## March 5, 2014

#### 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? 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 (20)

## February 21, 2014

### Metric Spaces, Generalized Logic, and Closed Categories

#### Posted by Emily Riehl

Guest post by Tom Avery

Before getting started, I’d like to thank Emily for organizing the seminar, as well as all the other participants. It’s been a lot of fun so far! I’d also like to thank my supervisor Tom Leinster for some very helpful suggestions when writing this post.

In the fourth instalment of the Kan Extension Seminar we’re looking at Lawvere’s paper “Metric spaces, generalized logic, and closed categories”. This is the paper that introduced the surprising description of metric spaces as categories enriched over a certain monoidal category $\mathbb{R}$. A lot of people find this very striking when they first see it, and it helps to drive home the point that enriched categories are not just ordinary categories with some extra structure on the hom-sets; in fact the hom-sets don’t have to be sets at all!

Lawvere also intended the paper to serve as an accessible introduction to enriched category theory, so it begins fairly gently with some basic definitions. For the purposes of this post however, I’ll assume the reader has at least seen the definitions of symmetric monoidal closed categories, $\mathcal{V}$-categories, and $\mathcal{V}$-functors. If not, everything you need can be found on the nlab.

Posted at 12:47 AM UTC | Permalink | Followups (42)

## February 13, 2014

### Relative Entropy

#### Posted by John Baez

You may recall how Tom Leinster, Tobias Fritz and I cooked up a neat category-theoretic characterization of entropy in a long conversation here on this blog. Now Tobias and I have a sequel giving a category-theoretic characterization of relative entropy. But since some people might be put off by the phrase ‘category-theoretic characterization’, it’s called:

• Relative Entropy (Part 1): how various structures important in probability theory arise naturally when you do linear algebra using only the nonnegative real numbers.
• Relative Entropy (Part 2): a category related to statistical inference, $\mathrm{FinStat},$ and how relative entropy defines a functor on this category.
• Relative Entropy (Part 3): statement of our main theorem, which characterizes relative entropy up to a constant multiple as the only functor $F : \mathrm{FinStat} \to [0,\infty)$ with a few nice properties.

But now the paper is actually done! Let me give a compressed version of the whole story here… with sophisticated digressions buried in some parenthetical remarks that you’re free to skip if you want.

Posted at 3:48 PM UTC | Permalink | Followups (40)

## February 12, 2014

### Call for Papers: Symposium on the Foundations of Mathematics

#### Posted by David Corfield

A friend, John Wigglesworth, has asked me to announce the following conference. Perhaps he’ll let us know if pluralist homotopy type theorists are welcome ;).

I wonder if any progress has been made on my MO ‘multiverse’ question.

## February 11, 2014

### The Deteriorating Relationship Between Academics and the NSA

#### Posted by Tom Leinster

Stefan Forcey just alerted me to a long and reflective piece (edit: now paywalled) in the Chronicle of Higher Education about the relationship between the NSA and American academics — mathematicians in particular.

Now many academics are trying to be heard from the outside, arguing that the NSA’s spying tactics are proving counterproductive and that university researchers have a duty to stop assisting them. […]

In the months since Edward J. Snowden fled the United States with tens of thousands of electronic documents describing NSA practices, mathematicians are realizing that they are in the same position as nuclear physicists in the middle of the last century, and business students in more recent times — suddenly needing to figure out the ethics behind what they do, said Edward Frenkel, a professor of mathematics at the University of California at Berkeley.

Posted at 12:04 AM UTC | Permalink | Followups (21)

## February 10, 2014

### Network Theory Talks at Oxford

#### Posted by John Baez

One of my dreams these days is to get people to apply modern math to ecology and biology, to help us design technologies that work with nature instead of against it. I call this dream ‘green mathematics’. But this will take some time to reach, since living systems are subtle, and most mathematicians are more familiar with physics.

So, I’ve been warming up by studying the mathematics of chemistry, evolutionary game theory, electrical engineering, control theory and information theory. There are a lot of ideas in common to all these fields, but making them clear requires some category theory. I call this project ‘network theory’. I’m giving some talks about it at Oxford.

