Applied Category Theory Seminar

14 December, 2018

We’re going to have a seminar on applied category theory here at U. C. Riverside! My students have been thinking hard about category theory for a few years, but they’ve decided it’s time to get deeper into applications. Christian Williams, in particular, seems to have caught my zeal for trying to develop new math to help save the planet.

We’ll try to videotape the talks to make it easier for you to follow along. I’ll also start discussions here and/or on the Azimuth Forum. It’ll work best if you read the papers we’re talking about and then join these discussions. Ask questions, and answer any questions you can!

Here’s how the schedule of talks is shaping up so far. I’ll add more information as it becomes available, either here or on a webpage devoted to the task.

January 8, 2019: John Baez – Mathematics in the 21st century

I’ll give an updated synthesized version of these earlier talks of mine, so check out these slides and the links:

The mathematics of planet Earth.

What is climate change?

Props in network theory.

January 15, 2019: Jonathan Lorand – Problems in symplectic linear algebra

Lorand is visiting U. C. Riverside to work with me on applications of symplectic geometry to chemistry. Here is the abstract of his talk:

In this talk we will look at various examples of classification problems in symplectic linear algebra: conjugacy classes in the symplectic group and its Lie algebra, linear lagrangian relations up to conjugation, tuples of (co)isotropic subspaces. I will explain how many such problems can be encoded using the theory of symplectic poset representations, and will discuss some general results of this theory. Finally, I will recast this discussion from a broader category-theoretic perspective.

January 22, 2019: Christina Vasilakopoulou – Wiring diagrams

Vasilakopoulou, a visiting professor here, previously worked with David Spivak. So, we really want to figure out how two frameworks for dealing with networks relate: Brendan Fong’s ‘decorated cospans’, and Spivak’s ‘monoidal category of wiring diagrams’. Since Fong is now working with Spivak they’ve probably figured it out already! But anyway, Vasilakopoulou will give a talk on systems as algebras for the wiring diagram monoidal category. It will be based on this paper:

• Patrick Schultz, David I. Spivak and Christina Vasilakopoulou, Dynamical systems and sheaves.

but she will focus more on the algebraic description (and conditions for deterministic/total systems) rather than the sheaf theoretic aspect of the input types. This work builds on earlier papers such as these:

• David I. Spivak, The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits.

• Dmitry Vagner, David I. Spivak and Eugene Lerman, Algebras of open dynamical systems on the operad of wiring diagrams.

January 29, 2019: Daniel Cicala – Dynamical systems on networks

Cicala will discuss a topic from this paper:

• Mason A. Porter and James P. Gleeson, Dynamical systems on networks: a tutorial.

His leading choice is a model for social contagion (e.g. opinions) which is discussed in more detail here:

• Duncan J. Watts, A simple model of global cascades on random networks.

Category Theory Course

13 October, 2018

I’m teaching a course on category theory at U.C. Riverside, and since my website is still suffering from reduced functionality I’ll put the course notes here for now. I taught an introductory course on category theory in 2016, but this one is a bit more advanced.

The hand-written notes here are by Christian Williams. They are probably best seen as a reminder to myself as to what I’d like to include in a short book someday.

Lecture 1: What is pure mathematics all about? The importance of free structures.

Lecture 2: The natural numbers as a free structure. Adjoint functors.

Lecture 3: Adjoint functors in terms of unit and counit.

Lecture 4: 2-Categories. Adjunctions.

Lecture 5: 2-Categories and string diagrams. Composing adjunctions.

Lecture 6: The ‘main spine’ of mathematics. Getting a monad from an adjunction.

Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

Lecture 8: The walking monad, the augmented simplex category and the simplex category.

Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

Lecture 10: The Dold-Thom theorem: the category of simplicial abelian groups is equivalent to the category of chain complexes of abelian groups. The homology of a chain complex.

Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

Lecture 8: The walking monad, the
augmented simplex category and the simplex category.

Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

Lecture 10: Chain complexes from simplicial abelian groups. The homology of a chain complex.

Lecture 12: The bar construction: getting a simplicial objects from an adjunction. The bar construction for G-sets, previewed.

Lecture 13: The adjunction between G-sets and sets.

