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Category: stories

The (somewhat less) Secret Bourbaki Archive

It has been many, many years since I’ve last visited the Bourbaki Archives.

The underground repository of the Bourbaki Secret Archives is a storage facility built beneath the cave of the former Capoulade Cafe. Given its sporadic use by staff and scholars, the entire space – including the Gallery of all intermediate versions of every damned Bourbaki book, the section reserved to Bourbaki’s internal notes, such as his Diktats, and all numbers of La Tribu, and the Miscellania, containing personal notes and other prullaria once belonging to its members – is illuminated by amber lighting activated only when movement is detected by strategically placed sensors, and is guarded by a private security firm, hired by the ACNB.

This description (based on that of the Vatican Secret Archives in the book The Magdalene Reliquary by Gary McAvoy) is far from the actual situation. The Bourbaki Archive has been pieced together from legates donated by some of its former members (including Delsarte, Weil, de Possel, Cartan, Samuel, and others), and consist of well over a hundredth labeled carton and plastic cases, fitting easily in a few standard white Billy Ikea bookcases.

The publicly available Bourbaki Archive is even much smaller. The Association des collaborateurs de Nicolas Bourbaki has strong opinions on which items can be put online. For years the available issues of La Tribu were restricted to those before 1953. I was once told that one of the second generation Bourbaki-members vetoed further releases.

As a result, we only had the fading (and often coloured) memories of Bourbaki-members to rely on if we wanted to reconstruct key events, for example, Bourbaki’s reluctance to include category theory in its works. Rather than to work on source material, we had to content ourselves with interviews, such as this one, the relevant part starts at 51.40 into the clip. See here for some other interesting time-slots.

On a recent visit to the Bourbaki Archives I was happy to see that all volumes of “La Tribu” (the internal newsletter of Bourbaki) are now online from 1940 until 1960.

Okay, it’s not the entire story yet but, for all you Grothendieck aficionados out there, it should be enough as G resigned from Bourbaki in 1960 with this letter (see here for a translation).

Grothendieck was present at just twelve Bourbaki congresses in the period between 1955 and 1960 (he was also present as a ‘cobaye’ at a 1951 congress in Nancy).

The period 1955-60 was crucial in the modern development of algebraic geometry. Serre’s ‘FAC’ was published, as was Grothendieck’s ‘Tohoku-paper’, there was the influential Chevalley seminar, and the internal Bourbaki-fight about categories and the functorial view.

Perhaps the definite paper on the later issue is Ralf Kromer’s La ‘Machine de Grothendieck’ se fonde-t-elle seulement sur les vocables metamathematiques? Bourbaki et les categories au cours des annees cinquante.

Kromer had access to most issues of La Tribu until 1962 (from the Delsarte archive in Nancy), but still felt the need to justify his use of these sources to the ACNB (footnote 9 of his paper):

“L’autorisation que j’ai obtenue par le Comité scientifique des Archives de la création des mathématiques, unité du CNRS qui fut chargée jusqu’en 2003 de la mise à disposition de ces archives, me donne également le droit d’utiliser les sources datant des années postérieures à l’année 1953, que j’avais consultées auparavant aux Archives Jean Delsarte, soit avant que l’ACNB (Association des Collaborateurs de Nicolas Bourbaki) ne rende publique sa décision d’ouvrir ses archives et ne décide des parties qui seraient consultables.

J’ai ainsi bénéficié d’une occasion qui ne se présenterait sans doute plus aujourd’hui, mais c’est en toute légitimité que je puis m’appuyer sur cette riche documentation. Toutefois, la collection des Archives Jean Delsarte étant à son tour limitée aux années antérieures à 1963, je n’ai pu étudier la discussion ultérieure.”

The Association des Collaborateurs de Nicolas Bourbaki made retirement from active B-membership mandatory at the age of 50. One might expect of it to open up all documents in its archives which are older than fifty years.

Meanwhile, we’ll have a go at the 1940-1960 issues of La Tribu.

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the topos of unconsciousness

Since wednesday, as mentioned last time, the book by Alain Connes and Patrick Gauthier-Lafaye: “A l’ombre de Grothendieck et de Lacan, un topos sur l’inconscient” is available in the better bookshops.



