Applied Category Theory 2019

7 February, 2019

I hope to see you at this conference, which will occur right before the associated school meets in Oxford:

Applied Category Theory 2019, July 15-19, 2019, Oxford, UK.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying systems of all sorts using category-theoretic tools. These systems are found in the natural sciences and social sciences, as well as in computer science, linguistics, and engineering. The background and experience of our members is as varied as the systems being studied. The goal of the ACT2019 Conference is to bring the majority of researchers in the field together and provide a platform for exposing the progress in the area. Both original research papers as well as extended abstracts of work submitted/accepted/published elsewhere will be considered.

There will be best paper award(s) and selected contributions will be awarded extended keynote slots.

The conference will include a business showcase and tutorials, and there also will be an adjoint school, the following week (see webpage).

Important dates

Submission of contributed papers: 3 May
Acceptance/Rejection notification: 7 June


Prospective speakers are invited to submit one (or more) of the following:

• Original contributions of high quality work consisting of a 5-12 page extended abstract that provides sufficient evidence of results of genuine interest and enough detail to allow the program committee to assess the merits of the work. Submissions of works in progress are encouraged but must be more substantial than a research proposal.

• Extended abstracts describing high quality work submitted/published elsewhere will also be considered, provided the work is recent and relevant to the conference. These consist of a maximum 3 page description and should include a link to a separate published paper or preprint.

The conference proceedings will be published in a dedicated Proceedings issue of the new Compositionality journal:

Only original contributions are eligible to be published in the proceedings.

Submissions should be prepared using LaTeX, and must be submitted in PDF format. Use of the Compositionality style is encouraged. Submission is done via EasyChair:

Program chairs

John Baez (U.C. Riverside)
Bob Coecke (University of Oxford)

Program committee

Bob Coecke (chair)
John Baez (chair)
Christina Vasilakopoulou
David Moore
Josh Tan
Stefano Gogioso
Brendan Fong
Steve Lack
Simona Paoli
Joachim Kock
Kathryn Hess Bellwald
Tobias Fritz
David I. Spivak
Ross Duncan
Dan Ghica
Valeria de Paiva
Jeremy Gibbons
Samuel Mimram
Aleks Kissinger
Jamie Vicary
Martha Lewis
Nick Gurski
Dusko Pavlovic
Chris Heunen
Corina Cirstea
Helle Hvid Hansen
Dan Marsden
Simon Willerton
Pawel Sobocinski
Dominic Horsman
Nina Otter
Miriam Backens

Steering committee

John Baez (U.C. Riverside)
Bob Coecke (University of Oxford)
David Spivak (M.I.T.)
Christina Vasilakopoulou (U.C. Riverside)

Applied Category Theory Course – Videos

15 January, 2019

Yay! David Spivak and Brendan Fong are teaching a course on applied category theory based on their book, and the lectures are on YouTube! Here are the first two videos:

Their book is free here:

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

If you’re in Boston you can actually go to the course. It’s at MIT January 14 – Feb 1, Monday-Friday, 14:00-15:00 in room 4-237.

They taught it last year too, and last year’s YouTube videos are on the same YouTube channel.

Also, I taught a course based on the first 4 chapters of their book, and you can read my “lectures”, see discussions and do problems here:

Applied category theory course.

So, there’s no excuse not to start applying category theory in your everday life!

Applied Category Theory 2019 School

5 January, 2019

Dear scientists, mathematicians, linguists, philosophers, and hackers:

We are writing to let you know about a fantastic opportunity to learn about the emerging interdisciplinary field of applied category theory from some of its leading researchers at the ACT2019 Adjoint School. It will begin February 18, 2019 and culminate in a meeting in Oxford, July 22–26, right after the associated conference. Applications are due January 30th; see below for details.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying systems of all sorts using category-theoretic tools. These systems are found in the natural sciences and social sciences, as well as in computer science, linguistics, and engineering. The background and experience of our community’s members is as varied as the systems being studied.

