This Week’s Finds – Lecture 9

29 November, 2022


In this talk I explained the quaternions and octonions, and showed how to multiply them using the dot product and cross product of vectors.

For more details, including a proof that octonion multiplication obeys |ab|=|a||b|, go here:

Octonions and the Standard Model (Part 2).

This was one of a series of lectures based on my column This Week’s Finds.



This Week’s Finds – Lecture 8

29 November, 2022


In this talk I explained the E8 root lattice and how it gives rise to the ‘octooctonionic projective plane’, a 128-dimensional manifold on which the compact Lie group called E8 acts as symmetries. I also discussed how some special root lattices give various notions of ‘integer’ for the real numbers, complex numbers, quaternions and octonions.

For more, read my paper Coxeter and Dynkin diagrams.

This was one of a series of lectures based on my column This Week’s Finds.

This Week’s Finds – Lecture 7

10 November, 2022


Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. In this talk I went through all the connected Dynkin diagrams and say how they correspond to compact simple Lie group— which happen to be act as symmetries of geometrical structures built using the real numbers, complex numbers, quaternions and octonions!

For more, read my paper Coxeter and Dynkin diagrams.

This was one of a series of lectures based on my column This Week’s Finds.

Seminar on “This Week’s Finds”

11 September, 2022

Here’s something new: I’ll be living in Edinburgh until January! I’m working with Tom Leinster at the University of Edinburgh, supported by a Leverhulme Fellowship.

One fun thing I’ll be doing is running seminars on some topics from my column This Week’s Finds. They’ll take place on Thursdays at 3:00 pm UK time in Room 6206 of the James Clerk Maxwell Building, home of the Department of Mathematics. The first will be on September 22nd, and the last on December 1st.

We’re planning to

1) make the talks hybrid on Zoom so that people can participate online:
Meeting ID: 822 7032 5098
Passcode: XXXXXX36

Here the X’s stand for the name of a famous lemma in category theory.

2) make lecture notes available on my website.

3) record them and eventually make them publicly available on my YouTube channel.

4) have a Zulip channel on the Category Theory Community Server dedicated to discussion of the seminars: it’s here.

More details soon!

The theme for these seminars is representation theory, interpreted broadly. The topics are:

• Young diagrams
• Dynkin diagrams
• q-mathematics
• The three-strand braid group
• Clifford algebras and Bott periodicity
• The threefold and tenfold way
• Exceptional algebras

Seven topics are listed, but there will be 11 seminars, so it’s not a one-to-one correspondence: each topic is likely to take one or two weeks. Here are more detailed descriptions:

Young diagrams

Young diagrams are combinatorial structures that show up in a myriad of applications. Among other things, they classify conjugacy classes in the symmetric groups Sn, irreducible representations of Sn, irreducible representations of the groups SL(n) over any field of characteristic zero, and irreducible unitary representations of the groups SU(n).

Dynkin diagrams

Coxeter and Dynkin diagrams classify a wide variety of structures, most notably Coxeter groups, lattices having such groups as symmetries, and simple Lie algebras. The simply laced Dynkin diagrams also classify the Platonic solids and quivers with finitely many indecomposable representations. This tour of Coxeter and Dynkin diagrams will focus on the connections between these structures.


A surprisingly large portion of mathematics generalizes to something called q-mathematics, involving a parameter q. For example, there is a subject called q-calculus that reduces to ordinary calculus at q = 1. There are important applications of q-mathematics to the theory of quantum groups and also to algebraic geometry over Fq, the finite field with q elements. These seminars will give an overview of q-mathematics and its

The three-strand braid group

The three-strand braid group has striking connections to the trefoil knot, rational tangles, the modular group PSL(2, Z), and modular forms. This group is also the simplest of the Artin–Brieskorn groups, a class of groups which map surjectively to the Coxeter groups. The three-strand braid group will be used as the starting point for a tour of these topics.

