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.

April 30, 2014

Should Mathematicians Cooperate with GCHQ? Part 2

Posted by Tom Leinster

A month ago, the newsletter of the London Mathematical Society published an opinion piece of mine, Should mathematicians cooperate with GCHQ?. It has just published an opposing opinion by Richard Pinch, GCHQ’s Strategic Advisor for Mathematics Research and formerly a number theorist at Cambridge.

Pinch’s reply is short and curiously insubstantial. First he makes a couple of general assertions in opposition to what I wrote. But unlike my piece, which linked heavily to sources, he provides no evidence for his assertions. Nor does he dispute any of the specific facts stated in my article. Then he quotes a politician and the director of GCHQ saying that they believe GCHQ operates with integrity. And that’s it.

So it’s almost too flimsy to be worth answering. However, it’s probably worth rebutting even insubstantial arguments when they come from people in positions of influence. Here’s my rebuttal.

Posted at 5:35 PM UTC | Permalink | Followups (18)

April 29, 2014

Homotopy Type Theory: Unified Foundations of Mathematics and Computation

Posted by David Corfield

Congratulations to Mike for being a part of a research team who will receive $7. 5 million to carry on the good work of the IAS Univalent Foundations program. Homotopy Type Theory: Unified Foundations of Mathematics and Computation will run for five years, organised by Steve Awodey at CMU. (Technical portion of the grant proposal is here.)

I hope they’re allowed to use some of that funding to spill into physics a little.

Posted at 12:48 PM UTC | Permalink | Followups (8)

April 28, 2014

On Two-Dimensional Monad Theory

Posted by Emily Riehl

Guest post by Sam van Gool

Monads provide a categorical setting for studying sets with additional structure. Similarly, 2-monads provide a 2-categorical setting for studying categories with additional structure. While there is really only one natural notion of algebra morphism in the context of monads, there are several choices of algebra morphism in the context of 2-monads. The interplay between these different kinds of morphisms is the main focus of the paper that I discuss in this post:

I will give an overview of the results and methods used in this paper. Also, especially towards the end of my post, I will also indicate some points that I think could still be clarified further by formulating some questions, which will hopefully lead to fruitful discussions below.

This post forms the 9th instalment of the series of posts written by participants of the Kan Extension Seminar, of which I’m very glad to be a part. In preparing the post I have greatly benefited from discussions with the other participants in the seminar, and of course with the seminar’s organizer, Emily Riehl. I am very grateful for the enthusiasm, encouragement and guidance that you all offered.

Posted at 9:31 PM UTC | Permalink | Followups (42)

April 27, 2014

New Scientist Article

Posted by Tom Leinster

I’ve got a full-page opinion piece in this week’s New Scientist, on why mathematicians should refuse to cooperate with agencies of mass surveillance. If you’re in the US, UK or Australia, it’s the print edition that came out yesterday.

Thumbnail of New Scientist

Posted at 5:19 AM UTC | Permalink | Followups (15)

April 20, 2014

Finite Products Theories

Posted by John Baez

Here’s a classic theorem about finite products theories, also known as algebraic theories or Lawvere theories. You can find it in Toposes, Triples and Theories, but it must go way back to Lawvere’s thesis. In my work with Nina Otter on phylogenetic trees, we need a slight generalization of it… and if true, this generalization must already be known. So I really just want a suitable reference!

Theorem. Suppose CC is a category with finite products that is single-sorted: every object is a finite product of copies of a single object xCx \in C. Let Mod(C)Mod(C) be the category of models of CC: that is, product-preserving functors

ϕ:CSet \phi : C \to Set

and natural transformations between these. Let

U:Mod(C)Set U : Mod(C) \to Set

be the functor sending any model to its underlying set:

U(ϕ)=ϕ(x) U(\phi) = \phi(x)