Lecture 14: The bar construction for groups.

Lecture 15: The simplicial set \mathbb{E}G obtained by applying the bar construction to the one-point G-set, its geometric realization EG = |\mathbb{E}G|, and the free simplicial abelian group \mathbb{Z}[\mathbb{E}G].

Lecture 16: The chain complex C(G) coming from the simplicial abelian group \mathbb{Z}[\mathbb{E}G], its homology, and the definition of group cohomology H^n(G,A) with coefficients in a G-module.

Lecture 17: Extensions of groups. The Jordan-Hölder theorem. How an extension of a group G by an abelian group A gives an action of G on A and a 2-cocycle c \colon G^2 \to A.

Lecture 18: Classifying abelian extensions of groups. Direct products, semidirect products, central extensions and general abelian extensions. The groups of order 8 as abelian extensions.

Lecture 19: Group cohomology. The chain complex for the cohomology of G with coefficients in A, starting from the bar construction, and leading to the 2-cocycles used in classifying abelian extensions. The classification of extensions of G by A in terms of H^2(G,A).

Lecture 20: Examples of group cohomology: nilpotent groups and the fracture theorem. Higher-dimensional algebra and homotopification: the nerve of a category and the nerve of a topological space. \mathbb{E}G as the nerve of the translation groupoid G/\!/G. BG = EG/G as the walking space with fundamental group G.

Lecture 21: Homotopification and higher algebra. Internalizing concepts in categories with finite products. Pushing forward internalized structures using functors that preserve finite products. Why the ‘discrete category on a set’ functor \mathrm{Disc} \colon \mathrm{Set} \to \mathrm{Cat}, the ‘nerve of a category’ functor \mathrm{N} \colon \mathrm{Cat} \to \mathrm{Set}^{\Delta^{\mathrm{op}}}, and the ‘geometric realization of a simplicial set’ functor |\cdot| \colon \mathrm{Set}^{\Delta^{\mathrm{op}}} \to \mathrm{Top} preserve products.

Lecture 22: Monoidal categories. Strict monoidal categories as monoids in \mathrm{Cat} or one-object 2-categories. The periodic table of strict n-categories. General ‘weak’ monoidal categories.

Lecture 23: 2-Groups. The periodic table of weak n-categories. The stabilization hypothesis. The homotopy hypothesis. Classifying 2-groups with G as the group of objects and A as the abelian group of automorphisms of the unit object in terms of H^3(G,A). The Eckmann–Hilton argument.

Applied Category Theory 2019

2 October, 2018


animation by Marius Buliga

I’m helping organize ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019.

More details will come later, but here’s the basic idea. If you’re a grad student interested in this subject, you should apply for the ‘school’. Not yet—we’ll let you know when.

Dear all,

As part of a new growing community in Applied Category Theory, now with a dedicated journal Compositionality, a traveling workshop series SYCO, a forthcoming Cambridge U. Press book series Reasoning with Categories, and several one-off events including at NIST, we launch an annual conference+school series named Applied Category Theory, the coming one being at Oxford, July 15-19 for the conference, and July 22-26 for the school. The dates are chosen such that CT 2019 (Edinburgh) and the ACT 2019 conference (Oxford) will be back-to-back, for those wishing to participate in both.

There already was a successful invitation-only pilot, ACT 2018, last year at the Lorentz Centre in Leiden, also in the format of school+workshop.

For the conference, for those who are familiar with the successful QPL conference series, we will follow a very similar format for the ACT conference. This means that we will accept both new papers which then will be published in a proceedings volume (most likely a Compositionality special Proceedings issue), as well as shorter abstracts of papers published elsewhere. There will be a thorough selection process, as typical in computer science conferences. The idea is that all the best work in applied category theory will be presented at the conference, and that acceptance is something that means something, just like in CS conferences. This is particularly important for young people as it will help them with their careers.

Expect a call for submissions soon, and start preparing your papers now!

The school in ACT 2018 was unique in that small groups of students worked closely with an experienced researcher (these were John Baez, Aleks Kissinger, Martha Lewis and Pawel Sobociński), and each group ended up producing a paper. We will continue with this format or a closely related one, with Jules Hedges and Daniel Cicala as organisers this year. As there were 80 applications last year for 16 slots, we may want to try to find a way to involve more students.

