Category: stories

NaNoWriMo (3)

Published November 30, 2016 by lievenlb

In 2001, Eugenia Cheng gave an interesting after-dinner talk Mathematics and Lego: the untold story. In it she compared math research to fooling around with lego. A quote:

“Lego: the universal toy. Enjoyed by people of all ages all over the place. The idea is simple and brilliant. Start with some basic blocks that can be joined together. Add creativity, imagination and a bit of ingenuity. Build anything.

Mathematics is exactly the same. We start with some basic building blocks and ways of joining them together. And then we use creativity, and, yes, imagination and certainly ingenuity, and try to build anything.”

She then goes on to explain category theory, higher dimensional topology, and the process of generalisation in mathematics, whole the time using lego as an analogy. But, she doesn’t get into the mathematics of lego, perhaps because the talk was aimed at students and researchers of all levels and all disciplines.

There are plenty of sites promoting lego in the teaching of elementary mathematics, here’s just one link-list-page: “27 Fantastic LEGO Math Learning Activities for All Ages”. I’m afraid ‘all ages’ here means: under 10…

$lego-math-teaching-children-alycia-zimmerman-fb__700-png$

Can one do better?

Everyone knows how to play with lego, which shapes you can build, and which shapes are simply impossible.

Can one tap into this subconscious geometric understanding to explain more advanced ideas such as symmetry, topological spaces, sheaves, categories, perhaps even topos theory… ?

Let’s continue our

[section_title text=”imaginary iterview”]

Question: What will be the opening scene of your book?

Alice posts a question on Lego-stackexchenge. She wants help to get hold of all imaginary lego shapes, including shapes impossible to construct in three-dimensional space, such as gluing two shapes over some internal common sub-shape, or Escher like constructions, and so on.

Question: And does she get help?

At first she only gets snide remarks, style: “brush off your French and wade through SGA4”.

Then, she’s advised to buy a large notebook and jot down whatever she can tell about shapes that one can construct.

If you think about this, you’ll soon figure out that you can only add new bricks along the upper or lower bricks of the shape. You may call these the boundary of the shape, and soon you’ll be doing topology, and forming coproducts.

These ‘legal’ lego shapes form what some of us would call a category, with a morphism from $A$ to $B$ for each different way one can embed shape $A$ into $B$.

Of course, one shouldn’t use this terminology, but rather speak of different instruction-manuals to get $B$ out of $A$ (the morphisms), stapling two sets of instructions together (the compositions), and the empty instruction-sheet (the identity morphism).

Question: But can one get to the essence of categorical results in this way?

Take Yoneda’s lemma. In the case of lego shapes it says that you know a shape once you know all morphisms into it from whatever shape.

For any coloured brick you’re given the number of ways this brick sits in that shape, so you know all the shape’s bricks. Then you may try for combination of two bricks, and so on. It sure looks like you’re going to be able to reconstruct the shape from all this info, but this quickly get rather messy.

But then, someone tells you the key argument in Yoneda’s proof: you only have to look for the shape to which the identity morphism is assigned. Bingo!

Question: Wasn’t your Alice interested in the ‘illegal’ or imaginary shapes?

Once you get to Yoneda, the rest follows routinely. You define presheaves on this category, figure out that you get a whole bunch of undesirable things, bring in Grothendieck topologies to be the policing agency weeding out that mess, and keep only the sheaves, which are exactly the desired imaginary shapes.

Question: Your book’s title is ‘Primes and other imaginary shapes’. How do you get from Lego shapes to prime numbers?

By the standard Gödelian trick: assign a prime number to each primitive coloured brick, and to a shape the product of the brick-primes.

That number is a sort of code of the shape. Shapes sharing the same code are made up from the same set of bricks.

Take the set of all strictly positive natural numbers partially ordered by divisibility, then this code is a functor from Lego shapes to numbers. If we extend this to imaginary shapes, we’ll rapidly end up at Connes’ arithmetic site, supernatural numbers, adeles and the recent realisation that the set of all prime numbers does have a geometric shape, but one with infinitely many dimensions.

Not sure yet how to include all of this, but hey, early days.

Question: So, shall we continue this interview at a later date?

