Compositionality – Now Open For Submissions

24 August, 2018

Our new journal Compositionality is now open for submissions!

It’s an 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.

Compositionality is free of cost for both readers and authors.



CALL FOR PAPERS

We invite you to submit a manuscript for publication in the first issue of Compositionality (ISSN: 2631-4444), a new open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline.

To submit a manuscript, please visit http://www.compositionality-journal.org/for-authors/.

SCOPE

Compositionality refers to complex things that can be built by sticking together simpler parts. We welcome papers using compositional ideas, most notably of a category-theoretic origin, in any discipline. This may concern foundational structures, an organising principle, a powerful tool, or an important application. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.

Related conferences and workshops that fall within the scope of Compositionality include the Symposium on Compositional Structures (SYCO), Categories, Logic and Physics (CLP), String Diagrams in Computation, Logic and Physics (STRING), Applied Category Theory (ACT), Algebra and Coalgebra in Computer Science (CALCO), and the Simons Workshop on Compositionality.

SUBMISSION AND PUBLICATION

Submissions should be original contributions of previously unpublished work, and may be of any length. Work previously published in conferences and workshops must be significantly expanded or contain significant new results to be accepted. There is no deadline for submission. There is no processing charge for accepted publications; Compositionality is free to read and free to publish in. More details can be found in our editorial policies at http://www.compositionality-journal.org/editorial-policies/.

STEERING BOARD

John Baez, University of California, Riverside, USA
Bob Coecke, University of Oxford, UK
Kathryn Hess, EPFL, Switzerland
Steve Lack, Macquarie University, Australia
Valeria de Paiva, Nuance Communications, USA

EDITORIAL BOARD

Corina Cirstea, University of Southampton, UK
Ross Duncan, University of Strathclyde, UK
Andree Ehresmann, University of Picardie Jules Verne, France
Tobias Fritz, Max Planck Institute, Germany
Neil Ghani, University of Strathclyde, UK
Dan Ghica, University of Birmingham, UK
Jeremy Gibbons, University of Oxford, UK
Nick Gurski, Case Western Reserve University, USA
Helle Hvid Hansen, Delft University of Technology, Netherlands
Chris Heunen, University of Edinburgh, UK
Aleks Kissinger, Radboud University, Netherlands
Joachim Kock, Universitat Autonoma de Barcelona, Spain
Martha Lewis, University of Amsterdam, Netherlands
Samuel Mimram, Ecole Polytechnique, France
Simona Paoli, University of Leicester, UK
Dusko Pavlovic, University of Hawaii, USA
Christian Retore, Universite de Montpellier, France
Mehrnoosh Sadrzadeh, Queen Mary University, UK
Peter Selinger, Dalhousie University, Canada
Pawel Sobocinski, University of Southampton, UK
David Spivak, MIT, USA
Jamie Vicary, University of Birmingham and University of Oxford, UK
Simon Willerton, University of Sheffield, UK

Sincerely,

The Editorial Board of Compositionality


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


Applied Category Theory Course: Databases

6 June, 2018

 

In my online course on applied category theory 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
Lecture 49 – Chapter 3: Kan Extensions
Lecture 50 – Chapter 3: Kan Extensions
Lecture 51 – Chapter 3: Right Kan Extensions
Lecture 52 – Chapter 3: The Hom-Functor
Lecture 53 – Chapter 3: Free and Forgetful Functors
Lecture 54 – Chapter 3: Tying Up Loose Ends


Applied Category Theory Course: Resource Theories

12 May, 2018

 

My course on applied category theory is continuing! After a two-week break where the students did exercises, I’m back to lecturing about Fong and Spivak’s book Seven Sketches. Now we’re talking about “resource theories”. Resource theories help us answer questions like this:

  1. Given what I have, is it possible to get what I want?
  2. Given what I have, how much will it cost to get what I want?
  3. Given what I have, how long will it take to get what I want?
  4. Given what I have, what is the set of ways to get what I want?

