer with us, it was mainly about his theories relating particle physics to quaternions.

Recently Mehrnoosh Sadrzadeh and Bob Coecke have taken up Lambek’s ideas, relating them to the category of finite-dimensional vector spaces. Choosing a monoidal functor from a pregroup grammar to this category allows one to study linguistics using linear algebra! This simplifies things, perhaps a bit too much—but it makes it easy to do massive computations, which is very popular in this age of “big data” and machine learning.

It also sets up a weird analogy between linguistics and quantum mechanics, which I’m a bit suspicious of. While the category of finite-dimensional vector spaces with its usual tensor product is monoidal, and has duals, it’s symmetric, so the difference between writing a word to the left of another and writing it to the right of another gets washed out! I think instead of using vector spaces one should use modules of some noncommutative Hopf algebra, or something like that. Hmm… I should talk to those folks.

To discuss this, please visit The n-Category Café, since there’s a nice conversation going on there and I don’t want to split it. There has also been a conversation on Google+, and I’ll quote some of it here, so you don’t have to run all over the internet.

Noam Zeilberger wrote:

You might have been simplifying things for the post, but a small comment anyways: what Lambek introduced in his original paper are these days usually called “Lambek grammars”, and not exactly the same thing as what Lambek later introduced as “pregroup grammars”. Lambek grammars actually correspond to monoidal biclosed categories in disguise (i.e., based on left/right division rather than left/right duals), and may also be considered without a unit (as in his original paper). (I only have a passing familiarity with this stuff, though, and am not very clear on the difference in linguistic expressivity between grammars based on division vs grammars based on duals.)

Noam Zeilberger wrote:

If you haven’t seen it before, you might also like Lambek’s followup paper “On the calculus of syntactic types”, which generalized his original calculus by dropping associativity (so that sentences are viewed as trees rather than strings). Here are the first few paragraphs from the introduction:

…and here is a bit near the end of the 1961 paper, where he made explicit how derivations in the (original) associative calculus can be interpreted as morphisms of a monoidal biclosed category:

John Baez wrote:

Noam Zeilberger wrote: “what Lambek introduced in his original paper are these days usually called “Lambek grammars”, and not exactly the same thing as what Lambek later introduced as “pregroup grammars”.”

Can you say what the difference is? I wasn’t simplifying things on purpose; I just don’t know this stuff. I think monoidal biclosed categories are great, and if someone wants to demand that the left or right duals be inverses, or that the category be a poset, I can live with that too…. though if I ever learned more linguistics, I might ask why those additional assumptions are reasonable. (Right now I have no idea how reasonable the whole approach is to begin with!)

Thanks for the links! I will read them in my enormous amounts of spare time. :-)

Noam Zeilberger wrote:

As I said it’s not clear to me what the linguistic motivations are, but the way I understand the difference between the original “Lambek” grammars and (later introduced by Lambek) pregroup grammars is that it is precisely analogous to the difference between a monoidal category with left/right residuals and a monoidal category with left/right duals. Lambek’s 1958 paper was building off the idea of “categorial grammar” introduced earlier by Ajdukiewicz and Bar-Hillel, where the basic way of combining types was left division A\B and right division B/A (with no product).

Noam Zeilberger wrote:

At least one seeming advantage of the original approach (without duals) is that it permits interpretations of the “semantics” of sentences/derivations in cartesian closed categories. So it’s in harmony with the approach of “Montague semantics” (mentioned by Richard Williamson over at the n-Cafe) where the meanings of natural language expressions are interpreted using lambda calculus. What I understand is that this is one of the reasons Lambek grammar started to become more popular in the 80s, following a paper by Van Benthem where he observed that such such lambda terms denoting the meanings of expressions could be computed via “homomorphism” from syntactic derivations in Lambek grammar.

Jason Nichols wrote:

John Baez, as someone with a minimal understanding of set theory, lambda calculus, and information theory, what would you recommend as background reading to try to understand this stuff?

It’s really interesting, and looks relevant to work I do with NLP and even abstract syntax trees, but I reading the papers and wiki pages, I feel like there’s a pretty big gap to cross between where I am, and where I’d need to be to begin to understand this stuff.

John Baez wrote:

Jason Nichols: I suggest trying to read some of Lambek’s early papers, like this one:

• Joachim Lambek, The mathematics of sentence structure, Amer. Math. Monthly 65 (1958), 154–170.