Clifford algebras and Bott periodicity

The Clifford algebra Cln is the associative real algebra freely generated by n anticommuting elements that square to -1. Iwill explain their role in geometry and superstring theory, and the origin of Bott periodicity in topology in facts about Clifford algebras.

The threefold and tenfold way

Irreducible real group representations come in three kinds, a fact arising from the three associative normed real division algebras: the real numbers, complex numbers and quaternions. Dyson called this the threefold way. When we generalize to superalgebras this becomes part of a larger classification, the tenfold way. We will examine these topics and their applications to representation theory, geometry and physics.

Exceptional algebras

Besides the three associative normed division algebras over the real numbers, there is a fourth one that is nonassociative: the octonions. They arise naturally from the fact that Spin(8) has three irreducible 8-dimensional representations. We will explain the octonions and sketch how the exceptional Lie algebras and the exceptional Jordan algebra can be constructed using octonions.

Joint Mathematics Meetings 2023

24 August, 2022

This is the biggest annual meeting of mathematicians:

Joint Mathematical Meetings 2023, Wednesday January 4 – Saturday January 7, 2023, John B. Hynes Veterans Memorial Convention Center, Boston Marriott Hotel, and Boston Sheraton Hotel, Boston, Massachusetts.

As part of this huge meeting, the American Mathematical Society is having a special session on Applied Category Theory on Thursday January 5th.

I hear there will be talks by Eugenia Cheng and Olivia Caramello!

You can submit an abstract to give a talk. The deadline is Tuesday, September 13, 2022.

It should be lots of fun. There will also be tons of talks on other subjects.

However, there’s a registration fee which is pretty big unless you’re a student or, even better, a ‘nonmathematician guest’. (I assume you’re not allowed to give a talk if you’re a nonmathematician.)

The special session is called SS 96 and it comes in two parts: one from 8 am to noon, and the other from 1 pm to 5 pm. It’s being run by these participants of this summer’s Mathematical Research Community on applied category theory:

• Charlotte Aten, University of Denver
• Pablo S. Ocal, University of California, Los Angeles
• Layla H. M. Sorkatti, Southern Illinois University
• Abigail Hickok, University of California, Los Angeles

This Mathematical Research Community was run by Daniel Cicala, Simon Cho, Nina Otter, Valeria de Paiva and me, and I think we’re all coming to the special session. At least I am!

Symposium on Compositional Structures 9

9 July, 2022

The Symposium on Compositional Structures is a nice informal conference series that happens more than once a year. You can now submit talks for this one.

Ninth Symposium on Compositional Structures (SYCO 9), Como, Italy, 8-9 September 2022. Deadline to submit a talk: Monday 1 August 2022.

Apparently you can attend online but to give a talk you have to go there. Here are some details:

The Symposium on Compositional Structures (SYCO) is an interdisciplinary series of meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. Previous SYCO events have been held in Birmingham, Strathclyde, Oxford, Chapman, Leicester and Tallinn.

We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering friendly discussion, disseminating new ideas, and spreading knowledge between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students. Submissions is easy, with no formatting or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere. You could submit work-in-progress, or a recently completed paper, or even a PhD or Masters thesis.

While no list of topics could be exhaustive, SYCO welcomes submissions with a compositional focus related to any of the following areas, in particular from the perspective of category theory:

• logical methods in computer science, including classical and quantum programming, type theory, concurrency, natural language processing and machine learning;
• graphical calculi, including string diagrams, Petri nets and reaction networks;
• languages and frameworks, including process algebras, proof nets, type theory and game semantics;
• abstract algebra and pure category theory, including monoidal category theory, higher category theory, operads, polygraphs, and relationships to homotopy theory;
• quantum algebra, including quantum computation and representation theory;
• tools and techniques, including rewriting, formal proofs and proof assistants, and game theory;
• industrial applications, including case studies and real-world problem descriptions.

Important dates

All deadlines are 23:59 Anywhere on Earth.