The goal of the ACT2019 School is to help grow this community by pairing ambitious young researchers together with established researchers in order to work on questions, problems, and conjectures in applied category theory.

Who should apply

Anyone from anywhere who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language—the definition of monoidal category for example—is encouraged.

We will consider advanced undergraduates, PhD students, and post-docs. We ask that you commit to the full program as laid out below.

Instructions for how to apply can be found below the research topic descriptions.

Senior research mentors and their topics

Below is a list of the senior researchers, each of whom describes a research project that their team will pursue, as well as the background reading that will be studied between now and July 2019.

Miriam Backens

Title: Simplifying quantum circuits using the ZX-calculus

Description: The ZX-calculus is a graphical calculus based on the category-theoretical formulation of quantum mechanics. A complete set of graphical rewrite rules is known for the ZX-calculus, but not for quantum circuits over any universal gate set. In this project, we aim to develop new strategies for using the ZX-calculus to simplify quantum circuits.

Background reading:

  1. Matthes Amy, Jianxin Chen, Neil Ross. A finite presentation of CNOT-Dihedral operators.
  2. Miriam Backens. The ZX-calculus is complete for stabiliser quantum mechanics.

Tobias Fritz

Title: Partial evaluations, the bar construction, and second-order stochastic dominance

Description: We all know that 2+2+1+1 evaluates to 6. A less familiar notion is that it can partially evaluate to 5+1. In this project, we aim to study the compositional structure of partial evaluation in terms of monads and the bar construction and see what this has to do with financial risk via second-order stochastic dominance.

Background reading:

  1. Tobias Fritz and Paolo Perrone. Monads, partial evaluations, and rewriting.
  2. Maria Manuel Clementino, Dirk Hofmann, George Janelidze. The monads of classical algebra are seldom weakly cartesian.
  3. Todd Trimble. On the bar construction.

Pieter Hofstra

Title: Complexity classes, computation, and Turing categories

Description: Turing categories form a categorical setting for studying computability without bias towards any particular model of computation. It is not currently clear, however, that Turing categories are useful to study practical aspects of computation such as complexity. This project revolves around the systematic study of step-based computation in the form of stack-machines, the resulting Turing categories, and complexity classes. This will involve a study of the interplay between traced monoidal structure and computation. We will explore the idea of stack machines qua programming languages, investigate the expressive power, and tie this to complexity theory. We will also consider questions such as the following: can we characterize Turing categories arising from stack machines? Is there an initial such category? How does this structure relate to other categorical structures associated with computability?

Background reading:

  1. J.R.B. Cockett and P.J.W. Hofstra. Introduction to Turing categories. APAL, Vol 156, pp. 183-209, 2008.
  2. J.R.B. Cockett, P.J.W. Hofstra and P. Hrubes. Total maps of Turing categories. ENTCS (Proc. of MFPS XXX), pp. 129-146, 2014.
  3. A. Joyal, R. Street and D. Verity. Traced monoidal categories. Mat. Proc. Cam. Phil. Soc. 3, pp. 447-468, 1996.

Bartosz Milewski

Title: Traversal optics and profunctors

Description: In functional programming, optics are ways to zoom into a specific part of a given data type and mutate it. Optics come in many flavors such as lenses and prisms and there is a well-studied categorical viewpoint, known as profunctor optics. Of all the optic types, only the traversal has resisted a derivation from first principles into a profunctor description. This project aims to do just this.

Background reading:

  1. Bartosz Milewski. Profunctor optics, categorical view.
  2. Craig Pastro, Ross Street. Doubles for monoidal categories.

Mehrnoosh Sadrzadeh

Title: Formal and experimental methods to reason about dialogue and discourse using categorical models of vector spaces

Description: Distributional semantics argues that meanings of words can be represented by the frequency of their co-occurrences in context. A model extending distributional semantics from words to sentences has a categorical interpretation via Lambek’s syntactic calculus or pregroups. In this project, we intend to further extend this model to reason about dialogue and discourse utterances where people interrupt each other, there are references that need to be resolved, disfluencies, pauses, and corrections. Additionally, we would like to design experiments and run toy models to verify predictions of the developed models.

