Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

February 23, 2018

Applied Category Theory at NIST

Posted by John Baez

I think it’s really cool how applied category theory is catching on. My former student Blake Pollard is working at the National Institute of Standards and Technology on applications of category theory to electrical engineering. He’s working with Spencer Breiner… and now Breiner is running a workshop on this stuff:

• Applied Category Theory: Bridging Theory & Practice, March 15–16, 2018, NIST, Gaithersburg, Maryland, USA.

Posted at 4:48 PM UTC | Permalink | Post a Comment

Higher Algebra and Mathematical Physics

Posted by John Baez

You all know about Homotopy Type Theory Electronic Seminar Talks. Here’s another way to cut carbon emissions: a double conference. The idea here is to have a conference in two faraway locations connected by live video stream, to reduce the amount of long-distance travel!

Even better, it’s about a great subject:

  • Higher algebra and mathematical physics, August 13–17, 2018, Perimeter Institute, Waterloo, Canada, and Max Planck Institute for Mathematics, Bonn, Germany. Organized by David Ayala, Lukas Brantner, Kevin Costello, Owen Gwilliam, Andre Henriques, Theo Johnson-Freyd, Aaron Mazel-Gee, and Peter Teichner.

Higher algebra, lower carbon emissions… what more could you want?

Posted at 6:43 AM UTC | Permalink | Followups (1)

February 19, 2018

Cartesian Bicategories

Posted by John Baez

guest post by Daniel Cicala and Jules Hedges

We continue the Applied Category Theory Seminar with a discussion of Carboni and Walters’ paper Cartesian Bicategories I. The star of this paper is the notion of ‘bicategories of relations’. This is an abstraction of relations internal to a category. As such, this paper provides excellent, if technical, examples of internal relations and other internal category theory concepts. In this post, we discuss bicategories of relations while occasionally pausing to enjoy some internal category theory such as relations, adjoints, monads, and the Kleisli construction.

We’d like to thank Brendan Fong and Nina Otter for running such a great seminar. We’d also like to thank Paweł Sobociński and John Baez for helpful discussions.

Posted at 7:50 PM UTC | Permalink | Followups (8)

February 15, 2018

Physics and 2-Groups

Posted by David Corfield

The founding vision of this blog, higher gauge theory, progresses. Three physicists have just brought out Exploring 2-Group Global Symmetries . Two of the authors have worked with Nathan Seiberg, a highly influential physicist, who, along with three others, had proposed in 2014 a program to study higher form field symmetries in QFT (Generalized Global Symmetries), without apparently being aware that this idea was considered before.

Eric Sharpe then published an article the following year, Notes on generalized global symmetries in QFT, explaining how much work had already been done along these lines:

The recent paper [1] proposed a more general class of symmetries that should be studied in quantum field theories: in addition to actions of ordinary groups, it proposed that we should also consider ‘groups’ of gauge fields and higher-form analogues. For example, Wilson lines can act as charged objects under such symmetries. By using various defects, the paper [1] described new characterizations of gauge theory phases.

Now, one can ask to what extent it is natural for n-forms as above to form a group. In particular, because of gauge symmetries, the group multiplication will not be associative in general, unless perhaps one restricts to suitable equivalence classes, which does not seem natural in general. A more natural understanding of such symmetries is in terms of weaker structures known as 2-groups and higher groups, in which associativity is weakened to hold only up to isomorphisms.

There are more 2-groups and higher groups than merely, ‘groups’ of gauge fields and higher-form tensor potentials (connections on bundles and gerbes), and in this paper we will give examples of actions of such more general higher groups in quantum field theory and string theory. We will also propose an understanding of certain anomalies as transmutations of symmetry groups of classical theories into higher group actions on quantum theories.

Posted at 9:18 AM UTC | Permalink | Followups (19)

February 14, 2018

Gradual Typing

Posted by Mike Shulman

(Guest post by Max New)

Dan Licata and I have just put up a paper on the arxiv with a syntax and semantics for a gradually typed programming language, which is a kind of synthesis of statically typed and dynamically typed programming styles. The central insight of the paper is to show that the dynamic type checking used in gradual typing has the structure of a proarrow equipment. Using this we can show that some traditional definitions of dynamic type checks can be proven to be in fact unique solutions to the specifications provided by the structure of an equipment. It’s a classic application of category theory: finding a universal property to better understand what was previously an ad-hoc construction.

The paper is written for computer scientists, so I’ll try to provide a more category-theorist-accessible intro here.

Posted at 11:56 PM UTC | Permalink | Followups (26)

February 8, 2018

Homotopy Type Theory Electronic Seminar

Posted by John Baez

What a great idea! A seminar on homotopy type theory, with talks by top experts, available to everyone with internet connection!

Posted at 5:11 AM UTC | Permalink | Followups (18)

February 6, 2018

Linguistics Using Category Theory

Posted by John Baez

guest post by Cory Griffith and Jade Master

Most recently, the Applied Category Theory Seminar took a step into linguistics by discussing the 2010 paper Mathematical Foundations for a Compositional Distributional Model of Meaning, by Bob Coecke, Mehrnoosh Sadrzadeh, and Stephen Clark.

Here is a summary and discussion of that paper.