Submission deadline: Monday 1 August
Author notification: Monday 8 August 2022
Symposium dates: Thursday 8 and Friday 9 September 2022

Submission instructions

Submissions are by EasyChair, via the SYCO 9 submission page:

Submission is easy, with no format requirements or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere. Think creatively: you could submit a recent paper, or notes on work in progress, or even a recent Masters or PhD thesis.

In the event that more good-quality submissions are received than can be accommodated in the timetable, the programme committee may choose to defer some submissions to a future meeting, rather than reject them. Deferred submissions can be re-submitted to any future SYCO meeting, where they will not need peer review, and where they will be prioritised for inclusion in the programme. Meetings will be held sufficiently frequently to avoid a backlog of deferred papers.

If you have a submission which was deferred from a previous SYCO meeting, it will not automatically be considered for SYCO 9; you still need to submit it again through EasyChair. When submitting, append the words “DEFERRED FROM SYCO X” to the title of your paper, replacing “X” with the appropriate meeting number. There is no need to attach any documents.

Programme committee

The PC chair is John van de Wetering, Radboud University. The Programme Committee will be announced soon.

Steering committee

Ross Duncan, University of Strathclyde
Chris Heunen, University of Edinburgh
Dominic Horsman, University of Oxford
Aleks Kissinger, University of Oxford
Samuel Mimram, École Polytechnique
Simona Paoli, University of Aberdeen
Mehrnoosh Sadrzadeh, University College London
Pawel Sobocinski, Tallinn University of Technology
Jamie Vicary, University of Cambridge

Categorical Semantics of Entropy

19 April, 2022

There will be a workshop on the categorical semantics of entropy at the CUNY Grad Center in Manhattan on Friday May 13th, organized by John Terilla. I was kindly invited to give an online tutorial beforehand on May 11, which I will give remotely to save carbon. Tai-Danae Bradley will also be giving a tutorial that day in person:

Tutorial: Categorical Semantics of Entropy, Wednesday 11 May 2022, 13:00–16:30 Eastern Time, Room 5209 at the CUNY Graduate Center and via Zoom. Organized by John Terilla. To attend, register here.

12:00-1:00 Eastern Daylight Time — Lunch in Room 5209.

1:00-2:30 — Shannon entropy from category theory, John Baez, University of California Riverside; Centre for Quantum Technologies (Singapore); Topos Institute.

Shannon entropy is a powerful concept. But what properties single out Shannon entropy as special? Instead of focusing on the entropy of a probability measure on a finite set, it can help to focus on the “information loss”, or change in entropy, associated with a measure-preserving function. Shannon entropy then gives the only concept of information loss that is functorial, convex-linear and continuous. This is joint work with Tom Leinster and Tobias Fritz.

2:30-3:00 — Coffee break.

3:00-4:30 — Operads and entropy, Tai-Danae Bradley, The Master’s University; Sandbox AQ.

This talk will open with a basic introduction to operads and their representations, with the main example being the operad of probabilities. I’ll then give a light sketch of how this framework leads to a small, but interesting, connection between information theory, abstract algebra, and topology, namely a correspondence between Shannon entropy and derivations of the operad of probabilities.

Symposium on Categorical Semantics of Entropy, Friday 13 May 2022, 9:30-3:15 Eastern Daylight Time, Room 5209 at the CUNY Graduate Center and via Zoom. Organized by John Terilla. To attend, register here.

9:30-10:00 Eastern Daylight Time — Coffee and pastries in Room 5209.

10:00-10:45 — Operadic composition of thermodynamic systems, Owen Lynch, Utrecht University.

The maximum entropy principle is a fascinating and productive lens with which to view both thermodynamics and statistical mechanics. In this talk, we present a categorification of the maximum entropy principle, using convex spaces and operads. Along the way, we will discuss a variety of examples of the maximum entropy principle and show how each application can be captured using our framework. This approach shines a new light on old constructions. For instance, we will show how we can derive the canonical ensemble by attaching a probabilistic system to a heat bath. Finally, our approach to this categorification has applications beyond the maximum entropy principle, and we will give an hint of how to adapt this categorification to the formalization of the composition of other systems.