Then UU has a left adjoint

F:SetMod(C) F : Set \to Mod(C)

and Mod(C)Mod(C) is equivalent to the category of algebras of the monad

UF:SetSet U F : Set \to Set

As a result, any model ϕ\phi can be written as a coequalizer

FUFU(ϕ)FU(ϕ)ϕ F U F U(\phi) \stackrel{\longrightarrow}{\longrightarrow} F U(\phi) \longrightarrow \phi

where the arrows are built from the counit of the adjunction

ε:FU1 Mod(C) \epsilon : F U \to 1_{Mod(C)}

in the obvious ways: FU(ε FU(ϕ)) F U (\epsilon_{F U (\phi)}), ε FUFU(ϕ)\epsilon_{F U F U(\phi)} and ε ϕ\epsilon_\phi.

Posted at 4:58 PM UTC | Permalink | Followups (22)

April 18, 2014

Elementary Observations on 2-Categorical Limits

Posted by Emily Riehl

Guest post by Christina Vasilakopoulou

In the eighth installment of the Kan Extension Seminar, we discuss the paper “Elementary Observations on 2-Categorical Limits” by G.M. Kelly, published in 1989. Even though Kelly’s classic book Basic Concepts of Enriched Category Theory, which contains the abstract theory related to indexed (or weighted) limits for arbitrary 𝒱\mathcal{V}-categories, was available since 1982, the existence of the present article is well-justifiable.

On the one hand, it constitutes an independent account of the fundamental case 𝒱\mathcal{V}=Cat\mathbf{Cat}, thus it motivates and exemplifies the more general framework through a more gentle, yet meaningful exposition of 2-categorical limits. The explicit construction of specific notable finite limits such as inserters, equifiers etc. promotes the comprehension of the definitions, via a hands-on description. Moreover, these finite limits and particular results concerning 2-categories rather than general enriched categories, such as the construction of the cotensor as a PIE limit, are central for the theory of 2-categories. Lastly, by introducing indexed lax and pseudo limits along with Street’s bilimits, and providing appropriate lax/ pseudo/ bicategorical completeness results, the paper serves also as an indespensable reference for the later “2-Dimensional Monad Theory” by Blackwell, Kelly and Power.

I would like to take this opportunity to thank Emily as well as all the other participants of the Kan Extension Seminar. This has been a unique experience of constant motivation and inspiration for me!

Posted at 8:34 PM UTC | Permalink | Followups (16)

April 16, 2014

Enrichment and the Legendre–Fenchel Transform I

Posted by Simon Willerton

The Legendre–Fenchel transform, or Fenchel transform, or convex conjugation, is, in its naivest form, a duality between convex functions on a vector space and convex functions on the dual space. It is of central importance in convex optimization theory and in physics it is used to switch between Hamiltonian and Lagrangian perspectives.


Suppose that VV is a real vector space and that f:V[,+] f\colon V\to [-\infty ,+\infty ] is a function then the Fenchel transform is the function f *:V #[,+] f^{\ast }\colon V^{#}\to [-\infty ,+\infty ] defined on the dual vector space V #V^{#} by f *(k)sup xV{k,xf(x)}. f^{\ast }(k)\coloneqq \sup _{x\in V}\big \{ \langle k,x\rangle -f(x)\big \} .

If you’re a regular reader then you will be unsurprised when I say that I want to show how it naturally arises from enriched category theory constructions. I’ll show that in the next post. In this post I’ll give a little introduction to the Legendre–Fenchel transform.

Posted at 10:59 AM UTC | Permalink | Followups (9)

April 14, 2014


Posted by Simon Willerton

A new Spanish language mathematical magazine has been launched: universo.math. Hispanophones should check out the first issue! There are some very interesting looking articles which cover areas from art through politics to research-level mathematics.

The editor-in-chief is my mathematical brother Jacob Mostovoy and he wants it to be a mix of Mathematical Intellingencer, Notices of the AMS and the New Yorker, together with less orthodox ingredients; the aim is to keep the quality high.

Besides Jacob, the contributors to the first issue that I recognise include Alberto Verjovsky, Ernesto Lupercio and Edward Witten, so universo.math seems to be off to a high quality start.

