Talk on the Tenfold Way

12 February, 2023

There are ten ways that a substance can have symmetry under time reversal, switching particles and holes, both or neither. But this fact turns out to extend far beyond condensed matter physics! It’s really built into the fabric of mathematics in a deep way.

I gave a talk on this at Nicohl Furey’s seminar Algebra, Particles and Quantum Theory, and you can see a video of my talk here:

You can also watch another version, where I explain this stuff to my friend James Dolan:

I like the idea of being able to watch an official talk but also watch the speaker chatting about the talk with a friend. It gives another view of the material. I skim over stuff Jim already knows, explain things I didn’t have time to get into in the actual talk, and emphasize the things I don’t understand. And he points out lots more patterns lurking in the tenfold way!

You can see my slides and more notes here.

The Joy of Abstraction

11 February, 2023

The Topos Institute is starting book club for The Joy of Abstraction by Eugenia Cheng. This book is an introduction to category theory for anyone who wants to get into the formality of the subject but does not necessarily have the mathematical background to read a standard textbook. Part 1 starts gently, so that readers can build up to the formality of rigorous mathematics even if they haven’t encountered it before. Part 2 gets into category theory, including limits and colimits, duality, functors, natural transformations, and the Yoneda lemma.

For more information about the book, try this:

Dr. Cheng would like to give people an opportunity to ask questions and get help with understanding the book. The book club will be hosted by the Topos Institute and will be run asynchronously. They will go at an approximate rate of one chapter per week. You can submit questions for each chapter according to the published schedule. Questions for the first chapter are due February 19, 2023. They will collate the questions and Dr. Cheng will make a video each week addressing the questions for that chapter. You will remain anonymous when asking the questions, so please don’t hesitate to ask questions that might feel “stupid”. They welcome any question that comes from you wanting to understand something better!

Each video will be posted at

during the week following the deadline for questions. If you are ahead of schedule you are welcome to submit questions in advance, but they will only be addressed in the video for that chapter.

To submit questions, please fill in this Google form. Include a page reference for your question, if relevant, so that Dr. Cheng can address the questions in order in the video.

This book club is open to everyone everywhere. Please spread the word!

Note that the schedule may change, but the deadlines for each chapter will only become later, never earlier.

Data Visualization Course

10 June, 2021

Are you a student interested in data analysis and sustainability? Or maybe you know some students interested in these things?

Then check this out: my former student Nina Otter, who now teaches at UCLA and Leipzig, is offering a short course on how to analyze and present data using modern methods like topological data analysis—with sustainable fishing as an example!

Students who apply before June 15 have a chance to learn a lot of cool stuff and get paid for it!

Call for Applications

We are advertising the following bootcamp, which will take place remotely on 22-25 June 2021.

If you are interested in participating, please apply here:

FishEthoBase data visualisation bootcamp: this is a 4-day bootcamp, organised by the DeMoS Institute, whose aim is to study ways to visualise scores and criteria from a fish ethology database. The database ( is an initiative led by the non-profits fair-fish international ( and FishEthoGroup ( The database is publicly accessible, it stores all currently available ethological knowledge on fish, with a specific focus on species farmed in aquacultures, with the goal of improving the welfare of fish.

The bootcamp will take place virtually on 22-25 June 2021, and will involve a maximum of eight students selected through an open call during the first half of June. The students will be guided by researchers in statistics and topological data analysis. During the first day of the bootcamp there will be talks given by researchers from FishEthoBase, as well as from the mentors. The next three days will be devoted to focused work in groups, with each day starting and ending with short presentations given by students about the progress of their work; after the presentations there will also be time for feedback and discussions from FishEthoBase researchers, and the mentors. Towards the end of August there will be a 2-hour follow-up meeting to discuss the implementation of the results from the bootcamp.

Target audience: we encourage applications from advanced undergraduate, master, and PhD students from a variety of backgrounds, including, but not limited to, computer science, mathematics, statistics, data analysis, computational biology, maritime sciences, and zoology.

Inclusivity: we encourage especially students from underrepresented groups to apply to this bootcamp.

Remuneration: The students who will be selected to participate in the bootcamp will be remunerated with a salary of 1400 euros.

When: 22-25 June 2021, approximately 11-18 CET each day

Where: remotely, on Zoom

I think it’s really cool that Nina Otter has started the DeMoS Institute. Here is the basic idea:

The institute carries out research on topics related to anti-democratic tendencies in our society, as well as on meta-scientific questions on how to make the scientific system more democratic. We believe that research must be done in the presence of those who bear their consequences. Therefore, we perform our research while at the same time implementing directly practices that promote inclusivity, interdisciplinarity, and in active engagement with society at large.