(If you have access to the version at the American Mathematical Monthly, it’s better typeset than this free version.) I don’t think you need to understand category theory to follow them, at least not this first one. At least for starters, knowing category theory mainly makes it clear that the structures he’s trying to use are not arbitrary, but “mathematically natural”. I guess that as the subject develops further, people take more advantage of the category theory and it becomes more important to know it. But anyway, I recommend Lambek’s papers!

Borislav Iordanov wrote:

Lambek was an amazing teacher, I was lucky to have him in my ungrad. There is a small and very approachable book on his pregroups treatment that he wrote shortly before he passed away: “From Word to Sentence: a computational algebraic approach to grammar”. It’s plain algebra and very fun. Sadly looks like out of print on Amazon, but if you can find it, well worth it.

Andreas Geisler wrote:

One immediate concern for me here is that this seems (don’t have the expertise to be sure) to repeat a very old mistake of linguistics, long abandoned :

Words do not have atomic meanings. They are not a part of some 1:1 lookup table.

The most likely scenario right now is that our brains store meaning as a continuously accumulating set of connections that ultimately are impacted by every instance of a form we’ve ever heard/seen.

So, you shall know a word by all the company you’ve ever seen it in.

Andreas Geisler wrote:

John Baez I am a linguist by training, you’re welcome to borrow my brain if you want. You just have to figure out the words to use to get my brain to index what you need, as I don’t know the category theory stuff at all.

It’s a question of interpretation. I am also a translator, so i might be of some small assistance there as well, but it’s not going to be easy either way I am afraid.

John Baez wrote:

Andreas Geisler wrote: “I might be of some small assistance there as well, but it’s not going to be easy either way I am afraid.”

No, it wouldn’t. Alas, I don’t really have time to tackle linguistics myself. Mehrnoosh Sadrzadeh is seriously working on category theory and linguistics. She’s one of the people leading a team of students at this Applied Category Theory 2018 school. She’s the one who assigned this paper by Lambek, which 2 students blogged about. So she would be the one to talk to.

So, you shall know a word by all the company you’ve ever seen it in.

Yes, that quote appears in the blog article by the students, which my post here was merely an advertisement for.

Applied Category Theory at UCR (Part 3)

13 November, 2017

We had a special session on applied category theory here at UCR:

Applied category theory, Fall Western Sectional Meeting of the AMS, 4-5 November 2017, U.C. Riverside.

A bunch of people stayed for a few days afterwards, and we had a lot of great discussions. I wish I could explain everything that happened, but I’m too busy right now. Luckily, even if you couldn’t come here, you can now see slides of almost all the talks… and videos of many!

Click on talk titles to see abstracts. For multi-author talks, the person whose name is in boldface is the one who gave the talk. For videos, go here: I haven’t yet created links to all the videos.

Saturday November 4, 2017

9:00 a.m.A higher-order temporal logic for dynamical systemstalk slides.
David I. Spivak, MIT.

10:00 a.m.
Algebras of open dynamical systems on the operad of wiring diagramstalk slides.
Dmitry Vagner, Duke University
David I. Spivak, MIT
Eugene Lerman, University of Illinois at Urbana-Champaign

10:30 a.m.
Abstract dynamical systemstalk slides.
Christina Vasilakopoulou, UCR
David Spivak, MIT
Patrick Schultz, MIT

3:00 p.m.
Decorated cospanstalk slides.
Brendan Fong, MIT

4:00 p.m.
Compositional modelling of open reaction networkstalk slides.
Blake S. Pollard, UCR
John C. Baez, UCR

4:30 p.m.
A bicategory of coarse-grained Markov processestalk slides.
Kenny Courser, UCR

5:00 p.m.
A bicategorical syntax for pure state qubit quantum mechanicstalk slides.
Daniel M. Cicala, UCR

5:30 p.m.
Open systems in classical mechanicstalk slides.
Adam Yassine, UCR

Sunday November 5, 2017

9:00 a.m.
Controllability and observability: diagrams and dualitytalk slides.
Jason Erbele, Victor Valley College

9:30 a.m.
Frobenius monoids, weak bimonoids, and corelationstalk slides.
Brandon Coya, UCR

10:00 a.m.
Compositional design and tasking of networks.
John D. Foley, Metron, Inc.
John C. Baez, UCR
Joseph Moeller, UCR
Blake S. Pollard, UCR