11:00-11:45 — Polynomial functors and Shannon entropy, David Spivak, MIT and the Topos Institute.

The category Poly of polynomial functors in one variable is extremely rich, brimming with categorical gadgets (e.g. eight monoidal products, two closures, limits, colimits, etc.) and applications including dynamical systems, databases, open games, and cellular automata. In this talk I’ll show that objects in Poly can be understood as empirical distributions. In part using the standard derivative of polynomials, we obtain a functor to Set × Setop which encodes an invariant of a distribution as a pair of sets. This invariant is well-behaved in the sense that it is a distributive monoidal functor: it acts on both distributions and maps between them, and it preserves both the sum and the tensor product of distributions. The Shannon entropy of the original distribution is then calculated directly from the invariant, i.e. only in terms of the cardinalities of these two sets. Given the many applications of polynomial functors and of Shannon entropy, having this link between them has potential to create useful synergies, e.g. to notions of entropic causality or entropic learning in dynamical systems.

12:00-1:30 — Lunch in Room 5209

1:30-2:15 — Higher entropy, Tom Mainiero, Rutgers New High Energy Theory Center.

Is the frowzy state of your desk no longer as thrilling as it once was? Are numerical measures of information no longer able to satisfy your needs? There is a cure! In this talk we’ll learn about: the secret topological lives of multipartite measures and quantum states; how a homological probe of this geometry reveals correlated random variables; the sly decategorified involvement of Shannon, Tsallis, Réyni, and von Neumann in this larger geometric conspiracy; and the story of how Gelfand, Neumark, and Segal’s construction of von Neumann algebra representations can help us uncover this informatic ruse. So come to this talk, spice up your entropic life, and bring new meaning to your relationship with disarray.

2:30-3:15 — On characterizing classical and quantum entropy, Arthur Parzygnat, Institut des Hautes Études Scientifiques.

In 2011, Baez, Fritz, and Leinster proved that the Shannon entropy can be characterized as a functor by a few simple postulates. In 2014, Baez and Fritz extended this theorem to provide a Bayesian characterization of the classical relative entropy, also known as the Kullback–Leibler divergence. In 2017, Gagné and Panangaden extended the latter result to include standard Borel spaces. In 2020, I generalized the first result on Shannon entropy so that it includes the von Neumann (quantum) entropy. In 2021, I provided partial results indicating that the Umegaki relative entropy may also have a Bayesian characterization. My results in the quantum setting are special applications of the recent theory of quantum Bayesian inference, which is a non-commutative extension of classical Bayesian statistics based on category theory. In this talk, I will give an overview of these developments and their possible applications in quantum information theory.

Wine and cheese reception to follow, Room 5209.

Applied Category Theory 2022

25 February, 2022

The Fifth International Conference on Applied Category Theory, ACT2022, will take place at the University of Strathclyde from 18 to 22 July 2022, preceded by the Adjoint School 2022 from 11 to 15 July. This conference follows previous events at Cambridge (UK), Cambridge (MA), Oxford and Leiden.

Applied category theory is important to a growing community of researchers who study computer science, logic, engineering, physics, biology, chemistry, social sciences, linguistics and other subjects using category-theoretic tools. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, strengthen the applied category theory community, disseminate the latest results, and facilitate further development of the field.


We accept submissions in English of original research papers, talks about work accepted/submitted/published elsewhere, and demonstrations of relevant software. Accepted original research papers will be published in a proceedings volume. The keynote addresses will be chosen from the accepted papers. The conference will include an industry showcase event and community meeting. We particularly encourage people from underrepresented groups to submit their work and the organizers are committed to non-discrimination, equity, and inclusion.

Submission formats

Extended Abstracts should be submitted describing the contribution and providing a basis for determining the topics and quality of the anticipated presentation (1-2 pages). These submissions will be adjudicated for inclusion as a talk at the conference. Such work should include references to any longer papers, preprints, or manuscripts providing additional details.