Applied Category Theory 2021 — Adjoint School

2 January, 2021

Do you want to get involved in applied category theory? Are you willing to do a lot of work and learn a lot? Then this is for you:

Applied Category Theory 2021 — Adjoint School. Applications due Friday 29 January 2021. Organized by David Jaz Myers, Sophie Libkind, and Brendan Fong.

There are four projects to work on with great mentors. You can see descriptions of them below!

By the way, it’s not yet clear if there will be an in-person component to this school —but if there is, it will happen at the University of Cambridge. ACT2021 is being organized by Jamie Vicary, who teaches in the computer science department there.

Who should apply?

Anyone, from anywhere in the world, 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, post-docs, as well as people working outside of academia. Members of groups which are underrepresented in the mathematics and computer science communities are especially encouraged to apply.

School overview

Participants are divided into four-person project teams. Each project is guided by a mentor and a TA. The Adjoint School has two main components: an Online Seminar that meets regularly between February and June, and an in-person Research Week in Cambridge, UK on July 5–9.

During the online seminar, we will read, discuss, and respond to papers chosen by the project mentors. Every other week, a pair of participants will present a paper which will be followed by a group discussion. Leading up to this presentation, study groups will meet to digest the reading in progress, and students will submit reading responses. After the presentation, the presenters will summarize the paper into a blog post for The n-Category Cafe.

The in-person research week will be held the week prior to the International Conference on Applied Category Theory and in the same location. During the week, participants work intensively with their research group under the guidance of their mentor. Projects from the Adjoint School will be presented during this conference. Both components of the school aim to develop a sense of belonging and camaraderie in students so that they can fully participate in the conference, for example by attending talks and chatting with other conference goers.

Projects to choose from

Here are the projects.

Topic: Categorical and computational aspects of C-sets

Mentors: James Fairbanks and Evan Patterson

Description: Applied category theory includes major threads of inquiry into monoidal categories and hypergraph categories for describing systems in terms of processes or networks of interacting components. Structured cospans are an important class of hypergraph categories. For example, Petri net-structured cospans are models of concurrent processes in chemistry, epidemiology, and computer science. When the structured cospans are given by C-sets (also known as co-presheaves), generic software can be implemented using the mathematics of functor categories. We will study mathematical and computational aspects of these categorical constructions, as well as applications to scientific computing.


Structured cospans, Baez and Courser.

An algebra of open dynamical systems on the operad of wiring diagrams, Vagner, Spivak, and Lerman.

Topic: The ubiquity of enriched profunctor nuclei

Mentor: Simon Willerton

Description: In 1964, Isbell developed a nice universal embedding for metric spaces: the tight span. In 1966, Isbell developed a duality for presheaves. These are both closely related to enriched profunctor nuclei, but the connection wasn’t spotted for 40 years. Since then, many constructions in mathematics have been observed to be enriched profunctor nuclei too, such as the fuzzy/formal concept lattice, tropical convex hull, and the Legendre–Fenchel transform. We’ll explore the world of enriched profunctor nuclei, perhaps seeking out further useful examples.


The Legendre–Fenchel transform from a category theoretic perspective, Willerton.

On the fuzzy concept complex (chapters 2-3), Elliot.

Topic: Double categories in applied category theory

Mentor: Simona Paoli

Description: Bicategories and double categories (and their symmetric monoidal versions) have recently featured in applied category theory: for instance, structured cospans and decorated cospans have been used to model several examples, such as electric circuits, Petri nets and chemical reaction networks.

An approach to bicategories and double categories is available in higher category theory through models that do not require a direct checking of the coherence axioms, such as the Segal-type models. We aim to revisit the structures used in applications in the light of these approaches, in the hope to facilitate the construction of new examples of interest in applications.


Structured cospans, Baez and Courser.

A double categorical model of weak 2-categories, Paoli and Pronk.

and introductory chapters of:

Simplicial Methods for Higher Categories: Segal-type Models of Weak n-Categories, Paoli.

Topic: Extensions of coalgebraic dynamic logic

Mentors: Helle Hvid Hansen and Clemens Kupke

Description: Coalgebra is a branch of category theory in which different types of state-based systems are studied in a uniform framework, parametric in an endofunctor F:C → C that specifies the system type. Many of the systems that arise in computer science, including deterministic/nondeterministic/weighted/probabilistic automata, labelled transition systems, Markov chains, Kripke models and neighbourhood structures, can be modeled as F-coalgebras. Once we recognise that a class of systems are coalgebras, we obtain general coalgebraic notions of morphism, bisimulation, coinduction and observable behaviour.

Modal logics are well-known formalisms for specifying properties of state-based systems, and one of the central contributions of coalgebra has been to show that modal logics for coalgebras can be developed in the general parametric setting, and many results can be proved at the abstract level of coalgebras. This area is called coalgebraic modal logic.

