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.

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.

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.

Do we need the sphere spectrum?

Last time I mentioned the talk “From noncommutative geometry to the tropical geometry of the scaling site” by Alain Connes, culminating in the canonical isomorphism (last slide of the talk)



Or rather, what is actually proved in his paper with Caterina Consani BC-system, absolute cyclotomy and the quantized calculus (and which they conjectured previously to be the case in Segal’s Gamma rings and universal arithmetic), is a canonical isomorphism between the $\lambda$-rings
\[
\mathbb{Z}[\mathbb{Q}/\mathbb{Z}] \simeq \mathbb{W}_0(\overline{\mathbb{S}}) \]
The left hand side is the integral groupring of the additive quotient-group $\mathbb{Q}/\mathbb{Z}$, or if you prefer, $\mathbb{Z}[\mathbf{\mu}_{\infty}]$ the integral groupring of the multiplicative group of all roots of unity $\mathbf{\mu}_{\infty}$.

The power maps on $\mathbf{\mu}_{\infty}$ equip $\mathbb{Z}[\mathbf{\mu}_{\infty}]$ with a $\lambda$-ring structure, that is, a family of commuting endomorphisms $\sigma_n$ with $\sigma_n(\zeta) = \zeta^n$ for all $\zeta \in \mathbf{\mu}_{\infty}$, and a family of linear maps $\rho_n$ induced by requiring for all $\zeta \in \mathbf{\mu}_{\infty}$ that
\[
\rho_n(\zeta) = \sum_{\mu^n=\zeta} \mu \]
The maps $\sigma_n$ and $\rho_n$ are used to construct an integral version of the Bost-Connes algebra describing the Bost-Connes sytem, a quantum statistical dynamical system.

On the right hand side, $\mathbb{S}$ is the sphere spectrum (an object from stable homotopy theory) and $\overline{\mathbb{S}}$ its ‘algebraic closure’, that is, adding all abstract roots of unity.

The ring $\mathbb{W}_0(\overline{\mathbb{S}})$ is a generalisation to the world of spectra of the Almkvist-ring $\mathbb{W}_0(R)$ defined for any commutative ring $R$, constructed from pairs $(E,f)$ where $E$ is a projective $R$-module of finite rank and $f$ an $R$-endomorphism on it. Addition and multiplication are coming from direct sums and tensor products of such pairs, with zero element the pair $(0,0)$ and unit element the pair $(R,1_R)$. The ring $\mathbb{W}_0(R)$ is then the quotient-ring obtained by dividing out the ideal consisting of all zero-pairs $(E,0)$.

The ring $\mathbb{W}_0(R)$ becomes a $\lambda$-ring via the Frobenius endomorphisms $F_n$ sending a pair $(E,f)$ to the pair $(E,f^n)$, and we also have a collection of linear maps on $\mathbb{W}_0(R)$, the ‘Verschiebung’-maps which send a pair $(E,f)$ to the pair $(E^{\oplus n},F)$ with
\[
F = \begin{bmatrix} 0 & 0 & 0 & \cdots & f \\
1 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & & \vdots \\
0 & 0 & 0 & \cdots & 1 \end{bmatrix} \]
Connes and Consani define a notion of modules and their endomorphisms for $\mathbb{S}$ and $\overline{\mathbb{S}}$, allowing them to define in a similar way the rings $\mathbb{W}_0(\mathbb{S})$ and $\mathbb{W}_0(\overline{\mathbb{S}})$, with corresponding maps $F_n$ and $V_n$. They then establish an isomorphism with $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$ such that the maps $(F_n,V_n)$ correspond to $(\sigma_n,\rho_n)$.

But, do we really have the go to spectra to achieve this?

All this reminds me of an old idea of Yuri Manin mentioned in the introduction of his paper Cyclotomy and analytic geometry over $\mathbb{F}_1$, and later elaborated in section two of his paper with Matilde Marcolli Homotopy types and geometries below $\mathbf{Spec}(\mathbb{Z})$.