Conference Papers should present original, high-quality work in the style of a computer science conference paper (up to 14 pages, not counting the bibliography; detailed proofs may be included in an appendix for the convenience of the reviewers). Such submissions should not be an abridged version of an existing journal article (see item 1) although pre-submission Arxiv preprints are permitted. These submissions will be adjudicated for both a talk and publication in the conference proceedings.

Software Demonstrations should be submitted in the format of an Extended Abstract (1-2 pages) giving the program committee enough information to assess the content of the demonstration. We are particularly interested in software that makes category theory research easier, or uses category theoretic ideas to improve software in other domains.

Extended abstracts and conference papers should be prepared with LaTeX. For conference papers please use the EPTCS style files available at

The submission link is

Important dates

The following dates are all in 2022, and Anywhere On Earth.

• Submission Deadline: Monday 9 May
• Author Notification: Tuesday 7 June
• Camera-ready version due: Tuesday 28 June
• Adjoint School: Monday 11 to Friday 15 July
• Main Conference: Monday 18 to Friday 22 July

Conference format

We hope to run the conference as a hybrid event with talks recorded or streamed for remote participation. However, due to the state of the pandemic, the possibility of in-person attendance is not yet confirmed. Please be mindful of changing conditions when booking travel or hotel accommodations.

Financial support

Limited financial support will be available. Please contact the organisers for more information.

Program committee

• Jade Master, University of Strathclyde (Co-chair)
• Martha Lewis, University of Bristol (Co-chair)

The full program committee will be announced soon.

Organizing committee

• Jules Hedges, University of Strathclyde
• Jade Master, University of Strathclyde
• Fredrik Nordvall Forsberg, University of Strathclyde
• James Fairbanks, University of Florida

Steering committee

• John Baez, University of California, Riverside
• Bob Coecke, Cambridge Quantum
• Dorette Pronk, Dalhousie University
• David Spivak, Topos Institute

Categories in Chemistry, Computing, and Social Networks

26 January, 2022

This article does two things:

• John Baez, Simon Cho, Daniel Cicala, Nina Otter, and Valeria de Paiva, Applied category theory in chemistry, computing, and social networks, Notices of the American Mathematical Society, February 2022.

It urges you — or your friends, or students — to apply for our free summer school in applied category theory run by the American Mathematical Society. It’s also a quick intro to some key ideas in applied category theory!

Applications are due Tuesday 2022 February 15 at 11:59 Eastern Time — go here for details. If you get in, you’ll get an all-expenses-paid trip to a conference center in upstate New York for a week in the summer. There will be a pool, bocci, lakes with canoes, woods to hike around in, campfires at night… and also whiteboards, meeting rooms, and coffee available 24 hours a day.

You can work with me on categories in chemistry, Nina on categories in the study of social networks, or Valeria on categories applied to concepts from computer science, like lenses.

There are also other programs to choose from. Read this, and click for more details:

Adjoint School 2022

2 January, 2022

Every year since 2018 we’ve been having annual courses on applied category theory where you can do research with experts. It’s called the Adjoint School.

You can apply to be a student at the 2022 Adjoint School now, and applications are due February 4th! Go here:

2022 Adjoint School: application.

The school will be run online from February to June, 2022, and then—coronavirus permitting—there will be in-person research at the University of Strathclyde in Glasgow, Scotland the week of July 11 – 15, 2022. This is also the location of the applied category theory conference ACT2022.

The 2022 Adjoint School is organized by Angeline Aguinaldo, Elena Di Lavore, Sophie Libkind, and David Jaz Myers. You can read more about how it works here:

About the Adjoint School.

There are four topics to work on, and you can see descriptions of them below.

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.

Also check out our inclusivity statement.