Posted at 6:04 PM UTC | Permalink | Followups (1)

April 7, 2014

The Modular Flow on the Space of Lattices

Posted by Simon Willerton

Guest post by Bruce Bartlett

The following is the greatest math talk I’ve ever watched!

  • Etienne Ghys (with pictures and videos by Jos Leys), Knots and Dynamics, ICM Madrid 2006. [See below the fold for some links.]

Etienne Ghys A modular knot

I wasn’t actually at the ICM; I watched the online version a few years ago, and the story has haunted me ever since. Simon and I have been playing around with some of this stuff, so let me share some of my enthusiasm for it!

The story I want to tell here is how, via modular flow of lattices in the plane, certain matrices in SL(2,)\SL(2,\mathbb{Z}) give rise to knots in the 3-sphere less a trefoil knot. Despite possibly sounding quite scary, this can be easily explained in an elementary yet elegant fashion.

Posted at 11:55 PM UTC | Permalink | Followups (28)

On a Topological Topos

Posted by Emily Riehl

Guest post by Sean Moss

In this post I shall discuss the paper “On a Topological Topos” by Peter Johnstone. The basic problem is that algebraic topology needs a “convenient category of spaces” in which to work: the category 𝒯\mathcal{T} of topological spaces has few good categorical properties beyond having all small limits and colimits. Ideally we would like a subcategory, containing most spaces of interest, which is at least cartesian closed, so that there is a useful notion of function space for any pair of objects. A popular choice for a “convenient category” is the full subcategory of 𝒯\mathcal{T} consisting of compactly-generated spaces. Another approach is to weaken the notion of topological space, i.e. to embed 𝒯\mathcal{T} into a larger category, hopefully with better categorical properties.

A topos is a category with enough good properties (including cartesian closedness) that it acts like the category of sets. Thus a topos acts like a mathematical universe with ‘sets’, ‘functions’ and its own internal logic for manipulating them. It is exciting to think that if a “convenient topos of spaces” could be found, then its logical aspects could be applied to the study of its objects. The set-like nature of toposes might make it seem unlikely that this can happen. For instance, every topos is balanced, but the category of topological spaces is famously not. However, sheaves (the objects in Grothendieck toposes) originate from geometry and already behave somewhat like generalized spaces.

I shall begin by elaborating on this observation about Grothendieck toposes, and briefly examine some previous attempts at a “topological topos”. I shall explore the idea of replacing open sets with convergent sequences and see how this leads us to Johnstone’s topos. Finally I shall describe how Johnstone’s topos is a good setting for homotopy theory.

I would to thank Emily Riehl for organizing the Kan Extension Seminar and for much useful feedback. Thanks go also to the other seminar participants for being a constant source of interesting perspectives, and to Peter Johnstone, for his advice and for writing this incredible paper.

Posted at 5:38 AM UTC | Permalink | Followups (30)

April 1, 2014

Big Data Power

Posted by Tom Leinster

Guest post by Nils Carqueville and Daniel Murfet

(My university probably isn’t alone in encouraging mathematicians and computer scientists to embrace the idea of “big data”, or in more sober terminology, “data science”. Here, Nils Carqueville and Daniel Murfet introduce their really excellent article on big data in whole-population surveillance. —TL)

In recent years Big Data has become an increasingly relevant topic in the economic sector, for intelligence agencies, and for the sciences.

A particularly far-reaching development made possible by Big Data is that of unprecedented mass surveillance. As Alexander Beilinson, Stefan Forcey, Tom Leinster and others have pointed out, the role that mathematicians and computer scientists play here is central. With this in mind last January we wrote an essay on

Big Data Power

stressing some of those aspects of the matter that we think deserve more attention or additional elaboration. We hope this to be a useful contribution to the necessary discussion on modern mass surveillance, and we thank Tom for his efforts in this direction, and for allowing us to post here.

Posted at 1:38 PM UTC | Permalink | Followups (13)