There’s no need to introduce Alain Connes on this blog. Patrick Gauthier-Lafaye is a French psychiatrist and psycho-analyst, working in Strassbourg.

The book is a lengthy dialogue in which the authors try to find a use for topos theory in Jaques Lacan’s psycho-analytical view of the unconscious.

If you are a complete Lacanian virgin, it may be helpful to browse through “Lacan, a beginners guide” (by Lionel Bailly) first.



If this left you bewildered, for example by Lacan’s strange (ab)use of mathematics, rest assured, you’re not alone.

It is no coincidence that Lacan’s works are the first case-study in the book “Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science” by Alan Sokal (the one of the hoax) and Jean Bricmont. You can download the book from this link.



If now you feel that Sokal and Bricmont are way too harsh on Lacan, I urge you to have a go at the book “Writing the structures of the subject, Lacan and topology” by Will Greenshields.



If you don’t have the time or energy for this, let me give you one illustrative example: the topological explanation of Lacan’s formula of fantasy:

\[
\$~\diamond~a \]

Loosely speaking this formula says “the barred subject stands within a circular relationship to the objet petit a (the object of desire), one part of which is determined by alienation, the other by separation”.

Lacan was obsessed with the immersion of the projective plane $\mathbb{P}^2(\mathbb{R})$ into $\mathbb{R}^3$ as the cross-cap. Here’s an image of it from his 1966-67 seminar on ‘Logique du fantasme’ (213 pages).



This image includes the position of the objet petit $a$ as the end point of the self-intersection curve, which itself is referred to as the ‘castration’, or the ‘phallus’, or whatever.

Brace yourself for the ‘explanation’ of $\$~\diamond~a$: if you walk twice around $a$ this divides the cross-cap into a disk and a Mobius-strip!

The mathematics is correct but I fail to see how this helps the psycho-analyst in her therapy. But hey, everyone will tell you I have absolutely no therapeutic talent.

Let’s return to the brand new book by Alain Connes and Patrick Gauthier-Lafaye: “A l’ombre de Grothendieck et de Lacan, un topos sur l’inconscient”.



It was to be expected that they would defend Lacan’s exploitation of (surface) topology by saying that he was just unfortunate not to have the more general notion of toposes available, as well as their much subtler logic. Perhaps someone should write a fictional parody on Greenshields book: “Lacan and the topos”…

Connes’ first attempt to construct the topos of unconsciousness was also not much of a surprise. According to Lacan the unconscious is ‘structured like a language’.

So, a natural approach might be to start with a ‘dictionary’-category (words and relations between them) or any other known use of a category in linguistics. A good starting point to read up on this is the blog post A new application of category theory in linguistics.

Eventually they settled for a much more ambitious project. To Connes and Gauthier-Lafaye every individual has her own topos and corresponding logic.

They don’t specify how to construct these individual toposes, but postulate that they are all connected to a classifying topos, which is their incarnation of the world of ‘myths’ and ‘fantasies’.

Surely an idea Lacan would have liked. Underlying the unconscious must be, according to Connes and Gauthier-Lafaye, a geometric theory! That is, it can be fully described by first order sentences.

Lacan himself used already some first order sequences in his teachings, such as in his logic of sexuation:

\[
\forall x~(\Phi~x)~\quad \text{but also} \quad \exists x~\neg~(\Phi~x) \]

where $\Phi~x$ is the phallic function. Quoting from Greenshield’s book:

“While all (the sons) are subject to ($\forall x$) the law of castration ($\Phi~x$), we also learn that this law nevertheless resides upon an exception: there exists a subject ($\exists x$) that is not subject to this law ($\neg \Phi~x$). This exception is embodied by the despotic father who, not being subject to the phallic function, experiences an impossible mode of totalised jouissance (he enjoys all the women). He is, quite simply, the exception that proves the law a necessary beyond that enables the law’s geometric bounds to be defined.”

It will be quite hard (but probably great fun for psycho-analysts) to turn the whole of Lacanian theory on the unconscious into a coherent geometric theory, construct its classifying topos, and apply the Joyal-Reyes theorem to get at the individual cases/toposes.