Take a manifold $M$ with a diffeomorphism $f$ and consider the corresponding discrete dynamical system by iterating the diffeomorphism. In such situations it is important to investigate the periodic orbits, or the fix-points $Fix(M,f^n)$ for all $n$. If we are in a situation that the number of fixed points is finite we can package these numbers in the Artin-Mazur zeta function
\[
\zeta_{AM}(M,f) = exp(\sum_{n=1}^{\infty} \frac{\# Fix(M,f^n)}{n}t^n) \]
and investigate the properties of this function.

To connect this type of problem to Almkvist-like rings, Manin considers the Morse-Smale dynamical systems, a structural stable diffeomorphism $f$, having a finite number of non-wandering points on a compact manifold $M$.



From Topological classification of Morse-Smale diffeomorphisms on 3-manifolds

In such a situation $f_{\ast}$ acts on homology $H_k(M,\mathbb{Z})$, which are free $\mathbb{Z}$-modules of finite rank, as a matrix $M_f$ having only roots of unity as its eigenvalues.

Manin argues that this action is similar to the action of the Frobenius on etale cohomology groups, in which case the eigenvalues are Weil numbers. That is, one might view roots of unity as Weil numbers in characteristic one.

Clearly, all relevant data $(H_k(M,\mathbb{Z}),f_{\ast})$ belongs to the $\lambda$-subring of $\mathbb{W}_0(\mathbb{Z})$ generated by all pairs $(E,f)$ such that $M_f$ is diagonalisable and all its eigenvalues are either $0$ or roots of unity.

If we denote for any ring $R$ by $\mathbb{W}_1(R)$ this $\lambda$-subring of $\mathbb{W}_0(R)$, probably one would obtain canonical isomorphisms

– between $\mathbb{W}_1(\mathbb{Z})$ and the invariant part of the integral groupring $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$ for the action of the group $Aut(\mathbb{Q}/\mathbb{Z}) = \widehat{\mathbb{Z}}^*$, and

– between $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$ and $\mathbb{W}_1(\mathbb{Z}(\mathbf{\mu}_{\infty}))$ where $\mathbb{Z}(\mathbf{\mu}_{\infty})$ is the ring obtained by adjoining to $\mathbb{Z}$ all roots of unity.

Alain Connes on his RH-project

In recent months, my primary focus was on teaching and family matters, so I make advantage of this Christmas break to catch up with some of the things I’ve missed.

Peter Woit’s blog alerted me to the existence of the (virtual) Lake Como-conference, end of september: Unifying themes in Geometry.

In Corona times, virtual conferences seem to sprout up out of nowhere, everywhere (zero costs), giving us an inflation of YouTubeD talks. I’m always grateful to the organisers of such events to provide the slides of the talks separately, as the generic YouTubeD-talk consists merely in reading off the slides.

Allow me to point you to one of the rare exceptions to this rule.

When I downloaded the slides of Alain Connes’ talk at the conference From noncommutative geometry to the tropical geometry of the scaling site I just saw a collage of graphics from his endless stream of papers with Katia Consani, and slides I’d seen before watching several of his YouTubeD-talks in recent years.

Boy, am I glad I gave Alain 5 minutes to convince me this talk was different.

For the better part of his talk, Alain didn’t just read off the slides, but rather tried to explain the thought processes that led him and Katia to move on from the results on this slide to those on the next one.

If you’re pressed for time, perhaps you might join in at 49.34 into the talk, when he acknowledges the previous (tropical) approach ran out of steam as they were unable to define any $H^1$ properly, and how this led them to ‘absolute’ algebraic geometry, meaning over the sphere spectrum $\mathbb{S}$.

Sadly, for some reason Alain didn’t manage to get his final two slides on screen. So, in this case, the slides actually add value to the talk…