Category: geometry

If you have neither the time nor energy to watch more than one interview or talk about Grothendieck’s life and mathematics, may I suggest to spare that privilege for Leila Schneps’ talk on ‘Le génie de Grothendieck’ in the ‘Thé & Sciences’ series at the Salon Nun in Paris.

I was going to add some ‘relevant’ time slots after the embedded YouTube-clip below, but I really think it is better to watch Leila’s interview in its entirety. Enjoy!

Last time, we’ve seen that the first time ‘schemes’ were introduced was in ‘La Tribu’ (the internal Bourbaki-account of their congresses) of the May-June 1955 congress in Chicago.

Here, we will focus on the events leading up to that event. If you always thought Grothendieck invented the word ‘schemes’, here’s what Colin McLarty wrote:

“A story says that in a Paris café around 1955 Grothendieck asked his friends “what is a scheme?”. At the time only an undefined idea of “schéma” was current in Paris, meaning more or less whatever would improve on Weil’s foundations.” (McLarty in The Rising Sea)

What were Weil’s foundations of algebraic geometry?

Well, let’s see how Weil defined an affine variety over a field $k$. First you consider a ‘universal field’ $K$ containing $k$, that is, $K$ is an algebraically closed field of infinite transcendence degree over $k$. A point of $n$-dimensional affine space is an $n$-tuple $x=(x_1,\dots,x_n) \in K^n$. For such a point $x$ you consider the field $k(x)$ which is the subfield of $K$ generated by $k$ and the coordinates $x_i$ of $x$.

Alternatively, the field $k(x)$ is the field of fractions of the affine domain $R=k[z_1,\dots,z_n]/I$ where $I$ is the prime ideal of all polynomials $f \in k[z_1,\dots,z_n]$ such that $f(x) = f(x_1,\dots,x_n)=0$.

An affine $k$-variety $V$ is associated to a ‘generic point’ $x=(x_1,\dots,x_n)$, meaning that the field $k(x)$ is a ‘regular extension’ of $k$ (that is, for all field-extensions $k’$ of $k$, the tensor product $k(x) \otimes_k k’$ does not contain zero-divisors.

The points of $V$ are the ‘specialisations’ of $x$, that is, all points $y=(y_1,\dots,y_n)$ such that $f(y_1,\dots,y_n)=0$ for all $f \in I$.

Perhaps an example? Let $k = \mathbb{Q}$ and $K=\mathbb{C}$ and take $x=(i,\pi)$ in the affine plane $\mathbb{C}^2$. What is the corresponding prime ideal $I$ of $\mathbb{Q}[z_1,z_2]$? Well, $i$ is a solution to $z_1^2+1=0$ whereas $\pi$ is transcendental over $\mathbb{Q}$, so $I=(z_1^2+1)$ and $R=\mathbb{Q}[z_1,z_2]/I= \mathbb{Q}(i)[z_2]$.

Is $x=(i,\pi)$ a generic point? Well, suppose it were, then the points of the corresponding affine variety $V$ would be all couples $(\pm i, \lambda)$ with $\lambda \in \mathbb{C}$ which is the union of two lines in $\mathbb{C}^2$. But then $i \otimes 1 + 1 \otimes i$ is a zero-divisor in $\mathbb{Q}(x) \otimes_{\mathbb{Q}} \mathbb{Q}(i)$. So no, it is not a generic point over $\mathbb{Q}$ and does not define an affine $\mathbb{Q}$-variety.

If we would have started with $k=\mathbb{Q}(i)$, then $x=(i,\pi)$ is generic and the corresponding affine variety $V$ consists of all points $(i,\lambda) \in \mathbb{C}^2$.

If this is new to you, consider yourself lucky to be young enough to have learned AG from Fulton’s Algebraic curves, or Hartshorne’s chapter 1 if you were that ambitious.

By 1955, Serre had written his FAC, and Bourbaki had developed enough commutative algebra to turn His attention to algebraic geometry.

La Ciotat congress (February 27th – March 6th, 1955)

With a splendid view on the mediterranean, a small group of Bourbaki members (Henri Cartan (then 51), with two of his former Ph.D. students: Jean-Louis Koszul (then 34), and Jean-Pierre Serre (then 29, and fresh Fields medaillist), Jacques Dixmier (then 31), and Pierre Samuel (then 34), a former student of Zariski’s) discussed a previous ‘Rapport de Geometrie Algebrique'(no. 206) and arrived at some unanimous decisions:

1. Algebraic varieties must be sets of points, which will not change at every moment.
2. One should include ‘abstract’ varieties, obtained by gluing (fibres, etc.).
3. All necessary algebra must have been previously proved.
4. The main application of purely algebraic methods being characteristic p, we will hide nothing of the unpleasant phenomena that occur there.