I’m sure there are much deeper insights to be gained from Connes’ and Gauthier-Lafaye’s book, but this is what i got from a first, fast, cursory reading of it.

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Grothendieck meets Lacan

Next month, a weekend-meeting is organised in Paris on Lacan et Grothendieck, l’impossible rencontre?.



Photo from Remembering my father, Jacques Lacan

Jacques Lacan was a French psychoanalyst and psychiatrist who has been called “the most controversial psycho-analyst since Freud”.

What’s the connection between Lacan and Grothendieck? Here’s Stephane Dugowson‘s take (G-translated):

“As we know, Lacan was passionate about certain mathematics, notably temporal logic and the theory of knots, where he thought he found material for advancing the theory of psychoanalysis. For his part, Grothendieck testifies in his non-strictly mathematical writings to his passion for the psyche, as shown by many pages of his Récoltes et Semailles just published by Gallimard (in January 2022), or even, among the tens of thousands of pages discovered at his death and of which we know almost nothing, the 3700 pages of mathematics grouped under the title ‘Structure of the Psyche’.

One might therefore be surprised that the two geniuses never met. In fact, a lunch did take place in the early 1970s organized by the mathematician and psychoanalyst Daniel Sibony. But a lunch does not necessarily make a meeting, and it seems that this one unfortunately did not happen.”

As it is ‘bon ton’ these days in Parisian circles to utter the word ‘topos’, several titles of the talks given at the meeting contain that word.

There’s Stephane Dugowson‘s talk on “Logique du topos borroméen et autres logiques à trois points”.

Lacan used the Borromean link to illustrate his concepts of the Real, Symbolic, and Imaginary (RSI). For more on this, please read chapter 6 of Lionel Baily’s excellent introduction to Lacan’s work Lacan, A Beginner’s Guide.

The Borromean topos is an example of Dugowson’s toposes associated to his ‘connectivity spaces’. From his paper Définition du topos d’un espace connectif I gather that the objects in the Borromean topos consist of a triple of set-maps from a set $A$ (the global sections) to sets $A_x,A_y$ and $A_z$ (the restrictions to three disconnected ‘opens’).

\[
\xymatrix{& A \ar[rd] \ar[d] \ar[ld] & \\ A_x & A_y & A_z} \]

This seems to be a topos with a Boolean logic, but perhaps there are other 3-point connectivity spaces with a non-Boolean Heyting subobject classifier.

There’s Daniel Sibony‘s talk on “Mathématiques et inconscient”. Sibony is a French mathematician, turned philosopher and psychoanalyst, l’inconscient is an important concept in Lacan’s work.

Here’s a nice conversation between Daniel Sibony and Alain Connes on the notions of ‘time’ and ‘truth’.

In the second part (starting around 57.30) Connes brings up toposes whose underlying logic is much subtler than brute ‘true’ or ‘false’ statements. He discusses the presheaf topos on the additive monoid $\mathbb{N}_+$ which leads to statements which are ‘one step from the truth’, ‘two steps from the truth’ and so on. It is also the example Connes used in his talk Un topo sur les topos.

Alain Connes himself will also give a talk at the meeting, together with Patrick Gauthier-Lafaye, on “Un topos sur l’inconscient”.

It appears that Connes and Gauthier-Lafaye have written a book on the subject, A l’ombre de Grothendieck et de Lacan : un topos sur l’inconscient. Here’s the summary (G-translated):

“The authors present the relevance of the mathematical concept of topos, introduced by A. Grothendieck at the end of the 1950s, in the exploration of the structure of the unconscious.”

The book will be released on May 11th.

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Yet more topos news

Every topos has its own internal language, the so called Mitchell-Bénabou language, allowing us to speak about formulas and their truth values.

Sadly, Jean Bénabou died last week.

Here’s a nice interview with Bénabou (in French) on category theory, Grothendieck, logic, and a rant on plagiarism among topos theorists (starting at 1:00:16).

Yesterday, France Culture’s ‘La methode scientifique’ hosted Alain Connes, Laurent Lafforgue and Olivia Caramello in a special programme Grothendieck: la moisson (Grothendieck, the harvest), dedicated to the recent publication of ‘Récoltes et Semailles’.

