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May 28, 2018

Applied Category Theory: Ordered Sets

Posted by John Baez

My applied category theory course based on Fong and Spivak’s book Seven Sketches is going well. Over 300 people have registered for the course, which allows them to ask question and discuss things. But even if you don’t register you can read my “lectures”.

Posted at 7:39 PM UTC | Permalink | Followups (5)

May 25, 2018

Laxification

Posted by John Baez

Talking to my student Joe Moeller today, I bumped into a little question that seems fun. If I’ve got a monoidal category AA, is there some bigger monoidal category A^\hat{A} such that lax monoidal functors out of AA are the same as strict monoidal functors out of A^\hat{A}?

Someone should know the answer already, but I’ll expound on it a little…

Posted at 9:40 PM UTC | Permalink | Followups (13)

May 24, 2018

Tropical Algebra and Railway Optimization

Posted by John Baez

Simon Willerton pointed out a wonderful workshop, which unfortunately neither he nor I can attend… nor Jamie Vicary, who is usually at Birmingham these days:

If you can go, please do — and report back!

Let me explain why it’s so cool…

Posted at 10:39 PM UTC | Permalink | Followups (5)

May 20, 2018

Postdoc at the Centre of Australian Category Theory

Posted by Emily Riehl

The Centre of Australian Category Theory is advertising for a postdoc. The position is for 2 years and the ad is here.

Applications close on 15 June. Most questions about the position would be best directed to Richard Garner or Steve Lack. You can also find out more about CoACT here.

This is a great opportunity to join a fantastic research group. Please help spread the word to those who might be interested!

Posted at 3:29 PM UTC | Permalink | Post a Comment

May 19, 2018

Circuits, Bond Graphs, and Signal-Flow Diagrams

Posted by John Baez

My student Brandon Coya finished his thesis, and successfully defended it last Tuesday!

• Brandon Coya, Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2018.

It’s about networks in engineering. He uses category theory to study the diagrams engineers like to draw, and functors to understand how these diagrams are interpreted.

His thesis raises some really interesting pure mathematical questions about the category of corelations and a ‘weak bimonoid’ that can be found in this category. Weak bimonoids were invented by Pastro and Street in their study of ‘quantum categories’, a generalization of quantum groups. So, it’s fascinating to see a weak bimonoid that plays an important role in electrical engineering!

However, in what follows I’ll stick to less fancy stuff: I’ll just explain the basic idea of Brandon’s thesis, say a bit about circuits and ‘bond graphs’, and outline his main results. What follows is heavily based on the introduction of his thesis, but I’ve baezified it a little.

Posted at 11:07 PM UTC | Permalink | Followups (4)

Linear Logic for Constructive Mathematics

Posted by Mike Shulman

Intuitionistic logic, i.e. logic without the principle of excluded middle (P¬PP \vee \neg P), is important for many reasons. One is that it arises naturally as the internal logic of toposes and more general categories. Another is that it is the logic traditionally used by constructive mathematicians — mathematicians following Brouwer, Heyting, and Bishop, who want all proofs to have “computational” or “algorithmic” content. Brouwer observed that excluded middle is the primary origin of nonconstructive proofs; thus using intuitionistic logic yields a mathematics in which all proofs are constructive.

However, there are other logics that constructivists might have chosen for this purpose instead of intuitionistic logic. In particular, Girard’s (classical) linear logic was explicitly introduced as a “constructive” logic that nevertheless retains a form of the law of excluded middle. But so far, essentially no constructive mathematicians have seriously considered replacing intuitionistic logic with any such alternative. I will refrain from speculating on why not. Instead, in a paper appearing on the arXiv today:

I argue that in fact, constructive mathematicians (going all the way back to Brouwer) have already been using linear logic without realizing it!

Let me explain what I mean by this and how it comes about — starting with an explanation, for a category theorist, of what linear logic is in the first place.

Posted at 4:53 AM UTC | Permalink | Followups (27)

May 14, 2018

Research Fellowship at the University of Leeds

Posted by Simon Willerton

João Faria Martins and Paul Martin at the University of Leeds are advertising a 2-year research fellowship in geometric topology, topological quantum field theory and applications to quantum computing. This is part of a Leverhulme funded project.

The deadline is Tuesday 29th May. Contact João or Paul with any informal inquiries.

Posted at 10:01 AM UTC | Permalink | Followups (1)

May 6, 2018

Compositionality

Posted by John Baez

A new journal! We’ve been working on it for a long time, but we finished sorting out some details at Applied Category Theory 2018, and now we’re ready to tell the world!

Posted at 5:11 PM UTC | Permalink | Followups (20)

May 5, 2018

The Fisher Metric Will Not Be Deformed

Posted by Tom Leinster

The pillars of society are those who cannot be bribed or bought, the upright citizens of integrity, the incorruptibles. Throw at them what you will, they never bend.

In the mathematical world, the Fisher metric is one such upstanding figure.

What I mean is this. The Fisher metric can be derived from the concept of relative entropy. But relative entropy can be deformed in various ways, and you might imagine that when you deform it, the Fisher metric gets deformed too. Nope. Bastion of integrity that it is, it remains unmoved.

You don’t need to know what the Fisher metric is in order to get the point: the Fisher metric is a highly canonical concept.

Posted at 12:00 AM UTC | Permalink | Followups (23)