Background reading:

  1. Gerhard Jager (1998): A multi-modal analysis of anaphora and ellipsis. University of Pennsylvania Working Papers in Linguistics 5(2), p. 2.
  2. Matthew Purver, Ronnie Cann, and Ruth Kempson. Grammars as parsers: meeting the dialogue challenge. Research on Language and Computation, 4(2-3):289–326, 2006.

David Spivak

Title: Toward a mathematical foundation for autopoiesis

Description: An autopoietic organization—anything from a living animal to a political party to a football team—is a system that is responsible for adapting and changing itself, so as to persist as events unfold. We want to develop mathematical abstractions that are suitable to found a scientific study of autopoietic organizations. To do this, we’ll begin by using behavioral mereology and graphical logic to frame a discussion of autopoeisis, most of all what it is and how it can be best conceived. We do not expect to complete this ambitious objective; we hope only to make progress toward it.

Background reading:

  1. Brendan Fong, David Jaz Myers, David Spivak. Behavioral mereology.
  2. Brendan Fong, David Spivak. Graphical regular logic.
  3. Luhmann. Organization and Decision, CUP. (Preface)

School structure

All of the participants will be divided up into groups corresponding to the projects. A group will consist of several students, a senior researcher, and a TA. Between January and June, we will have a reading course devoted to building the background necessary to meaningfully participate in the projects. Specifically, two weeks are devoted to each paper from the reading list. During this two week period, everybody will read the paper and contribute to discussion in a private online chat forum. There will be a TA serving as a domain expert and moderating this discussion. In the middle of the two week period, the group corresponding to the paper will give a presentation via video conference. At the end of the two week period, this group will compose a blog entry on this background reading that will be posted to the n-category cafe.

After all of the papers have been presented, there will be a two-week visit to Oxford University, 15–26 July 2019. The second week is solely for participants of the ACT2019 School. Groups will work together on research projects, led by the senior researchers.

The first week of this visit is the ACT2019 Conference, where the wider applied category theory community will arrive to share new ideas and results. It is not part of the school, but there is a great deal of overlap and participation is very much encouraged. The school should prepare students to be able to follow the conference presentations to a reasonable degree.

To apply

To apply please send the following to by January 30th, 2019:

  • Your CV
  • A document with:
    • An explanation of any relevant background you have in category theory or any of the specific projects areas
    • The date you completed or expect to complete your Ph.D and a one-sentence summary of its subject matter.
  • Order of project preference
  • To what extent can you commit to coming to Oxford (availability of funding is uncertain at this time)
  • A brief statement (~300 words) on why you are interested in the ACT2019 School. Some prompts:
    • how can this school contribute to your research goals?
    • how can this school help in your career?

Also have sent on your behalf to a brief letter of recommendation confirming any of the following:

  • your background
  • ACT2019 School’s relevance to your research/career
  • your research experience


For more information, contact either
– Daniel Cicala. cicala (at) math (dot) ucr (dot) edu
– Jules Hedges. julian (dot) hedges (at) cs (dot) ox (dot) ac (dot) uk

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. The talks are on Tuesdays 3:34–5:00 pm in Room 268 of Skye Hall, the mathematics building at U. C. Riverside.

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

Abstract. The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics – and be affected by it – just as dramatically as the agricultural revolution or industrial revolution. We should get ready!

Talk slides: Mathematics in the 21st century.

Also try these slides and videos from related talks:

The mathematics of planet Earth.

Props in network theory.

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

Abstract. 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.

Talk slides: Problems in symplectic linear algebra.

Reading material:

• Jonathan Lorand, Classifying linear canonical relations.

• Jonathan Lorand and Alan Weinstein, Decomposition of (co)isotropic relations.