10:30 a.m.
Operads for modeling networkstalk slides.
Joseph Moeller, UCR
John Foley, Metron Inc.
John C. Baez, UCR
Blake S. Pollard, UCR

2:00 p.m.
Reeb graph smoothing via cosheavestalk slides.
Vin de Silva, Department of Mathematics, Pomona College

3:00 p.m.
Knowledge representation in bicategories of relationstalk slides.
Evan Patterson, Stanford University, Statistics Department

3:30 p.m.
The multiresolution analysis of flow graphstalk slides.
Steve Huntsman, BAE Systems

4:00 p.m.
Data modeling and integration using the open source tool Algebraic Query Language (AQL)talk slides.
Peter Y. Gates, Categorical Informatics
Ryan Wisnesky, Categorical Informatics

Applied Category Theory 2018 — Adjoint School

22 October, 2017

There’s a ‘school’ on applied category theory one week before the workshop Applied Category Theory 2018. The deadline for applying to this school is Wednesday November 1st.

Applied Category Theory: Adjoint School: online sessions starting in January 2018, followed by a meeting 23–27 April 2018 at the Lorentz Center in Leiden, the Netherlands. Organized by Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford).

The name ‘adjoint school’ is a bad pun, but the school should be great. Here’s how it works:


The Workshop on Applied Category Theory 2018 takes place in May 2018. A principal goal of this workshop is to bring early career researchers into the applied category theory community. Towards this goal, we are organising the Adjoint School.

The Adjoint School will run from January to April 2018. By the end of the school, each participant will:

  • be familiar with the language, goals, and methods of four prominent, current research directions in applied category theory;
  • have worked intensively on one of these research directions, mentored by an expert in the field; and
  • know other early career researchers interested in applied category theory.

They will then attend the main workshop, well equipped to take part in discussions across the diversity of applied category theory.


The Adjoint School comprises (1) an Online Reading Seminar from January to April 2018, and (2) a four day Research Week held at the Lorentz Center, Leiden, The Netherlands, from Monday April 23rd to Thursday April 26th.

In the Online Reading Seminar we will read papers on current research directions in applied category theory. The seminar will consist of eight iterations of a two week block. Each block will have one paper as assigned reading, two participants as co-leaders, and three phases:

  • A presentation (over WebEx) on the assigned reading delivered by the two block co-leaders.
  • Reading responses and discussion on a private forum, facilitated by Brendan Fong and Nina Otter.
  • Publication of a blog post on the n-Category Café written by the co-leaders.

Each participant will be expected to co-lead one block.

The Adjoint School is taught by mentors John Baez, Aleks Kissinger, Martha Lewis, and Pawel Sobocinski. Each mentor will mentor a working group comprising four participants. During the second half of the Online Reading Seminar, these working groups will begin to meet with their mentor (again over video conference) to learn about open research problems related to their reading.

In late April, the participants and the mentors will convene for a four day Research Week at the Lorentz Center. After opening lectures by the mentors, the Research Week will be devoted to collaboration within the working groups. Morning and evening reporting sessions will keep the whole school informed of the research developments of each group.

The following week, participants will attend Applied Category Theory 2018, a discussion-based 60-attendee workshop at the Lorentz Center. Here they will have the chance to meet senior members across the applied category theory community and learn about ongoing research, as well as industry applications.

Following the school, successful working groups will be invited to contribute to a new, to be launched, CUP book series.

Reading list

Meetings will be on Mondays; we will determine a time depending on the locations of the chosen participants.

Research projects

John Baez: Semantics for open Petri nets and reaction networks
Petri nets and reaction networks are widely used to describe systems of interacting entities in computer science, chemistry and other fields, but the study of open Petri nets and reaction networks is new, and raise many new questions connected to Lawvere’s “functorial semantics”.
Reading: Fong; Baez and Pollard.

Aleks Kissinger: Unification of the logic of causality
Employ the framework of (pre-)causal categories to unite notions of causality and techniques for causal reasoning which occur in classical statistics, quantum foundations, and beyond.
Reading: Kissinger and Uijlen; Henson, Lal, and Pusey.

Martha Lewis: Compositional approaches to linguistics and cognition
Use compact closed categories to integrate compositional models of meaning with distributed, geometric, and other meaning representations in linguistics and cognitive science.
Reading: Coecke, Sadrzadeh, and Clark; Bolt, Coecke, Genovese, Lewis, Marsden, and Piedeleu.