In this project, we will focus on coalgebraic dynamic logic, a coalgebraic framework that encompasses Propositional Dynamic Logic (PDL) and Parikh’s Game Logic. The aim is to extend coalgebraic dynamic logic to system types with probabilities. As a concrete starting point, we aim to give a coalgebraic account of stochastic game logic, and apply the coalgebraic framework to prove new expressiveness and completeness results.

Participants in this project would ideally have some prior knowledge of modal logic and PDL, as well as some familiarity with monads.


Parts of these:

Universal coalgebra: a theory of systems, Rutten.

Coalgebraic semantics of modal logics: an overview, Kupke and Pattinson.

Strong completeness of iteration-free coalgebraic dynamic logics, Hansen, Kupke, and Leale.

Categorical Statistics Group

10 June, 2020

As a spinoff of the workshop Categorical Probability and Statistics, Oliver Shetler has organized a reading group on category theory applied to statistics. The first meeting is Saturday June 27th at 17:00 UTC.

You can sign up for the group here, and also read more about it there. We’re discussing the group on the Category Theory Community Server, so if you want to join the reading group should probably also join that.

Here is a reading list. I’m sure the group won’t cover all these papers—we’ll start with the first one and see how it goes from there. But it’s certainly helpful to have a list like this.

• McCullagh, What is a statistical model?

• Morse and Sacksteder, Statistical isomorphism.

• Simpson, Probability sheaves and the Giry monad.

• Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics.

• Jacobs, Probabilities, distribution monads, and convex categories.

• Keimel, The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras.

• McCullaugh, Di Nardo, Senato, Natural statistics for spectral samples.

• Perrone, Categorical Probability and Stochastic Dominance in Metric Spaces. (Ph.D. thesis)

• Patterson, The Algebra and Machine Representation of Statistical Models. (Ph.D. thesis)

• Tuyeras, A category theoretical argument for causal inference.

• Culbertson and Sturtz, A categorical foundation for Bayesian probability.

• Fong, Causal Theories: A Categorical Perspective on Bayesian Networks. (Masters thesis)

• Fritz and Perrone, A probability monad as the colimit of spaces of finite samples.

• Fritz and Perrone, Bimonoidal structure of probability monads.

• Fritz, A presentation of the category of stochastic matrices.

• Jacobs and Furber, Towards a categorical account of conditional probability.

• Bradley, At the Interface of Algebra and Statistics. (Ph.D. Thesis)

• Bradley, Stoudenmire and Terilla, Modeling sequences with quantum states.

• Jacobs, Categorical aspects of parameter learning.

• Jacobs, Parameters and parameterization in specification, using distributive categories.

• Parzygnat, Inverses, disintegrations, and Bayesian inversion in quantum Markov categories.

• Jacobs, A channel-based perspective on conjugate priors.

ACT@UCR Seminar (Part 2)

7 June, 2020

The spring 2020 seminar on applied category theory at U.C. Riverside is done! Here you can see videos of all the talks, along with talk slides, discussions and more:

• John Baez: Structured cospans and double categories.

• Prakash Panangaden: A categorical view of conditional expectation.

• Jules Hedges: Open games: the long road to practical applications.

• Michael Shulman: Star-autonomous envelopes.

• Gershom Bazerman: A localic approach to the semantics of dependency, conflict, and concurrency.

• Sarah Rovner-Frydman: Separation logic through a new lens.

• Tai-Danae Bradley: Formal concepts vs. eigenvectors of density operators.

• Gordon Plotkin: A complete axiomatisation of partial differentiation.

• Simon Willerton: The Legendre–Fenchel transform from a category theoretic perspective.

• Nina Otter: Values and inclusivity in the applied category theory community.

Thanks to everyone for participating in this!

Values and Inclusivity in the ACT Community

27 May, 2020

In the tenth and final talk of this spring’s ACT@UCR seminar, Nina Otter led a discussion about diversity in the applied category theory community, with these speakers:

• Nina Otter: introduction, and some potential initiatives

• Jade Master: Experience in setting up an online research community for minorities in ACT

• Brendan Fong: Statement of values for ACT community

• Emily Riehl: Experience at MATRIX institute

• Christian Williams: Quick overview of ACT server

This is a change from her originally scheduled talk, due to the killing of George Floyd and ensuing events.

The discussion took place at the originally scheduled time on Wednesday June 3rd. Afterwards we had discussions at the Theory Community Server, here:

You can join the conversation there if you sign in.

You can see her slides here, or download a video here, or watch the video here:

• Nina Otter, Values and inclusivity in the applied category theory community.