(Henri Cartan and Jean-Pierre Serre, photo by Paul Halmos)

The approach the propose is clearly based on Serre’s FAC. The points of an affine variety are the maximal ideals of an affine $k$-algebra, this set is equipped with the Zariski topology such that the local rings form a structure sheaf. Abstract varieties are then constructed by gluing these topological spaces and sheaves.

At the insistence of the ‘specialistes’ (Serre, and Samuel who had just written his book ‘Méthodes d’algèbre abstraite en géométrie algébrique’) two additional points are adopted, but with some hesitation. The first being a jibe at Weil:
1. …The congress, being a little disgusted by the artificiality of the generic point, does not want $K$ to be always of infinite transcendent degree over $k$. It admits that generic points are convenient in certain circumstances, but refuses to see them put to all the sauces: one could speak of a coordinate ring or of a functionfield without stuffing it by force into $K$.
2. Trying to include the arithmetic case.

The last point was problematic as all their algebras were supposed to be affine over a field $k$, and they wouldn’t go further than to allow the overfield $K$ to be its algebraic closure. Further, (and this caused a lot of heavy discussions at coming congresses) they allowed their varieties to be reducible.

The Chicago congress (May 30th – June 2nd 1955)

Apart from Samuel, a different group of Bourbakis gathered for the ‘second Caucus des Illinois’ at Eckhart Hall, including three founding members Weil (then 49), Dixmier (then 49) and Chevalley (then 46), and two youngsters, Armand Borel (then 32) and Serge Lang (then 28).

Their reaction to the La Ciotat meeting (the ‘congress of the public bench’) was swift:

(page 1) : “The caucus discovered a public bench near Eckhart Hall, but didn’t do much with it.”
(page 2) : “The caucus did not judge La Ciotat’s plan beyond reproach, and proposed a completely different plan.”

They wanted to include the arithmetic case by defining as affine scheme the set of all prime ideals (or rather, the localisations at these prime ideals) of a finitely generated domain over a Dedekind domain. They continue:

(page 4) : “The notion of a scheme covers the arithmetic case, and is extracted from the illustrious works of Nagata, themselves inspired by the scholarly cogitations of Chevalley. This means that the latter managed to sell all his ideas to the caucus. The Pope of Chicago, very happy to be able to reject very far projective varieties and Chow coordinates, willingly rallied to the suggestions of his illustrious colleague. However, we have not attempted to define varieties in the arithmetic case. Weil’s principle is that it is unclear what will come out of Nagata’s tricks, and that the only stable thing in arithmetic theory is reduction modulo $p$ a la Shimura.”

“Contrary to the decisions of La Ciotat, we do not want to glue reducible stuff, nor call them varieties. … We even decide to limit ourselves to absolutely irreducible varieties, which alone will have the right to the name of varieties.”

The insistence on absolutely irreducibility is understandable from Weil’s perspective as only they will have a generic point. But why does he go along with Chevalley’s proposal of an affine scheme?

In Weil’s approach, a point of the affine variety $V$ determined by a generic point $x=(x_1,\dots,x_n)$ determines a prime ideal $Q$ of the domain $R=k[x_1,\dots,x_n]$, so Chevalley’s proposal to consider all prime ideals (rather than only the maximal ideals of an affine algebra) seems right to Weil.

However in Weil’s approach there are usually several points corresponding to the same prime ideal $Q$ of $R$, namely all possible embeddings of the ring $R/Q$ in that huge field $K$, so whenever $R/Q$ is not algebraic over $k$, there are infinitely Weil-points of $V$ corresponding to $Q$ (whence the La Ciotat criticism that points of a variety were not supposed to change at every moment).

According to Ralf Krömer in his book Tool and Object – a history and philosophy of category theory this shift from Weil-points to prime ideals of $R$ may explain Chevalley’s use of the word ‘scheme’:

(page 164) : “The ‘scheme of the variety’ denotes ‘what is invariant in a variety’.”

Another time we will see how internal discussion influenced the further Bourbaki congresses until Grothendieck came up with his ‘hyperplan’.

Wikipedia claims:

“The word scheme was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski’s ideas.”

and refers to the lecture by Chevalley ‘Les schemas’, given on December 12th, 1955 at the ENS-based ‘Seminaire Henri Cartan’ (in fact, that year it was called the Cartan-Chevalley seminar, and the next year Chevalley set up his own seminar at the ENS).

Items recently added to the online Bourbaki Archive give us new information on time and place of the birth of the concept of schemes.

From May 30th till June 2nd 1955 the ‘second caucus des Illinois’ Bourbaki-congress was held in ‘le grand salon d’Eckhart Hall’ at the University of Chicago (Weil’s place at that time).