An interesting item is ‘le reportage du jour’ by Céline Loozen in which she manages to have a look at the 60.000 pages of Grothendieck’s Lasserre notes, stocked in the cellars of the Librairie Alain Brieux, and talks to Jean-Bernard Gillot who is commissioned by Grothendieck’s son to appraise the work (starts at 36:40).

Perhaps the publication of ‘Récoltes et Semailles’ is part of a deal with the family to make these notes available, at last.

Towards the end of the programme Connes, Caramello and Lafforgue lament that topos theory is still not taken seriously by the mathematical community at large, whereas it is welcomed warmly by the engineers of Huawei.

In more topos news, I learn from the blog of Olivia Caramello, that Laurent Lafforgue is going to give an online course on toposes as ‘bridges’ at the University of Warwick, the first talk starts today at 14hrs London time.

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The hype cycle of an idea

These three ideas (re)surfaced over the last two decades, claiming to have potential applications to major open problems:

  • (2000) $\mathbb{F}_1$-geometry tries to view $\mathbf{Spec}(\mathbb{Z})$ as a curve over the field with one element, and mimic Weil’s proof of RH for curves over finite fields to prove the Riemann hypothesis.
  • (2012) IUTT, for Inter Universal Teichmuller Theory, the machinery behind Mochizuki’s claimed proof of the ABC-conjecture.
  • (2014) topos theory : Connes and Consani redirected their RH-attack using arithmetic sites, while Lafforgue advocated the use of Caramello’s bridges for unification, in particular the Langlands programme.

It is difficult to voice an opinion about the (presumed) current state of such projects without being accused of being either a believer or a skeptic, resorting to group-think or being overly critical.

We lack the vocabulary to talk about the different phases a mathematical idea might be in.

Such a vocabulary exists in (information) technology, the five phases of the Gartner hype cycle to represent the maturity, adoption, and social application of a certain technology :

  1. Technology Trigger
  2. Peak of Inflated Expectations
  3. Trough of Disillusionment
  4. Slope of Enlightenment
  5. Plateau of Productivity

This model can then be used to gauge in which phase several emerging technologies are, and to estimate the time it will take them to reach the stable plateau of productivity. Here’s Gartner’s recent Hype Cycle for emerging Artificial Intelligence technologies.



Picture from Gartner Hype Cycle for AI 2021

What might these phases be in the hype cycle of a mathematical idea?

  1. Technology Trigger: a new idea or analogy is dreamed up, marketed to be the new approach to that problem. A small group of enthusiasts embraces the idea, and tries to supply proper definitions and the very first results.
  2. Peak of Inflated Expectations: the idea spreads via talks, blogposts, mathoverflow and twitter, and now has enough visibility to justify the first conferences devoted to it. However, all this activity does not result in major breakthroughs and doubt creeps in.
  3. Trough of Disillusionment: the project ran out of steam. It becomes clear that existing theories will not lead to a solution of the motivating problem. Attempts by key people to keep the idea alive (by lengthy papers, regular meetings or seminars) no longer attract new people to the field.
  4. Slope of Enlightenment: the optimistic scenario. One abandons the original aim, ditches the myriad of theories leading nowhere, regroups and focusses on the better ideas the project delivered.

    A negative scenario is equally possible. Apart for a few die-hards the idea is abandoned, and on its way to the graveyard of forgotten ideas.

  5. Plateau of Productivity: the polished surviving theory has applications in other branches and becomes a solid tool in mathematics.

It would be fun so see more knowledgable people draw such a hype cycle graph for recent trends in mathematics.

Here’s my own (feeble) attempt to gauge where the three ideas mentioned at the start are in their cycles, and here’s why:

  • IUTT: recent work of Kirti Joshi, for example this, and this, and that, draws from IUTT while using conventional language and not making exaggerated claims.
  • $\mathbb{F}_1$: the preliminary programme of their seminar shows little evidence the $\mathbb{F}_1$-community learned from the past 20 years.
  • Topos: Developing more general theory is not the way ahead, but concrete examples may carry surprises, even though Gabriel’s topos will remain elusive.

