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March 31, 2014

Operads and Trees

Posted by John Baez

Nina Otter is a master’s student in mathematics at ETH Zürich who has just gotten into the PhD program at Oxford. She and I are writing a paper on operads and the tree of life.

Anyone who knows about operads knows that they’re related to trees. But I’m hoping someone has proved some precise theorems about this relationship, so that we don’t have to.

Posted at 9:02 AM UTC | Permalink | Followups (15)

March 30, 2014

Fourier Series and Flipped Classrooms

Posted by Tom Leinster

Term is nearly over, which for me means the end of the 4th year Fourier Analysis course I’ve been teaching for the last couple of years.

I was fortunate enough to take over the course from Jim Wright, a genuine expert on the subject, and I inherited a great set of notes from him. But I felt the need to make the course my own, so I’ve been writing my own notes, which I’ve just finished: notes here, plus accompanying problem sheets. They’re mostly about convergence of Fourier series, with a delicious dessert of Fourier analysis on finite abelian groups.

But what I wanted to write about here — and get your opinions on — was not Fourier analysis, but some questions of teaching. This year, I’ve been (in the jargon) “flipping the classroom”, or at least partially flipping it (which reminds me of that mysterious substance, partially inverted sugar syrup, that you sometimes see on ingredients lists). I’d like to hear about other people’s similar experiences.

Posted at 1:32 PM UTC | Permalink | Followups (16)

March 24, 2014

An Exegesis of Yoneda Structures

Posted by Emily Riehl

Guest post by Alexander Campbell

We want to develop category theory in a general 2-category, in order to both generalise and clarify our understanding of category theory. The key to this endeavour is to express the basic notions of the theory of categories in a natural 2-categorical language. In this way we are continuing a theme present in previous posts from the Kan Extension Seminar, wherein monads and adjunctions were given a 2-categorical setting, and by analogy, in our very first paper, whose purpose was to express basic notions of the theory of sets in a natural categorical language. In this post we consider a concept very central and special to category theory: the Yoneda lemma.

So what’s the Yoneda Lemma again?

The Yoneda lemma says that for any object aa of a category AA, the diagram 1 a A * ι A(a,) Set \begin{matrix} 1 & \overset{a}{\rightarrow} & A \\ {}_{\ast} \searrow & \overset{\iota}{\Rightarrow} & \swarrow_{A(a,-)} \\ & Set \\ \end{matrix} is a left extension.

In this post I will give a motivation for the notion of Yoneda structure, as defined in the paper Yoneda Structures on 2-Categories of Ross Street and Bob Walters. But before we begin I would like to take this opportunity to thank Emily for inviting me to join the Kan Extension Seminar and for her support and encouragement throughout the course. This has been and continues to be a singularly valuable experience in my first year as a category theorist.

Posted at 5:31 AM UTC | Permalink | Followups (43)

March 18, 2014

Translating Grothendieck’s Biography into English

Posted by John Baez

Leila Schneps is trying to raise $6,000 for what sounds like a good cause: translating a biography of Grothendieck into English:

As of this moment she’s raised $350… including $100 of her own money.

Posted at 4:02 PM UTC | Permalink | Followups (4)

March 15, 2014

Fuzzy Logic and Enriching Over the Category [0,1]

Posted by Simon Willerton

Standard logic involving the truth values ‘true’ and ‘false’ can make it difficult to model some of the fuzziness we use in everyday speech. If you’d bought a bike yesterday then today it would be truthful to say “This bike is new”, but it wouldn’t be truthful so say it in 20 years’ time. However, between now and then there won’t be a specific day on which the statement “This bike is new” suddenly switches from being true to being false. How can you model this situation?

An old bike

One approach to modelling this situation is with fuzzy logic where you allow your truth values to be things other than just true and false. For instance, you can take the interval [0,1][0,1] as the set of truth values with 00 representing false and 11 representing true. So the truth degree of the statement “This bike is new” would vary, being 11 today and decreasing to something very close to 00 in 20 years’ time.

This post is an attempt by me to understand this fuzzy logic in the context of enriched category theory, in particular, using [0,1][0,1] as a monoidal category to enrich over. We will see that categories enriched over [0,1][0,1] can be interpreted as fuzzy posets or fuzzy preorders.

This was going to be a comment on Tom Avery’s Kan Extension Seminar post on Metric Spaces, Generalized Logic, and Closed Categories but grew too big!

Posted at 5:51 PM UTC | Permalink | Followups (36)

March 9, 2014

Review of the Elements of 2-Categories

Posted by Emily Riehl

Guest post by Dimitri Zaganidis

First of all, I would like to thank Emily for organizing the Kan extension seminar. It is a pleasure to be part of it. I want also to thank my advisor Kathryn Hess and my office mate Martina Rovelli for their revisions.

In the fifth installment of the Kan Extension Seminar we read the paper “Review of the Elements of 2-categories” by G.M Kelly and Ross Street. This article was published in the Proceedings of the Sydney Category Theory Seminar, and its purpose is to “serve as a common introduction to the authors’ paper in this volume”.

The article has three main parts, the first of them being definitions in elementary terms of double categories and 2-categories, together with the notion of pasting. In a second chapter, they review adjunctions in 2-categories with a nice expression of the naturality of the bijection given by mates using double categories. The last part of the article introduces monads in 2-categories, and specializing to 2-monads towards the end.

Posted at 7:50 AM UTC | Permalink | Followups (39)

March 5, 2014

Operads of Finite Groups

Posted by Tom Leinster

Guest post by Nick Gurski

I have been thinking about various sorts of operads with my PhD student Alex Corner, and have become interested in the following very concrete question: what are examples of operads in the category of finite groups under the cartesian product? I don’t know any really interesting examples, but maybe you do! After the break I will explain why I got interested in this question, and tell you about some examples that I do know.

Posted at 4:48 PM UTC | Permalink | Followups (5)

March 2, 2014

Should Mathematicians Cooperate with GCHQ?

Posted by Tom Leinster

I’ve just submitted a piece for the new Opinions section of the monthly LMS Newsletter: Should mathematicians cooperate with GCHQ? (Update: now available (p.34).) The LMS is the London Mathematical Society, which is the UK’s national mathematical society. My piece should appear in the April edition of the newsletter, and you can read it below.

Here’s the story. Since November, I’ve been corresponding with people at the LMS, trying to find out what connections there are between it and GCHQ. Getting the answer took nearly three months and a fair bit of pushing. In the process, I made some criticisms of the LMS’s total silence over the GCHQ/NSA scandal:

GCHQ is a major employer of mathematicians in the UK. The NSA is said to be the largest employer of mathematicians in the world. If there had been a major scandal at the heart of the largest publishing houses in the world, unfolding constantly over the last eight months, wouldn’t you expect it to feature prominently in every issue of the Society of Publishers’ newsletter?

To its credit, the LMS responded by inviting me to write an inaugural piece for a new Opinions section of the newsletter. Here it is.

Posted at 7:09 PM UTC | Permalink | Followups (46)