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February 28, 2018

Hypergraph Categories of Cospans

Posted by John Baez

guest post by Jonathan Lorand and Fabrizio Genovese

In the Applied Category Theory Seminar, we most recently read Brendan Fong’s article on decorated cospans. This construction is part of a larger framework, developed in Brendan Fong’s PhD thesis, for studying interconnected, open, network-style systems. A paradigmatic example: systems composed of electric circuits having input and output terminals, allowing for composition of smaller circuits into larger. An aim of Brendan’s framework is to give, for any such kind of system, a unified categorical way to describe both the formal, symbolic language of such systems (their “syntax”) as well as the behavior of the systems that these formal symbols represent (the “semantics”). For circuits: syntax is formal rules for combining circuit diagram nomenclature; semantics is a (mathematical) description of how real-life circuits behave in the presence of voltages, currents, etc.. Decorated cospans are a tool ideal for “syntax”; decorated corelations are designed to handle “semantics” and are flexible enough to model any so-called hypergraph category. We’ll focus on the former, and hint at the latter.

Posted at 4:19 PM UTC | Permalink | Followups (4)

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 | Followups (3)

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 (18)

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 (21)

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 (33)

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

mLab

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:

mmLab.

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)