Clearly, you don’t agree, and that’s fine. We now have a common terminology, and you can point me to results or events I must have missed, forcing me to redraw my graph.

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Chevalley’s circle of friends

Last week, Danielle Couty ArXiVed her paper Friendly views on Claude Chevalley (in French).

From the abstract: “We propose to follow the itinerary of Claude Chevalley during the last twenty years of his life, through the words of Jacques Roubaud, Denis Guedj and Alexander Grothendieck. Our perspective is that of their testimonies filled with friendship.”

Claude Chevalley was one of the founding fathers of Bourbaki. Two of the four pre-WW2 Bourbaki-congresses were held in “La Massoterie”, the Chevalley family domain in Chancay (see this post, update: later I learned from Liliane Beaulieu that the original house was destroyed by fire).

In 1938 he left for Princeton and stayed there during the war, making it impossible to return to a position in France for a very long time. Only in 1957 he could return to Paris where he led a seminar which proved to be essential for the development of algebraic groups and algebraic geometry.



Picture from N. Bourbaki, an interview with C. Chevalley

The Couty paper focusses on the post-1968 period in which Chevalley distanced himself from Bourbaki (some of its members, he thought, had become ‘mandarins’ and ‘reactionaires’), became involved with the ecological movement ‘Survivre et vivre’ and started up the maths department of a new university at Vincennes.

The paper is based on the recollections of three of his friends.

1. Jacques Roubaud is a French poet, writer and mathematician.

On this blog you may have run into Roubaud as the inventor of Bourbaki’s death announcement, and the writer of the book with title $\in$.

He’s also a member of Oulipo, a loose gathering of (mainly) French-speaking writers and mathematicians. Famous writers such as Georges Perec and Italo Calvino were also Oulipo-members (see also Ouilpo’s use of the Tohoku paper).

Chevalley introduced Roubaud (and others) to the game of Go. From Couty’s paper this quote from Roubaud (G-translated):

“. . . it turns out that he had learned to play go in Japan and then, in Paris, he could not find a player […] I played go with him […] and then at a certain moment , we thought, Pierre Lusson and myself, it would still be good to create circumstances such that Chevalley could have players. And so, we had a lot of ambition, we said to ourselves: “We’re going to write a treatise on go, and then lots of people will start playing go”. »

The resulting Go-book is A short treatise inviting the reader to discover the subtle art of Go. Here’s Georges Perec (left) and Jacques Roubaud playing a game.



Picture from Petit traite invitant a la decouverte de l’art subtil du Go

2. Denis Guedj was a French novelist, mathematician and historian of science professor, perhaps best known for his book The Parrot’s Theorem.

In May 1968, Guedj was a PhD-student of Jean-Paul Benzecri (the one defining God as the Alexandroff compactification of the univers), working in the building where ‘Le Comité de Grève’ installed itself. Here he met Chevalley. A Guedj-quote from Couty’s paper (G-translated):

“Claude Chevalley was one of the three professors of the Faculty of Science to commit himself totally to the adventure until the end, occupying the premises with the students on the Quai Saint-Bernard […] and sleeping there frequently . That’s where I met him.

We had taken possession of this universe which until then had only been a place of study and knowledge, and which, in the mildness of this month of May, had become a place of life, of a life wonderfully exhilarating. The college was ours. At night we walked down the aisles yet? lined with tall trees, entered the empty lecture halls, slept under the stars. Needless to say that at the beginning of the school year, in the fall of 1968, it was impossible for us to find our place in these undressed spaces from which the magic had withdrawn. »



Picture from Décès de l’écrivain et universitaire Denis Guedj

In June 2008, Guedj was one of the guests at the special edition of France Culture on the occasion of Grothendieck’s 80th birthday, Autour d’Alexandre Grothendieck.

3. Alexander Grothendieck, mathematician and misogynist, deified by some of today’s ‘mandarins’.

The paper by Danielle Couty may shed additional light on Grothendieck’s withdrawal from Bourbaki and mathematics as a whole. A G-translated Grothendieck quote from the paper:

“It was Chevalley who was one of the first, with Denis Guedj whom I also met through Survivre, to draw my attention to this ideology (they called it “meritocracy” or a name like that), and what there was in her of violence, of contempt. It was because of that, Chevalley told me […] that he could no longer bear the atmosphere in Bourbaki and had stopped setting foot there. »

Claude Chevalley stayed at Vincennes until his retirement in 1978, he died on June 28th 1984.