No way, I’d better start writing.

NaNoWriMo (2)

Published November 28, 2016 by lievenlb

Two more days to go in the NaNoWriMo 2016 challenge. Alas, it was clear from the outset that I would fail, bad.

I didn’t have a sound battle plan. Hell, I didn’t even have a a clue which book to write…

But then, I may treat myself to a SloWriMo over the Christmas break.

For I’ve used this month to sketch the vaguest possible outlines of an imaginary book.

[section_title text=”An imaginary interview”]

Question: What is the title of your book?

I don’t know for sure, but my working title is Primes and other imaginary shapes.

Question: What will the cover-illustration look like?

At the moment I would settle for something like this:

Question: Does your book have an epigraph?

That’s an easy one. Whenever this works out, I’ll use for the opening quote:

[quote name=”David Spivak in ‘Presheaf, the cobbler'”]God willing, I will get through SGA 4 and Lurie’s book on Higher Topos Theory.
[/quote]

Question: Any particular reason?

Sure. That’s my ambition for this book, but perhaps I’ll save Lurie’s stuff for the sequel.

Question: As you know, Emily Riehl has a textbook out: Category Theory in Context. Here’s a recent tweet of hers:

Brilliant #thanksgiving question from my uncle: "Does this book have a protagonist?" #CategoryTheoryInContext @doverpubs

— Emily Riehl (@emilyriehl) November 24, 2016

Whence the question: does your book have a protagonist?

Well, I hope someone gave Emily the obvious reply: Yoneda! As you know, category theory is a whole bunch of definitions, resulting in one hell of a lemma.

But to your question, yes there’ll be a main character and her name is Alice.

I know, i know, an outrageous cliché, but at least I can guarantee there’ll be no surprise appearances of Bob.

These days, Alices don’t fall in rabbit holes, or crawl through looking-glasses. They just go online and encounter weird and wondrous creatures. I need her to be old enough to set up a Facebook and other social accounts.

My mental image of Alice is that of the archetypical STEM-girl

In her younger years she was a lot like Lewis Carroll’s Alice. In ten years time she’ll be a copy-cat Alice Butler, the heroine of Scarlett Thomas’ novel PopCo.

Question: What will be the opening scene of your book?

Alice will post a question on Lego-stackexchenge, and yes, to my surprise such a site really exists…

(to be continued, perhaps)

le lemme de la Gare du Nord

Published November 18, 2016 by lievenlb

Theorems have the tendency to pop into existence when you least expect them: taking a bath, during your sleep, dreaming away during a dull lecture, waiting for an airplane, bicycling, whatever.

One of the most famous (and useful) lemmas was dreamed up in the Parisian Gare du Nord station, during a conversation between Saunders Mac Lane (then in his mid 40ties) and a young Japanese mathematician, half his age, Nobuo Yoneda.

Here’s the story:

Yoneda’s story

In the announcement of the death of his friend Yoneda on the catlist, the computer scientist Yoshiki Kinoshita writes:

“Prof. Yoneda was born on 28 March, 1930. He studied mathematics in the University of Tokyo; in the last year of his studies he followed the seminar of Prof. Shokiti Iyanaga, where he became interested in algebraic topology.

Soon after that (or maybe when he was still an undergraduate), Prof. Samuel Eilenberg visited Japan, and Yoneda traveled around Japan with him, as a translator and guide. At that time, he was exposed to the Cartan-Eilenberg book, which was still in galley form.

Later, he got a Fulbright scholarship and he visited Princeton, to study with Eilenberg. Those days, the Yen was still very weak, so even the Fulbright money was not enough to support even a modest living in U.S. Therefore, he worked as a proofreader at a European publisher (Elsevier? NorthHolland? maybe American), to earn money for the study in U.S. Even 25 years later, he seemed to be proud of his professional skill as a proofreader, which he gained then.

When he arrived in Princeton, Eilenberg had moved (sabbatical?) to France (or maybe, Eilenberg left US just after Yoneda’s arrival). So, Yoneda went to France a year later.

At that time, Saunders Mac Lane was visiting category theorists, apparently to obtain information to write his book (or former survey), and he met the young Yoneda, among others.