Resource theories in their modern form were arguably born in these papers:

• Bob Coecke, Tobias Fritz and Robert W. Spekkens, A mathematical theory of resources.

• Tobias Fritz, Resource convertibility and ordered commutative monoids.

We are lucky to have Tobias in our course, helping the discussions along! He’s already posted some articles on resource theory here on this blog:

• Tobias Fritz, Resource convertibility (part 1), Azimuth, 7 April 2015.

• Tobias Fritz, Resource convertibility (part 2), Azimuth, 10 April 2015.

• Tobias Fritz, Resource convertibility (part 3), Azimuth, 13 April 2015.

We’re having fun bouncing between the relatively abstract world of monoidal preorders and their very concrete real-world applications to chemistry, scheduling, manufacturing and other topics. Here are the lectures so far:

Lecture 18 – Chapter 2: Resource Theories
Lecture 19 – Chapter 2: Chemistry and Scheduling
Lecture 20 – Chapter 2: Manufacturing
Lecture 21 – Chapter 2: Monoidal Preorders
Lecture 22 – Chapter 2: Symmetric Monoidal Preorders
Lecture 23 – Chapter 2: Commutative Monoidal Posets
Lecture 24 – Chapter 2: Pricing Resources
Lecture 25 – Chapter 2: Reaction Networks
Lecture 26 – Chapter 2: Monoidal Monotones
Lecture 27 – Chapter 2: Adjoints of Monoidal Monotones
Lecture 28 – Chapter 2: Ignoring Externalities
Lecture 29 – Chapter 2: Enriched Categories
Lecture 30 – Chapter 2: Preorders as Enriched Categories
Lecture 31 – Chapter 2: Lawvere Metric Spaces
Lecture 32 – Chapter 2: Enriched Functors
Lecture 33 – Chapter 2: Tying Up Loose Ends

 


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”.

We study the applications to logic—both classical logic based on subsets, and a 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. But the real payoff comes in Chapter 2, where we discuss “resource theories”.

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+. 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.

Preface

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.


Azimuth Backup Project (Part 5)

5 October, 2017

I haven’t spoken much about the Azimuth Climate Data Backup Project, but it’s going well, and I’ll be speaking about it soon, here:

International Open Access Week, Wednesday 25 October 2017, 9:30–11:00 a.m., University of California, Riverside, Orbach Science Library, Room 240.

“Open in Order to Save Data for Future Research” is the 2017 event theme.

Open Access Week is an opportunity for the academic and research community to learn about the potential benefits of sharing what they’ve learned with colleagues, and to help inspire wider participation in helping to make “open access” a new norm in scholarship, research and data planning and preservation.

The Open Access movement is made of up advocates (librarians, publishers, university repositories, etc.) who promote the free, immediate, and online publication of research.

The program will provide information on issues related to saving open data, including climate change and scientific data. The panelists also will describe open access projects in which they have participated to save climate data and to preserve end-of-term presidential data, information likely to be and utilized by the university community for research and scholarship.

The program includes:

• Brianna Marshall, Director of Research Services, UCR Library: Brianna welcomes guests and introduces panelists.

• John Baez, Professor of Mathematics, UCR: John will describe his activities to save US government climate data through his collaborative effort, the Azimuth Climate Data Backup Project. All of the saved data is now open access for everyone to utilize for research and scholarship.

• Perry Willett, Digital Preservation Projects Manager, California Digital Library: Perry will discuss the open data initiatives in which CDL participates, including the end-of-term presidential web archiving that is done in partnership with the Library of Congress, Internet Archive and University of North Texas.

• Kat Koziar, Data Librarian, UCR Library: Kat will give an overview of DASH, the UC system data repository, and provide suggestions for researchers interested in making their data open.

This will be the eighth International Open Access Week program hosted by the UCR Library.

The event is free and open to the public. Light refreshments will be served.