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Grothendieck stuff

January 13th, Gallimard published Grothendieck’s text Recoltes et Semailles in a fancy box containing two books.



Here’s a G-translation of Gallimard’s blurb:

“Considered the mathematical genius of the second half of the 20th century, Alexandre Grothendieck is the author of Récoltes et semailles, a kind of “monster” of more than a thousand pages, according to his own words. The mythical typescript, which opens with a sharp criticism of the ethics of mathematicians, will take the reader into the intimate territories of a spiritual experience after having initiated him into radical ecology.

In this literary braid, several stories intertwine, “a journey to discover a past; a meditation on existence; a picture of the mores of a milieu and an era (or the picture of the insidious and implacable shift from one era to another…); an investigation (almost police at times, and at others bordering on the swashbuckling novel in the depths of the mathematical megapolis…); a vast mathematical digression (which will sow more than one…); […] a diary ; a psychology of discovery and creation; an indictment (ruthless, as it should be…), even a settling of accounts in “the beautiful mathematical world” (and without giving gifts…)”.”

All literary events, great or small, are cause for the French to fill a radio show.

January 21st, ‘Le grand entretien’ on France Inter invited Cedric Villani and Jean-Pierre Bourguignon to talk about Grothendieck’s influence on mathematics (h/t Isar Stubbe).

The embedded YouTube above starts at 12:06, when Bourguignon describes Grothendieck’s main achievements.

Clearly, he starts off with the notion of schemes which, he says, proved to be decisive in the further development of algebraic geometry. Five years ago, I guess he would have continued mentioning FLT and other striking results, impossible to prove without scheme theory.

Now, he goes on saying that Grothendieck laid the basis of topos theory (“to define it, I would need not one minute and a half but a year and a half”), which is only now showing its first applications.

Grothendieck, Bourguignon goes on, was the first to envision the true potential of this theory, which we should take very seriously according to people like Lafforgue and Connes, and which will have applications in fields far from algebraic geometry.

Topos20 is spreading rapidly among French mathematicians. We’ll have to await further results before Topos20 will become a pandemic.

Another interesting fragment starts at 16:19 and concerns Grothendieck’s gribouillis, the 50.000 pages of scribblings found in Lasserre after his death.

Bourguignon had the opportunity to see them some time ago, and when asked to describe them he tells they are in ‘caisses’ stacked in a ‘libraire’.

Here’s a picture of these crates taken by Leila Schneps in Lasserre around the time of Grothendieck’s funeral.



If you want to know what’s in these notes, and how they ended up at that place in Paris, you might want to read this and that post.

If Bourguignon had to consult these notes at the Librairie Alain Brieux, it seems that there is no progress in the negotiations with Grothendieck’s children to make them public, or at least accessible.

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Poly

Following up on the deep learning and toposes-post, I was planning to do something on the logic of neural networks.

Prepping for this I saw David Spivak’s paper Learner’s Languages doing exactly that, but in the more general setting of ‘learners’ (see also the deep learning post).

And then … I fell under the spell of $\mathbf{Poly}$.

Spivak is a story-telling talent. A long time ago I copied his short story (actually his abstract for a talk) “Presheaf, the cobbler” in the Children have always loved colimits-post.

Last week, he did post Poly makes me happy and smart on the blog of the Topos Institute, which is another great read.

If this is way too ‘fluffy’ for you, perhaps you should watch his talk Poly: a category of remarkable abundance.

If you like (applied) category theory and have some days to waste, you can binge-watch all 15 episodes of the Poly-course Polynomial Functors: A General Theory of Interaction.

If you are more the reading-type, the 273 pages of the Poly-book will also kill a good number of your living hours.

Personally, I have no great appetite for category theory, I prefer to digest it in homeopathic doses. And, I’m allergic to co-terminology.

So then, how to define $\mathbf{Poly}$ for the likes of me?