Pawel Sobocinski: Modelling of open and interconnected systems
Use Carboni and Walters’ bicategories of relations as a multidisciplinary algebra of open and interconnected systems.
Reading: Carboni and Walters; Willems.


We hope that each working group will comprise both participants who specialise in category theory and in the relevant application field. As a prerequisite, those participants specialising in category theory should feel comfortable with the material found in Categories for the Working Mathematician or its equivalent; those specialising in applications should have a similar graduate-level introduction.

To apply, please fill out the form here. You will be asked to upload a single PDF file containing the following information:

  • Your contact information and educational history.
  • A brief paragraph explaining your interest in this course.
  • A paragraph or two describing one of your favorite topics in category theory, or your application field.
  • A ranked list of the papers you would most like to present, together with an explanation of your preferences. Note that the paper you present determines which working group you will join.

You may add your CV if you wish.

Anyone is welcome to apply, although preference may be given to current graduate students and postdocs. Women and members of other underrepresented groups within applied category theory are particularly encouraged to apply.

Some support will be available to help with the costs (flights, accommodation, food, childcare) of attending the Research Week and the Workshop on Applied Category Theory; please indicate in your application if you would like to be considered for such support.

If you have any questions, please feel free to contact Brendan Fong (bfo at mit dot edu) or Nina Otter (otter at maths dot ox dot ac dot uk).

Application deadline: November 1st, 2017.

Applied Category Theory at UCR (Part 2)

21 September, 2017

I’m running a special session on applied category theory, and now the program is available:

Applied category theory, Fall Western Sectional Meeting of the AMS, 4-5 November 2017, U.C. Riverside.

This is going to be fun.

My former student Brendan Fong is now working with David Spivak at MIT, and they’re both coming. My collaborator John Foley at Metron is also coming: we’re working on the CASCADE project for designing networked systems.

Dmitry Vagner is coming from Duke: he wrote a paper with David and Eugene Lerman on operads and open dynamical systems. Christina Vaisilakopoulou, who has worked with David and Patrick Schultz on dynamical systems, has just joined our group at UCR, so she will also be here. And the three of them have worked with Ryan Wisnesky on algebraic databases. Ryan will not be here, but his colleague Peter Gates will: together with David they have a startup called Categorical Informatics, which uses category theory to build sophisticated databases.

That’s not everyone—for example, most of my students will be speaking at this special session, and other people too—but that gives you a rough sense of some people involved. The conference is on a weekend, but John Foley and David Spivak and Brendan Fong and Dmitry Vagner are staying on for longer, so we’ll have some long conversations… and Brendan will explain decorated corelations in my Tuesday afternoon network theory seminar.

Here’s the program. Click on talk titles to see abstracts. For a multi-author talk, the person with the asterisk after their name is doing the talking. All the talks will be in Room 268 of the Highlander Union Building or ‘HUB’.

Saturday November 4, 2017, 9:00 a.m.-10:50 a.m.

9:00 a.m.
A higher-order temporal logic for dynamical systems.
David I. Spivak, MIT

10:00 a.m.
Algebras of open dynamical systems on the operad of wiring diagrams.
Dmitry Vagner*, Duke University
David I. Spivak, MIT
Eugene Lerman, University of Illinois at Urbana-Champaign

10:30 a.m.
Abstract dynamical systems.
Christina Vasilakopoulou*, University of California, Riverside
David Spivak, MIT
Patrick Schultz, MIT

Saturday November 4, 2017, 3:00 p.m.-5:50 p.m.

3:00 p.m.
Black boxes and decorated corelations.
Brendan Fong, MIT

4:00 p.m.
Compositional modelling of open reaction networks.
Blake S. Pollard*, University of California, Riverside
John C. Baez, University of California, Riverside

4:30 p.m.
A bicategory of coarse-grained Markov processes.
Kenny Courser, University of California, Riverside

5:00 p.m.
A bicategorical syntax for pure state qubit quantum mechanics.
Daniel M. Cicala, University of California, Riverside

5:30 p.m.
Open systems in classical mechanics.
Adam Yassine, University of California Riverside

Sunday November 5, 2017, 9:00 a.m.-10:50 a.m.