(This diagram is written in Systems Biology Graphical Notation.)

Posted at 10:34 AM UTC | Permalink | Followups (9)

## February 5, 2014

### Galois Correspondences and Enriched Adjunctions

#### Posted by Simon Willerton

This is the fourth post in a series on categorical ideas related to formal concept analysis (the other posts are linked below). I want to bring together the ideas of the previous posts and tell you what relations and Galois correspdonences have to do with profunctors and nuclei. To do that I have to tell you what happens when you think of posets as categories enriched over the category of truth values.

Here is a picture of a poset, with $x\le y$ if you can climb up the arrows from $x$ to $y$.

The picture can also be interpreted as representing a thin category, that is a category with at most one morphism between each pair of objects.

Most people who know what a category is know that a thin category is the same thing as a poset; with an arrow $x\to y$ corresponding to the relation $x\le y$. However, despite being well-known, this fact is also not strictly true. You have to be wary of the antisymmetry axiom.

In a poset the antisymmetry axiom asserts that if $a\le b$ and $b\le a$ then $a=b$, but this translates in a thin category to having morphisms $a\to b$ and $b\to a$ implying that $a=b$ and that is not true in general, all you can say is that $a$ is isomorphic to $b$.

A poset-without-antisymmetry is called a preorder, and the true fact is that thin categories correspond to preorders (and posets correspond to skeletal, thin categories).

Less well-known than the above not-entirely-true fact, at least according to my unscientific straw poll of mathematicians, is that a category enriched over truth values is the same thing as a preorder. Although this seems, superficially, to be the same as the previous fact, it does have some deeper connotations due to the depths of enriched category theory. It is some of these depths that I want to start to explore here.

Posted at 11:04 PM UTC | Permalink | Followups (14)

### Categories of Continuous Functors

#### Posted by Emily Riehl

Guest post by Fosco Loregian

The aim of this note is to give a short account of

Freyd, P. J., & Kelly, G. M. (1972). Categories of continuous functors, I. JPAA, 2(3), 169-191.

as part of the Kan Extension Seminar series of lectures. I warmly thank all the participants and the organizer Emily Riehl for giving me this way of escape to the woeful solitude a “baby” category theorist (like I am preparing to become) suffers here in Italy. It is an amazing and overwhelming experience, I can’t even estimate the amount of things I already learned after these three lectures. There are two other people without whom this wouldn’t have been possible: my current advisor, D. Fiorenza, who patiently helped me to polish the exposition you are about to read, and my friend Paolo, for which the words “I don’t want to learn this” are meaningless.

That said, let’s begin with the real discussion.

Freyd and Kelly’s paper was the first to raise and solve in a very elegant way some fundamental questions in elementary Category Theory, the so-called Orthogonal subcategory problem, and Continuous functor problem.

Posted at 3:11 PM UTC | Permalink | Followups (25)

## February 2, 2014

### An Emerging Pattern in Algebra and Topology II

#### Posted by Emily Riehl

Last time, I described an emerging pattern in algebra and topology, exemplified by the cohomology of the space of configurations of ordered points in the plane. In language I’ll introduce below, cohomology with rational coefficients defines a uniformly representation stable sequence, the fundamental data of which is an $S_n$-representation for each $n$.

In this post, I want to tell you in more detail about the representation stable sequences introduced in a paper by Tom Church and Benson Farb. I’ll also discuss the second generation approach to representation theory, which centers of the functor category of what they call FI-modules. Jordan Ellenberg, the third author of this paper, has blogged about this also. The first post in his series can be found here. In contrast with Part I, my exposition here will be more explicitly directed at the categorically-minded reader.

Posted at 3:24 AM UTC | Permalink | Followups (16)

## February 1, 2014

### An Emerging Pattern in Algebra and Topology I

#### Posted by Emily Riehl

At the Joint Mathematics Meetings in Baltimore, I saw Benson Farb deliver a joint invited address on representation stability, the above eponymous “emerging pattern.” He began this work in 2010 with Tom Church, then a PhD student at the University of Chicago. I’ve been a fan for a few years now, and Benson’s beautiful talk has inspired me to write a brief summary.