$\mathbf{Poly}$, you might have surmised, is a category. So, we need ‘objects’ and ‘morphisms’ between them.

Any set $A$ has a corresponding ‘representable functor’ sending a given set $S$ to the set of all maps from $A$ to $S$
\[
y^A~:~\mathbf{Sets} \rightarrow \mathbf{Sets} \qquad S \mapsto S^A=Maps(A,S) \]
This looks like a monomial in a variable $y$ ($y$ for Yoneda, of course), but does it work?

What is $y^1$, where $1$ stands for the one-element set $\{ \ast \}$? $Maps(1,S)=S$, so $y^1$ is the identity functor sending $S$ to $S$.

What is $y^0$, where $0$ is the empty set $\emptyset$? Well, for any set $S$ there is just one map $\emptyset \rightarrow S$, so $y^0$ is the constant functor sending any set $S$ to $1$. That is, $y^0=1$.

Going from monomials to polynomials we need an addition. We add such representable functors by taking disjoint unions (finite or infinite), that is
\[
\sum_{i \in I} y^{A_i}~:~\mathbf{Sets} \rightarrow \mathbf{Sets} \qquad S \mapsto \bigsqcup_{i \in I} Maps(A_i,S) \]
If all $A_i$ are equal (meaning, they have the same cardinality) we use the shorthand $Iy^A$ for this sum.

The objects in $\mathbf{Poly}$ are exactly these ‘polynomial functors’
\[
p = \sum_{i \in I} y^{p[i]} \]
with all $p[i] \in \mathbf{Sets}$. Remark that $p(1)=I$ as for any set $A$ there is just one map to $1$, that is $y^A(1) = Maps(A,1) = 1$, and we can write
\[
p = \sum_{i \in p(1)} y^{p[i]} \]
An object $p \in \mathbf{Poly}$ is thus described by the couple $(p(1),p[-])$ with $p(1)$ a set, and a functor $p[-] : p(1) \rightarrow \mathbf{Sets}$ where $p(1)$ is now a category with objects the elements of $p(1)$ and no morphisms apart from the identities.

We can depict $p$ by a trimmed down forest, Spivak calls it the corolla of $p$, where the tree roots are the elements of $p(1)$ and the tree with root $i \in p(1)$ has one branch from the root for any element in $p[i]$. The corolla of $p=y^2+2y+1$ looks like



If $M$ is an $m$-dimensional manifold, then you might view its tangent bundle $TM$ set-theoretically as the ‘corolla’ of the polynomial functor $M y^{\mathbb{R}^m}$, the tree-roots corresponding to the points of the manifold, and the branches to the different tangent vectors in these points.

Morphisms in $\mathbf{Poly}$ are a bit strange. For two polynomial functors $p=(p(1),p[-])$ and $q=(q(1),q[-])$ a map $p \rightarrow q$ in $\mathbf{Poly}$ consists of

  • a map $\phi_1 : p(1) \rightarrow q(1)$ on the tree-roots in the right direction, and
  • for any $i \in p(1)$ a map $q[\phi_1(i)] \rightarrow p[i]$ on the branches in the opposite direction

In our manifold/tangentbundle example, a morphism $My^{\mathbb{R}^m} \rightarrow y^1$ sends every point $p \in M$ to the unique root of $y^1$ and the unique branch in $y^1$ picks out a unique tangent-vector for every point of $M$. That is, vectorfields on $M$ are very special (smooth) morphisms $Mu^{\mathbb{R}^m} \rightarrow y^1$ in $\mathbf{Poly}$.

A smooth map between manifolds $M \rightarrow N$, does not determine a morphism $My^{\mathbb{R}^m} \rightarrow N y^{\mathbb{R}^n}$ in $\mathbf{Poly}$ because tangent vectors are pushed forward, not pulled back.

If instead we view the cotangent bundle $T^*M$ as the corolla of the polynomial functor $My^{\mathbb{R}^m}$, then everything works well.

But then, I promised not to use co-terminology…

Another time I hope to tell you how $\mathbf{Poly}$ helps us to understand the logic of learners.

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Grothendieck talks

In 2017-18, the seminar Lectures grothendieckiennes took place at the ENS in Paris. Among the speakers were Alain Connes, Pierre Cartier, Laurent Lafforgue and Georges Maltsiniotis.