9:00 a.m.
Controllability and observability: diagrams and duality.
Jason Erbele, Victor Valley College

9:30 a.m.
Frobenius monoids, weak bimonoids, and corelations.
Brandon Coya, University of California, Riverside

10:00 a.m.
Compositional design and tasking of networks.
John D. Foley*, Metron, Inc.
John C. Baez, University of California, Riverside
Joseph Moeller, University of California, Riverside
Blake S. Pollard, University of California, Riverside

10:30 a.m.
Operads for modeling networks.
Joseph Moeller*, University of California, Riverside
John Foley, Metron Inc.
John C. Baez, University of California, Riverside
Blake S. Pollard, University of California, Riverside

Sunday November 5, 2017, 2:00 p.m.-4:50 p.m.

2:00 p.m.
Reeb graph smoothing via cosheaves.
Vin de Silva, Department of Mathematics, Pomona College

3:00 p.m.
Knowledge representation in bicategories of relations.
Evan Patterson*, Stanford University, Statistics Department

3:30 p.m.
The multiresolution analysis of flow graphs.
Steve Huntsman*, BAE Systems

4:00 p.m.
Data modeling and integration using the open source tool Algebraic Query Language (AQL).
Peter Y. Gates*, Categorical Informatics
Ryan Wisnesky, Categorical Informatics

Applied Category Theory 2018

12 September, 2017

There will be a workshop on applied category theory!

Applied Category Theory (ACT 2018). School 23–27 April 2018 and workshop 30 April–4 May 2018 at the Lorentz Center in Leiden, the Netherlands. Organized by Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford).

The plenary speakers will be:

• Samson Abramsky (Oxford)
• John Baez (UC Riverside)
• Kathryn Hess (EPFL)
• Mehrnoosh Sadrzadeh (Queen Mary)
• David Spivak (MIT)

One week before the workshop there will be a ‘school’ for students: the ACT 2018 Adjoint School.

There will be a lot more to say as this progresses, but for now let me just quote from the conference website:

Applied Category Theory (ACT 2018) is a five-day workshop on applied category theory running from April 30 to May 4 at the Lorentz Center in Leiden, the Netherlands.

Towards an Integrative Science: in this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one scientific discipline can be reused in another. The aim of the workshop is to (1) explore the use of category theory within and across different disciplines, (2) create a more cohesive and collaborative ACT community, especially among early-stage researchers, and (3) accelerate research by outlining common goals and open problems for the field.

While the workshop will host discussions on a wide range of applications of category theory, there will be four special tracks on exciting new developments in the field:

1. Dynamical systems and networks
2. Systems biology
3. Cognition and AI
4. Causality

Accompanying the workshop will be an Adjoint Research School for early-career researchers. This will comprise a 16 week online seminar, followed by a 4 day research meeting at the Lorentz Center in the week prior to ACT 2018. Applications to the school will open prior to October 1, and are due November 1. Admissions will be notified by November 15.

The organizers

Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford)

We welcome any feedback! Please send comments to this link.

About Applied Category Theory

Category theory is a branch of mathematics originally developed to transport ideas from one branch of mathematics to another, e.g. from topology to algebra. Applied category theory refers to efforts to transport the ideas of category theory from mathematics to other disciplines in science, engineering, and industry.

This site originated from discussions at the Computational Category Theory Workshop at NIST on Sept. 28-29, 2015. It serves to collect and disseminate research, resources, and tools for the development of applied category theory, and hosts a blog for those involved in its study.

The proposal: Towards an Integrative Science

Category theory was developed in the 1940s to translate ideas from one field of mathematics, e.g. topology, to another field of mathematics, e.g. algebra. More recently, category theory has become an unexpectedly useful and economical tool for modeling a range of different disciplines, including programming language theory [10], quantum mechanics [2], systems biology [12], complex networks [5], database theory [7], and dynamical systems [14].

A category consists of a collection of objects together with a collection of maps between those objects, satisfying certain rules. Topologists and geometers use category theory to describe the passage from one mathematical structure to another, while category theorists are also interested in categories for their own sake. In computer science and physics, many types of categories (e.g. topoi or monoidal categories) are used to give a formal semantics of domain-specific phenomena (e.g. automata [3], or regular languages [11], or quantum protocols [2]). In the applied category theory community, a long-articulated vision understands categories as mathematical workspaces for the experimental sciences, similar to how they are used in topology and geometry [13]. This has proved true in certain fields, including computer science and mathematical physics, and we believe that these results can be extended in an exciting direction: we believe that category theory has the potential to bridge specific different fields, and moreover that developments in such fields (e.g. automata) can be transferred successfully into other fields (e.g. systems biology) through category theory. Already, for example, the categorical modeling of quantum processes has helped solve an important open problem in natural language processing [9].