Posted at 7:18 PM UTC | Permalink | Followups (38)

February 5, 2018


Posted by John Baez

Since nothing get parodied until it’s sufficiently well-known to make it worth the effort, this proves the nnLab is a success:


Click on the links!

Posted at 6:23 AM UTC | Permalink | Followups (16)

February 2, 2018

A Problem on Pushouts and Pullbacks

Posted by John Baez

I have a problem involving pullbacks and pushouts. This problem arose in work with Kenny Courser on an application of category theory. But you don’t need to understand anything about that application to understand — and I hope solve! —our problem.

If you can solve it, we will credit you in our paper.

Posted at 1:15 AM UTC | Permalink | Followups (29)

January 29, 2018

The Stable Homotopy Hypothesis and Categorified Abelian Groups

Posted by Tom Leinster

guest post by Nick Gurski

In December, Niles Johnson, Angélica Osorno, and I posted a proof of the stable homotopy hypothesis in dimension two on the arxiv. This is the fourth paper we have written together on this project (one also joint with Marc Stephan), and since proving the SHH in dimension two was the first step in a long line of research which you might call “an introduction to twice-categorified abelian groups,” I thought now was a good time to try to explain what we have been up to and why.

Pears by Edouard Manet

Posted at 9:37 PM UTC | Permalink | Followups (11)

January 25, 2018

Announcing the 2018 Talbot Workshop: Model-Independent Theory of Infinity-Categories

Posted by Emily Riehl

The Talbot Workshop is a 1-week learning workshop for roughly 35 graduate students and a few postdocs. Most of the talks will be given by participants, and will be expository in nature. This year’s workshop will be held from May 28 - June 2, 2018 at Government Camp, Oregon USA and Dominic Verity and I will be the mentors. The topic is on our joint work developing a theory of \infty-categories from first principles in a model-independent fashion, that is, using a common axiomatic framework that is satisfied by a variety of models. The goal is to demonstrate that theorems proven using the combinatorics of a particular model transfer across specified “change of model” functors. More details about the program, including a preliminary list of talks and references, can be found here.

Applications are now open, and close on Wednesday, February 28 at 11:59PM. You can apply online here.

Talbot is meant to encourage collaboration among young researchers, with an emphasis on graduate students. We also aim to gather participants with a diverse array of knowledge and interests, so applicants need not be an expert in the field– in particular, students at all levels of graduate education are encouraged to apply. As we are committed to promoting diversity in mathematics, we also especially encourage women and minorities to apply.

The Talbot Workshop grant will cover all local expenses including lodging and food and also offer partial funding for participants’ travel costs.

I’m happy to answer questions about the scientific content of this workshop on this blog - please comment below. If you have logistical questions, please do not hesitate to e-mail the organizers at talbotworkshop (at)

On behalf of the organizers: Eva Belmont, Calista Bernard, Inbar Klang, Morgan Opie, Sean Pohorence

Posted at 11:31 PM UTC | Permalink | Post a Comment

More Secrets of the Associahedra

Posted by John Baez

The associahedra are wonderful things discovered by Jim Stasheff around 1963 but even earlier by Dov Tamari in his thesis. They hold the keys to understanding ‘associativity up to coherent homotopy’ in exquisite combinatorial detail.

But do they still hold more secrets? I think so!

Posted at 7:05 PM UTC | Permalink | Followups (35)

January 23, 2018

Statebox: A Universal Language of Distributed Systems

Posted by John Baez

We’re getting a lot of great posts here this week, but I also want to point out this, by one grad students:

A brief teaser follows, in case you’re wondering what this is about.

Posted at 7:55 PM UTC | Permalink | Post a Comment


Posted by Tom Leinster

guest post by Chris Heunen

I don’t have to tell you, dear nn-Category Café reader, that string diagrams are extremely useful. They speed up computations massively, reveal what makes a proof tick without an impenetrable forest of details, and suggest properties that you might not even have thought about in algebraic notation. They also make your paper friendlier to read.

However, string diagrams are also a pain to typeset. First, there is the entrance fee of learning a modern LaTeX drawing package, like TikZ. Then, there is the learning period of setting up macros tailored to string diagrams and internalizing them in your muscle memory. But even after all that investment, it still takes a lot of time. And unlike that glorious moment when you realise that you have cycled about twice the circumference of the Earth in your life, this time is mostly wasted. I estimate I’ve wasted over 2000 string diagrams’ worth of time by now.

Wouldn’t it be great if you could simply draw your diagram by hand, and have it magically converted into TikZ? Now you can!

Posted at 2:01 PM UTC | Permalink | Followups (30)

January 22, 2018

A Categorical Semantics for Causal Structure

Posted by John Baez

guest post by Joseph Moeller and Dmitry Vagner

We begin the Applied Category Theory Seminar by discussing the paper A categorical semantics for causal structure by Aleks Kissinger and Sander Uijlen.

Compact closed categories have been used in categorical quantum mechanics to give a structure for talking about quantum processes. However, they prove insufficient to handle higher order processes, in other words, processes of processes. This paper offers a construction for a *\ast-autonomous extension of a given compact closed category which allows one to reason about higher order processes in a non-trivial way.

We would like to thank Brendan Fong, Nina Otter, Joseph Hirsh and Tobias Heindel as well as the other participants for the discussions and feedback.

Posted at 9:26 PM UTC | Permalink | Followups (35)