January 22, 2019: Christina Vasilakopoulou
Systems as wiring diagram algebras

Abstract. We will start by describing the monoidal category of labeled boxes and wiring diagrams and its induced operad. Various kinds of systems such as discrete and continuous dynamical systems have been expressed as algebras for that operad, namely lax monoidal functors into the category of categories. A major advantage of this approach is that systems can be composed to form a system of the same kind, completely determined by the specific way the composite systems are interconnected (‘wired’ together). We will then introduce a generalized system, called a machine, again as a wiring diagram algebra. On the one hand, this abstract concept is all-inclusive in the sense that discrete and continuous dynamical systems are sub-algebras; on the other hand, we can specify succinct categorical conditions for totality and/or determinism of systems that also adhere to the algebraic description.

Christina Vasilakopoulou’s talk was based on this paper:

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

but she focused 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
Social contagion modeled on random networks

Abstract. A social contagion may manifest as a cultural trend, a spreading opinion or idea or belief. In this talk, we explore a simple model of social contagion on a random network. We also look at the effect that network connectivity, edge distribution, and heterogeneity has on the diffusion of a contagion.

Reading material:

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

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

February 5, 2019: Jade Master
Backprop as functor: a compositional perspective on supervised learning

Abstract. Fong, Spivak and Tuyéras have found a categorical framework in which gradient descent algorithms can be constructed in a compositional way. To explain this, we first give a brief introduction to backprogation and gradient descent. We then describe their monoidal category Learn, where the morphisms are given by abstract learning algorithms. Finally, we show how gradient descent can be realized as a monoidal functor from Para, the category of Euclidean spaces with differentiable parameterized functions between them, to Learn.

Reading material:

• Brendan Fong, David I. Spivak and Rémy Tuyéras, Backprop as functor: a compositional perspective on supervised learning.

February 12, 2019: Christian Williams
The pi calculus: towards global computing

Abstract. Historically, code represents a sequence of instructions for a single machine. Each computer is its own world, and only interacts with others by sending and receiving data through external ports. As society becomes more interconnected, this paradigm becomes more inadequate – these virtually isolated nodes tend to form networks of great bottleneck and opacity. Communication is a fundamental and integral part of computing, and needs to be incorporated in the theory of computation.

To describe systems of interacting agents with dynamic interconnection, in 1980 Robin Milner invented the pi calculus: a formal language in which a term represents an open, evolving system of processes (or agents) which communicate over names (or channels). Because a computer is itself such a system, the pi calculus can be seen as a generalization of traditional computing languages; there is an embedding of lambda into pi – but there is an important change in focus: programming is less like controlling a machine and more like designing an ecosystem of autonomous organisms.

We review the basics of the pi calculus, and explore a variety of examples which demonstrate this new approach to programming. We will discuss some of the history of these ideas, called “process algebra”, and see exciting modern applications in blockchain and biology.

“… as we seriously address the problem of modelling mobile communicating systems we get a sense of completing a model which was previously incomplete; for we can now begin to describe what goes on outside a computer in the same terms as what goes on inside – i.e. in terms of interaction. Turning this observation inside-out, we may say that we inhabit a global computer, an informatic world which demands to be understood just as fundamentally as physicists understand the material world.” — Robin Milner

Reading material:

• Robin Milner, The polyadic pi calculus: a tutorial.

• Robin Milner, Communicating and Mobile Systems.

• Joachim Parrow, An introduction to the pi calculus.

February 19, 2019: Kenny Courser
Category theory for genetics

Abstract. Rémy Tuyéras has developed a categorical framework aimed at handling various commonly studied subjects from the theory of genetics. Some of these include alignment methods, CRISPR, homologous recombination, haplotypes, and genetic linkage. In this talk, I will introduce the foundations on which this framework is built and give a few examples related to DNA and RNA sequencing which are able to be described in this environment.

Reading material:

• Rémy Tuyéras, Category theory for genetics.

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 11: Comonads from adjunctions. The walking comonad. The bar construction.

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.

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