In this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one discipline can be reused in another. Tangibly and in the short-term, we will bring together people from different disciplines in order to write an expository survey paper that grounds the varied research in applied category theory and lays out the parameters of the research program.

In formulating this research program, we are motivated by recent successes where category theory was used to model a wide range of phenomena across many disciplines, e.g. open dynamical systems (including open Markov processes and open chemical reaction networks), entropy and relative entropy [6], and descriptions of computer hardware [8]. Several talks will address some of these new developments. But we are also motivated by an open problem in applied category theory, one which was observed at the most recent workshop in applied category theory (Dagstuhl, Germany, in 2015): “a weakness of semantics/CT is that the definitions play a key role. Having the right definitions makes the theorems trivial, which is the opposite of hard subjects where they have combinatorial proofs of theorems (and simple definitions). […] In general, the audience agrees that people see category theorists only as reconstructing the things they knew already, and that is a disadvantage, because we do not give them a good reason to care enough” [1, pg. 61].

In this workshop, we wish to articulate a natural response to the above: instead of treating the reconstruction as a weakness, we should treat the use of categorical concepts as a natural part of transferring and integrating knowledge across disciplines. The restructuring employed in applied category theory cuts through jargon, helping to elucidate common themes across disciplines. Indeed, the drive for a common language and comparison of similar structures in algebra and topology is what led to the development category theory in the first place, and recent hints show that this approach is not only useful between mathematical disciplines, but between scientific ones as well. For example, the ‘Rosetta Stone’ of Baez and Stay demonstrates how symmetric monoidal closed categories capture the common structure between logic, computation, and physics [4].

[1] Samson Abramsky, John C. Baez, Fabio Gadducci, and Viktor Winschel. Categorical methods at the crossroads. Report from Dagstuhl Perspectives Workshop 14182, 2014.

[2] Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. In Handbook of Quantum Logic and Quantum Structures. Elsevier, Amsterdam, 2009.

[3] Michael A. Arbib and Ernest G. Manes. A categorist’s view of automata and systems. In Ernest G. Manes, editor, Category Theory Applied to Computation and Control. Springer, Berlin, 2005.

[4] John C. Baez and Mike STay. Physics, topology, logic and computation: a Rosetta stone. In Bob Coecke, editor, New Structures for Physics. Springer, Berlin, 2011.

[5] John C. Baez and Brendan Fong. A compositional framework for passive linear networks. arXiv e-prints, 2015.

[6] John C. Baez, Tobias Fritz, and Tom Leinster. A characterization of entropy in terms of information loss. Entropy, 13(11):1945–1957, 2011.

[7] Michael Fleming, Ryan Gunther, and Robert Rosebrugh. A database of categories. Journal of Symbolic Computing, 35(2):127–135, 2003.

[8] Dan R. Ghica and Achim Jung. Categorical semantics of digital circuits. In Ruzica Piskac and Muralidhar Talupur, editors, Proceedings of the 16th Conference on Formal Methods in Computer-Aided Design. Springer, Berlin, 2016.

[9] Dimitri Kartsaklis, Mehrnoosh Sadrzadeh, Stephen Pulman, and Bob Coecke. Reasoning about meaning in natural language with compact closed categories and Frobenius algebras. In Logic and Algebraic Structures in Quantum Computing and Information. Cambridge University Press, Cambridge, 2013.

[10] Eugenio Moggi. Notions of computation and monads. Information and Computation, 93(1):55–92, 1991.

[11] Nicholas Pippenger. Regular languages and Stone duality. Theory of Computing Systems 30(2):121–134, 1997.

[12] Robert Rosen. The representation of biological systems from the standpoint of the theory of categories. Bulletin of Mathematical Biophysics, 20(4):317–341, 1958.

[13] David I. Spivak. Category Theory for Scientists. MIT Press, Cambridge MA, 2014.

[14] David I. Spivak, Christina Vasilakopoulou, and Patrick Schultz. Dynamical systems and sheaves. arXiv e-prints, 2016.