We are fortunate to have a number of private sector companies closely associated in some way or another, who will also participate, with Cambridge Quantum Computing Inc. and StateBox having already made major financial/logistic contributions.

On behalf of the ACT Steering Committee,

John Baez, Bob Coecke, David Spivak, Christina Vasilakopoulou

What is Applied Category Theory?

18 September, 2018

Tai-Danae Bradley has a new free “booklet” on applied category theory. It was inspired by the workshop Applied Category Theory 2018, which she attended, and I think it makes a great complement to Fong and Spivak’s book Seven Sketches and my online course based on that book:

• Tai-Danae Bradley, What is Applied Category Theory?

Abstract. This is a collection of introductory, expository notes on applied category theory, inspired by the 2018 Applied Category Theory Workshop, and in these notes we take a leisurely stroll through two themes (functorial semantics and compositionality), two constructions (monoidal categories and decorated cospans) and two examples (chemical reaction networks and natural language processing) within the field.

Check it out!

Applied Category Theory Course: Collaborative Design

13 July, 2018

Our course on applied category theory is now starting the fourth chapter of Fong and Spivak’s book Seven Sketches. Chapter 4 is about collaborative design: building big projects from smaller parts. This is based on work by Andrea Censi:

• Andrea Censi, A mathematical theory of co-design.

The main mathematical content of this chapter is the theory of enriched profunctors. We’ll mainly talk about enriched profunctors between categories enriched in monoidal preorders. The picture above shows what one of these looks like!

Here are the lectures:

Lecture 55 – Chapter 4: Enriched Profunctors and Collaborative Design
Lecture 56 – Chapter 4: Feasibility Relations
Lecture 57 – Chapter 4: Feasibility Relations
Lecture 58 – Chapter 4: Composing Feasibility Relations
Lecture 59 – Chapter 4: Cost-Enriched Profunctors
Lecture 60 – Chapter 4: Closed Monoidal Preorders
Lecture 61 – Chapter 4: Closed Monoidal Preorders
Lecture 62 – Chapter 4: Constructing Enriched Categories
Lecture 63 – Chapter 4: Composing Enriched Profunctors
Lecture 64 – Chapter 4: The Category of Enriched Profunctors
Lecture 65 – Chapter 4: Collaborative Design
Lecture 66 – Chapter 4: Collaborative Design
Lecture 67 – Chapter 4: Feedback in Collaborative Design
Lecture 68 – Chapter 4: Feedback in Collaborative Design
Lecture 69 – Chapter 4: Feedback in Collaborative Design
Lecture 70 – Chapter 4: Tensoring Enriched Profunctors
Lecture 71 – Chapter 4: Caps and Cups for Enriched Profunctors
Lecture 72 – Chapter 4: Monoidal Categories
Lecture 73 – Chapter 4: String Diagrams and Strictification
Lecture 74 – Chapter 4: Compact Closed Categories
Lecture 75 – Chapter 4: The Grand Synthesis
Lecture 76 – Chapter 4: The Grand Synthesis
Lecture 77 – Chapter 4: The End? No, the Beginning!

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 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 2018 – Videos

30 April, 2018

Some of the talks at Applied Category Theory 2018 were videotaped by the Statebox team. You can watch them on YouTube:

• David Spivak, A higher-order temporal logic for dynamical systems. Book available here and slides here.

• Fabio Zanasi and Bart Jacobs, Categories in Bayesian networks. Paper available here. (Some sound missing; when you hit silence skip forwards to about 15:00.)

• Bob Coecke and Aleks Kissinger, Causality. Paper available here.

• Samson Abramsky, Games and constraint satisfaction, Part 1 and Part 2. Paper available here.

• Dan Ghica, Diagrammatic semantics for digital circuits. Paper available here.

• Kathryn Hess, Towards a categorical approach to neuroscience.

• Tom Leinster, Biodiversity and the theory of magnitude. Papers available here and here.

• John Baez, Props in network theory. Slides available here, paper here and blog article here.