Topic 1: Compositional Thermodynamics

Mentors: Spencer Breiner and Joe Moeller

TA: Owen Lynch

Description: Thermodynamics is the study of the relationships between heat, energy, work, and matter. In category theory, we model flows in physical systems using string diagrams, allowing us to formalize physical axioms as diagrammatic equations. The goal of this project is to establish such a compositional framework for thermodynamical networks. A first goal will be to formalize the laws of thermodynamics in categorical terms. Depending on the background and interest of the participants, further topics may include the Carnot and Otto engines, more realistic modeling for real-world systems, and software implementation within the AlgebraicJulia library.


• John C. Baez, Owen Lynch, and Joe Moeller, Compositional thermostatics.

• F. William Lawvere, State categories, closed categories and the existence of semi-continuous entropy functions.

Topic 2: Fuzzy Type Theory for Opinion Dynamics

Mentor: Paige North

TA: Hans Reiss

Description: When working in type theory (or most logics), one is interested in proving propositions by constructing witnesses to their incontrovertible truth. In the real world, however, we can often only hope to understand how likely something is to be true, and we look for evidence that something is true. For example, when a doctor is trying to determine if a patient has a certain condition, they might ask certain questions and perform certain tests, each of which constitutes a piece of evidence that the patient does or does not have that condition. This suggests that a fuzzy version of type theory might be appropriate for capturing and analyzing real-world situations. In this project, we will explore the space of fuzzy type theories which can be used to reason about the fuzzy propositions of disease and similar dynamics.


• Daniel R. Grayson, An introduction to univalent foundations for mathematicians.

• Jakob Hansen and Robert Ghrist, Opinion dynamics on discourse sheaves.

Topic 3: A Compositional Theory of Timed and Probabilistic Processes: CospanSpan(Graph)

Mentor: Nicoletta Sabadini

TA: Mario Román

Description: Span(Graph), introduced by Katis, Sabadini and Walters as a categorical algebra for automata with interfaces, provides, in a very intuitive way, a compositional description of hierarchical networks of interacting components with fixed topology. The algebra also provides a calculus of connectors, with an elegant description of signal broadcasting. In particular, the operations of “parallel with communication” (that allows components to evolve simultaneously, like connected gears), and “non-sequential feedback” (not considered in Kleene’s algebra for classical automata) are fundamental in modelling complex distributed systems such as biological systems. Similarly, the dual algebra Cospan(Graph) allows us to compose systems sequentially. Hence, the combined algebra CospanSpan(Graph), which extends Kleene’s algebra for classical automata, is a general algebra for reconfigurable networks of interacting components. Still, some very interesting aspects and possible applications of this model deserve a better understanding:

• How can timed actions and probability be combined in CospanSpan(Graph)?

• If not, can we describe time-varying probability in a compositional setting?

• Which is the possible role of “parallel with communication” in understanding causality?


• L. de Francesco Albasini, N. Sabadini, and R.F.C. Walters, The compositional construction of Markov processes II.

• A. Cherubini, N. Sabadini, and R.F.C. Walters, Timing in the Cospan-Span model.

Topic 4: Algebraic Structures in Logic and Relations

Mentor: Filippo Bonchi

Description: Fox’s theorem provides a bridge between structures defined by universal properties (products in a category) and structures specified by algebraic means (comonoids in a symmetric monoidal category). Such a theorem has recently received a renewed interest as the algebraic structures allows for reasoning in terms of string diagrams. While the universal properties underlying logical theories have been extensively studied in categorical logic, their algebraic counterparts have been the objects of fewer investigations. This raises a natural question: can we capture the universal content of logical theories algebraically? In other words, what are the ‘Fox theorems’ for logic? In this project, we attempt to answer to this question by taking as starting point Cartesian bicategories which serves as algebraic setting for regular logic.


• Aurelio Carboni and R. F. C. Walters, Cartesian bicategories I.

• Filippo Bonchi, Jens Seeber and Pawel Sobocinski, Graphical conjunctive queries.

• Filippo Bonchi, Dusko Pavlovic and Pawel Sobocinski, Functorial semantics for relational theories.