This post is divided in two parts. In Part I, I’ll tell you about the talk, which was largely accessible to anyone with an undergraduate math background. In Part II, I’ll say a bit about the technical details and write about more recent developments, joint also with Jordan Ellenberg, in which some categorical ideas enable a simplified conceptual understanding of the patterns that frequently arise in practice.

Posted at 1:32 AM UTC | Permalink | Followups (2)

## January 30, 2014

### Weightings for Compact Metric Spaces

#### Posted by Tom Leinster

Guest post by Mark Meckes

A recent paper of Mark’s proved the very substantial result that Minkowski dimension (one of the most important types of fractal dimension) can be derived from magnitude (an invariant ultimately coming from category theory). More exactly, he showed that the Minkowski dimension of a compact subset of $\mathbb{R}^n$ is exactly equal to its “magnitude dimension” (Cor 7.4). Here, Mark explains not this specific result, but the overall framework that makes such theorems possible. —TL

I’d like to report on some recentish progress in understanding the magnitude of compact metric spaces. To set the stage I’ll recap some background, which will repeat a number of things which have already appeared in posts by Tom and Simon. For those who want to be reminded of more, Tom has provided a reading list.

Posted at 4:45 PM UTC | Permalink | Followups (15)

### Tensor Categories and TQFT in Erlangen

#### Posted by John Baez

There’s a meeting in Erlangen coming up!

Structures on Tensor Categories and Topological Field Theories, Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, March 4-6, 2014, organized by Catherine Meusburger and Christoph Schweigert.

Posted at 11:33 AM UTC | Permalink | Followups (4)

## January 27, 2014

### Formal Theory of Monads (Following Street)

#### Posted by Emily Riehl

Guest post by Eduard Balzin

The Kan extension seminar continues, and with it we now come to the paper by Ross Street. Published in 1972, this paper is one of the first instances where the notion of a monad was made relative to an arbitrary 2-category. A lot of aspects, such as algebras over a monad, Eilenberg-Moore and Kleisli categories, the relation with adjunctions, were generalised accordingly. Other topics involve representability, duality, and the example of $\mathbf{Cat}$. I will therefore try to talk about this paper in detail, and add something of my own.

Myself, in the course of my (higher)-categorical education, I have not learned in full about monads. I do not mean definitions or main theorems, but rather the answer to the question why monads. While preparing for the seminar, I’ve managed to give a series of answers to this (my own) question, and I hope that the discussion will take me (and everyone else) even further.

Let me sincerely thank Emily Riehl for organising the seminar. I truly hope it will be a new castle on the landscape of categorical life. I am also quite grateful for the participants of the seminar, who have greatly contributed during the online discussion and in their readers discussion. It all makes me happy.

Posted at 6:53 PM UTC | Permalink | Followups (40)

## January 22, 2014

### The Magnitude of a Graph

#### Posted by Tom Leinster

I’ve just arXived a new paper about a new invariant, The magnitude of a graph. Much of the development of this invariant has taken place at this blog, with two previous posts and crucial contributions from David Speyer and Simon Willerton. Your comments, critical or otherwise, would be very welcome.

Magnitude seems to be orthogonal to other graph invariants. You can’t derive from it such classic invariants as the Tutte polynomial or the chromatic number, nor the girth nor the clique number nor even the number of connected components. And conversely, you can’t derive the magnitude from these or any other well-known graph invariants. Apparently, it captures information of a different kind.

Its most appealing characteristic is that it behaves like cardinality. It’s multiplicative with respect to cartesian product, additive with respect to disjoint union, and obeys a restricted version of the inclusion-exclusion principle. And it also behaves a bit like the Tutte polynomial. For instance, it’s often invariant under Whitney twists — though not always, as I’ll explain.

Posted at 12:19 AM UTC | Permalink | Followups (15)