Olivia Caramello, who also contributed to the seminar, posts on her blog Around Toposes that the proceedings of this lectures series is now available from the SMF.

Olivia’s blogpost links also to the YouTube channel of the seminar. Several of these talks are well worth your time watching.

If you are at all interested in toposes and their history, and if you have 90 minutes to kill, I strongly recommend watching Colin McLarthy’s talk Grothendieck’s 1973 topos lectures:

In 1973, Grothendieck gave three lectures series at the Department of Mathematics of SUNY at Buffalo, the first on ‘Algebraic Geometry’, the second on ‘The Theory of Algebraic Groups’ and the third one on ‘Topos Theory’.

All of these Grothendieck talks were audio(!)-taped by John (Jack) Duskin, who kept and preserved them with the help of William Lawvere. They constitute more than 100 hours of rare recordings of Grothendieck.

This MathOverflow (soft) question links to this page stating:

“The copyright of all these recordings is that of the Department of Mathematics of SUNY at Buffalo to whose representatives, in particular Professors Emeritus Jack DUSKIN and Bill LAWVERE exceptional thanks are due both for the preservation and transmission of this historic archive, the only substantial archive of recordings of courses given by one of the greatest mathematicians of all time, whose work and ideas exercised arguably the most profound influence of any individual figure in shaping the mathematics of the second half od the 20th Century. The material which it is proposed to make available here, with their agreement, will form a mirror site to the principal site entitled “Grothendieck at Buffalo” (url: ).”

Sadly, the URL is still missing.

Fortunately, another answer links to the Grothendieck project Thèmes pour une Harmonie by Mateo Carmona. If you scroll down to the 1973-section, you’ll find there all of the recordings of these three Grothendieck series of talks!

To whet your appetite, here’s the first part of his talk on topos theory on April 4th, 1973:

For all subsequent recordings of his talks in the Topos Theory series on May 11th, May 18th, May 25th, May 30th, June 4th, June 6th, June 20th, June 27th, July 2nd, July 10th, July 11th and July 12th, please consult Mateo’s website (under section 1973).

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Huawei and topos theory

Apart from the initiatives I mentioned last time, Huawei set up a long term collaboration with the IHES, the Huawei Young Talents Program.

“Every year, the Huawei Young Talents Program will fund on average 7 postdoctoral fellowships that will be awarded by the Institute’s Scientific Council, only on the basis of scientific excellence. The fellows will collaborate with the Institute’s permanent professors and work on topics of their interest.”

Over the next ten years, Huawei will invest 5 million euros in this program, and an additional 1 million euros goes into the creation of the ‘Huawei Chair in Algebraic Geometry’. It comes as no particular surprise that the first chairholder is Laurent Lafforgue.

At the launch of this Young Talents Program in November 2020, Lafforgue gave a talk on The creative power of categories: History and some new perspectives.

The latter part of the talk (starting at 47:50) clarifies somewhat Huawei’s interest in topos theory, and what Lafforgue (and others) hope to get out of their collaboration with the telecom company.

Clearly, Huawei is interested in deep neural networks, and if you can convince them your expertise is useful in that area, perhaps they’ll trow some money at you.

Jean-Claude Belfiore, another mathematician turned Huaweian, is convinced topos theory is the correct tool to study DNNs. Here’s his Huawei-clip from which it is clear he was originally hired to improve Huawei’s polar code.

At the 2018 IHES-Topos conference he gave the talk Toposes for Wireless Networks: An idea whose time has come, and recently he arXived the paper Topos and Stacks of Deep Neural Networks, written jointly with Daniel Bennequin. Probably, I’ll come back to this paper another time, for now, the nForum has this page on it.

Towards the end of his talk, Lafforgue suggests the idea of creating an institute devoted to toposes and their applications, endorsed by IHES and supported by Huawei. Surely he knows that the Topos Institute already exists.

And, if you wonder why Huawei trows money at IHES rather than your university, I leave you with Lafforgue’s parting words:

“IHES professors are able to think and evaluate for themselves, whereas most mathematicians just follow ‘group thinking'”

Ouch!

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