The interview started in a Cafe at Gare du Nord, and went on and on, and was continued even in Yoneda’s train until its departure. The contents of this talk was later named by Mac Lane as Yoneda lemma. So, the famous Yoneda lemma was born in Gare du Nord.

This must have been a good memory for Yoneda; I heard him tell this story many times. I do not know whether Mac Lane managed to leave the train before departure!”

Mac Lane’s story

In his obituary of Yoneda, Saunders Mac Lane also recalls the story as I learned from this tweet by Emily Riehl:

A personal account of the history of the Yoneda lemma, from Mathematica Japonica, found in Cabot science library. pic.twitter.com/Rc7NyboFPH

— Emily Riehl (@emilyriehl) February 24, 2015

Emily also quotes part of the article in the section on Yoneda’s lemma in her excellent book Categories in Context.

At the time, Mac Lane was in his late 80ties and it is not clear whether this obituary is based on his own recollections or on the catlist-posting by Kinoshita.

Here’s the full version:

“Yoneda enjoyed relating the story of the origins of this lemma, as follows.

He had guided Samuel Eilenberg during Eilenberg’s visit to Japan, and in the process learned homological algebra.

Soon Yoneda spent a year in France (apparently in 1954 and 1955).

There he met Saunders Mac Lane. Mac Lane, then visiting Paris, was anxious to learn from Yoneda, and commenced an interview with Yoneda in a café at Gare du Nord.

The interview was continued on Yoneda’s train until its departure. In its course, Mac Lane learned about the lemma and subsequently baptized it.”

When was the Yoneda lemma born?

Remains the problem of putting an approximate date on this interview.

We know that Mac Lane was in France in the summer and fall of 1954. Thanks to this page maintained by John Baez we know that Mac Lane gave two lectures at the Colloque Henri Poincaré in Paris at the Institut Henri Poincaré in oktober 1954.

We also know that Mac Lane was a guest at the summer Bourbaki meeting which was held from august 17 till august 31, 1954 in Murols.

In this photograph taken at the meeting we see from left to right: Roger Godement, Jean Dieudonné, André Weil, Saunders Mac Lane and Jean-Pierre Serre.

Sadly, the Bourbaki-report (la Tribu) of the Murols-congress is not yet in the public domain (the pre-1954 reports can be found here), so it is a bit difficult to get hold of a full list of people present, or what exactly was discussed at that meeting.

Still, in a footnote to a paper on Gerhard Hochschild we find that Gerhard Hochschild and John Tate were in Murols as “foreigh visitors”, Saunders Mac Lane attended as “efficiency expert” (more on this below), and … Iyanaga (Yoneda’s Ph.D. adviser) and Yoshida (probably they meant Kosaku Yosida) attended as “honorable foreign visitors”.

Both Japanese mathematicians were on their way to the ICM in Amsterdam, which was held from september 2nd till september 9th 1954. Incidentally, Iyanaga chaired the invited address given by Andre Weil on wednesday 8th, titled “Abstract vs. classical algebraic geometry”.

Presumably, Yoneda was taking a train at the gare du Nord on his way to meat his adviser Iyanaga, to or from the Bourbaki meeting in Murols. My best guess is that the Yoneda lemma was born in the gare du Nord mid august 1954.

Mac Lane, the efficiency expert

What was Mac Lane’s role at the Bourbaki Murols-congress?

According to him there had been a heated argument at a previous Bourbaki-congress on using the terminology “exact sequence”. Mac Lane took it upon himself to convince the Bourbaki-members to embrace the use of category theory, perhaps by explaining the brand new Yoneda-lemma.

He failed miserably. Here’s what Weil wrote to Chevalley about it:

“As you know, my honourable colleague Mac Lane supports the idea that every structural notion necessarily comes equipped with a notion of homomorphism. […] What on earth does he hope to deduce from this kind of considerations?”

Mac Lane attributes his failure to convince Bourbaki (or at least Weil) to “perhaps my command of the French language was inadequate to the task of persuasion”.

I find it striking that top-mathematicians such as Andre Weil failed to see the importance of morphisms between objects even as late as the mid 50ties.

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