The n-Category Café

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

$$\phi : C \to Set$$

and natural transformations between these. Let

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

be the functor sending any model to its underlying set:

$$U(\phi) = \phi(x)$$

Then $U$ has a left adjoint

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

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

$$U F : Set \to Set$$

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

$$F U F U(\phi) \stackrel{\longrightarrow}{\longrightarrow} F U(\phi) \longrightarrow \phi$$

where the arrows are built from the counit of the adjunction

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

in the obvious ways: $F U (\epsilon_{F U (\phi)})$, $\epsilon_{F U F U(\phi)}$ and $\epsilon_\phi$.

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 $\mathcal{V}$-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 $\mathcal{V}$=$\mathbf{Cat}$, 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!

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

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.

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.

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.]

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 $\SL(2,\mathbb{Z})$ 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.

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 $\mathcal{T}$ 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 $\mathcal{T}$ consisting of compactly-generated spaces. Another approach is to weaken the notion of topological space, i.e. to embed $\mathcal{T}$ 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.

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.

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.

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.

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 $a$ of a category $A$, the diagram $$\begin{matrix} 1 & \overset{a}{\rightarrow} & A \\ {}_{\ast} \searrow & \overset{\iota}{\Rightarrow} & \swarrow_{A(a,-)} \\ & Set \\ \end{matrix}$$ 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.

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

• Translation of Grothendieck Biography.

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

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?

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 $[0,1]$ as the set of truth values with $0$ representing false and $1$ representing true. So the truth degree of the statement “This bike is new” would vary, being $1$ today and decreasing to something very close to $0$ 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 $[0,1]$ as a monoidal category to enrich over. We will see that categories enriched over $[0,1]$ 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!

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.

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.

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.