Abstract. Saddened by the current events, we are taking this opportunity to pause and reflect on what we can do to change the status quo and try to bring about real and long-lasting change. Thus, we are holding a discussion aimed at finding concrete solutions to make the Applied Category Theory community more inclusive, and also to reflect about the values that our community would like to stand for and endorse, in particular, in terms of which sources of funding go against our values. While this discussion is specific to the applied category theory community, we believe that many of the topics will be of interest also to people in other fields, and thus we welcome anybody with an interest to attend. The discussion will consist of two parts: we will have first several people give short talks to discuss common issues that we need to address, as well as present specific plans for initiatives that we could take. We believe that the current pandemic, and the fact that all activities are now taking place remotely, gives us the opportunity to involve people who would otherwise find it difficult to travel, because of disabilities, financial reasons or care-taking responsibilities. Thus, now we have the opportunity to come up with new types of mentoring, collaborations, and many other initiatives that might have been difficult to envision until just a couple of months ago. The second part of the discussion will take place on the category theory community server, and its purpose is to allow for a broader participation in the discussion, and ideally during this part we will be able to flesh out in detail the specific initiatives that have been proposed in the talks.

The Legendre Transform: a Category Theoretic Perspective

26 May, 2020

In the ninth talk of the ACT@UCR seminar, Simon Willerton told us about a categorical approach to the Legendre transform, and its connection to tropical algebra.

He gave his talk on Wednesday May 27th. Afterwards we discussed it on the Category Theory Community Server, here:

You can view or join the conversation there if you sign in.

You can see his slides here, or download a video here, or watch the video here:

• Simon Willerton, The Legendre–Fenchel transform from a category theoretic perspective.

Abstract. The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this talk I’ll show how it arises in the context of category theory using categories enriched over the extended real numbers \overline{ \mathbb{R}}:=[-\infty,+\infty]. It turns out that it arises out of nothing more than the pairing between a vector space and its dual in the same way that the many classical dualities (e.g. in Galois theory or algebraic geometry) arise from a relation between sets.

I won’t assume knowledge of the Legendre-Fenchel transform.

The talk is based on this paper:

• Simon Willerton, The Legendre-Fenchel transform from a category theoretic perspective.

Also see his blog article:

• Simon Willerton, The nucleus of a profunctor: some categorified linear algebra, The n-Category Café.

A Complete Axiomatisation of Partial Differentiation

18 May, 2020

In the eighth talk of the ACT@UCR seminar, Gordon Plotkin told us about partial differentiation, viewed as a logical theory.

He gave his talk on Wednesday May 20th. Afterwards we discussed it on the Category Theory Community Server, here:

You can view or join the conversation there if you sign in.

You can see his slides here, or download a video of his talk here, or watch his video here:

• Gordon Plotkin, A complete axiomatisation of partial differentiation.

Abstract. We formalise the well-known rules of partial differentiation in a version of equational logic with function variables and binding constructs. We prove the resulting theory is complete with respect to polynomial interpretations. The proof makes use of Severi’s theorem that all multivariate Hermite problems are solvable. We also hope to present a number of related results, such as decidability and Hilbert–Post completeness.


Formal Concepts vs Eigenvectors of Density Operators

7 May, 2020

In the seventh talk of the ACT@UCR seminar, Tai-Danae Bradley told us about applications of categorical quantum mechanics to formal concept analysis.

She gave her talk on Wednesday May 13th. Afterwards we discussed her talk at the Category Theory Community Server. You can see those discussions here if you become a member:

You can see her slides here, or download a video here, or watch the video here:

• Tai-Danae Bradley: Formal concepts vs. eigenvectors of density operators.

Abstract. In this talk, I’ll show how any probability distribution on a product of finite sets gives rise to a pair of linear maps called density operators, whose eigenvectors capture “concepts” inherent in the original probability distribution. In some cases, the eigenvectors coincide with a simple construction from lattice theory known as a formal concept. In general, the operators recover marginal probabilities on their diagonals, and the information stored in their eigenvectors is akin to conditional probability. This is useful in an applied setting, where the eigenvectors and eigenvalues can be glued together to reconstruct joint probabilities. This naturally leads to a tensor network model of the original distribution. I’ll explain these ideas from the ground up, starting with an introduction to formal concepts. Time permitting, I’ll also share how the same ideas lead to a simple framework for modeling hierarchy in natural language. As an aside, it’s known that formal concepts arise as an enriched version of a generalization of the Isbell completion of a category. Oftentimes, the construction is motivated by drawing an analogy with elementary linear algebra. I like to think of this talk as an application of the linear algebraic side of that analogy.

Her talk is based on her thesis:

• Tai-Danae Bradley, At the Interface of Algebra and Statistics.