Only six of the Bourbaki members were present:

• Jean Dieudonne (then 49), the scribe of the Bourbaki-gang.
• Andre Weil (then 49), called ‘Le Pape de Chicago’ in La Tribu, and responsible for his ‘Foundations of Algebraic Geometry’.
• Claude Chevalley (then 46), who wanted a better, more workable version of algebraic geometry. He was just nominated professor at the Sorbonne, and was prepping for his seminar on algebraic geometry (with Cartan) in the fall.
• Pierre Samuel (then 34), who studied in France but got his Ph.D. in 1949 from Princeton under the supervision of Oscar Zariski. He was a Bourbaki-guinea pig in 1945, and from 1947 attended most Bourbaki congresses. He just got his book Methodes d’algebre abstraite en geometrie algebrique published.
• Armand Borel (then 32), a Swiss mathematician who was in Paris from 1949 and obtained his Ph.D. under Jean Leray before moving on to the IAS in 1957. He was present at 9 of the Bourbaki congresses between 1955 and 1960.
• Serge Lang (then 28), a French-American mathematician who got his Ph.D. in 1951 from Princeton under Emil Artin. In 1955, he just got a position at the University of Chicago, which he held until 1971. He attended 7 Bourbaki congresses between 1955 and 1960.

The issue of La Tribu of the Eckhart-Hall congress is entirely devoted to algebraic geometry, and starts off with a bang:

“The Caucus did not judge the plan of La Ciotat above all reproaches, and proposed a completely different plan.

I – Schemes
II – Theory of multiplicities for schemes
III – Varieties
IV – Calculation of cycles
V – Divisors
VI – Projective geometry
etc.”

In the spring of that year (February 27th – March 6th, 1955) a Bourbaki congress was held ‘Chez Patrice’ at La Ciotat, hosting a different group of Bourbaki members (Samuel was the singleton intersection) : Henri Cartan (then 51), Jacques Dixmier (then 31), Jean-Louis Koszul (then 34), and Jean-Pierre Serre (then 29, and fresh Fields medaillist).

In the La Ciotat-Tribu,nr. 35 there are also a great number of pages (page 14 – 25) used to explain a general plan to deal with algebraic geometry. Their summary (page 3-4):

“Algebraic Geometry : She has a very nice face.

Chap I : Algebraic varieties
Chap II : The rest of Chap. I
Chap III : Divisors
Chap IV : Intersections”

There’s much more to say comparing these two plans, but that’ll be for another day.

We’ve just read the word ‘schemes’ for the first (?) time. That unnumbered La Tribu continues on page 3 with “where one explains what a scheme is”:

So, what was their first idea of a scheme?

Well, you had your favourite Dedekind domain $D$, and you considered all rings of finite type over $D$. Sorry, not all rings, just all domains because such a ring $R$ had to have a field of fractions $K$ which was of finite type over $k$ the field of fractions of your Dedekind domain $D$.

They say that Dedekind domains are the algebraic geometrical equivalent of fields. Yeah well, as they only consider $D$-rings the geometric object associated to $D$ is the terminal object, much like a point if $D$ is an algebraically closed field.

But then, what is this geometric object associated to a domain $R$?

In this stage, still under the influence of Weil’s focus on valuations and their specialisations, they (Chevalley?) take as the geometric object $\mathbf{Spec}(R)$, the set of all ‘spots’ (taches), that is, local rings in $K$ which are the localisations of $R$ at prime ideals. So, instead of taking the set of all prime ideals, they prefer to take the set of all stalks of the (coming) structure sheaf.

But then, speaking about sheaves is rather futile as there is no trace of any topology on this set, then. Also, they make a big fuss about not wanting to define a general schema by gluing together these ‘affine’ schemes, but then they introduce a notion of ‘apparentement’ of spots which basically means the same thing.

It is still very early days, and there’s a lot more to say on this, but if no further documents come to light, I’d say that the birthplace of ‘schemes’, that is , the place where the first time there was a documented consensus on the notion, is Eckhart Hall in Chicago.

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.

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. 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. Until now, we’ve looked at actions of groups (such as the$T/I$or$PLR$-group) or (transformation) monoids (such as Noll’s monoid) on special sets of musical elements, in particular the twelve pitch classes$\mathbb{Z}_{12}$, or the set of all$24$major and minor chords. Elephant-lovers recognise such settings as objects in the presheaf topos on the one-object category$\mathbf{M}$corresponding to the group or monoid. That is, we look at contravariant functors$\mathbf{M} \rightarrow \mathbf{Sets}$. Last time we’ve encountered the ‘Cube Dance Grap’ which depicts a particular relation among the major, minor, and augmented chords. Recall that the twelve major chords (numbered for$1$to$12$) are the ordered triples of tones in$\mathbb{Z}_{12}$of the form$(n,n+4,n+7)$(such as the triangle on the left). The twelve minor chords (numbered from$13$to$24$) are the ordered triples$(n,n+3,n+7)$(such as the middle triangle). The four augmented chords (numbered from$25$to$28$) are the triples of the form$(n,n+4,n+8)$(such as the rightmost triangle). The Cube Dance Graph relates two of these chords when they share two tones (pitch classes) whereas the remaining tones differ by a halftone. Picture modified from this post. We can separate this symmetric binary relation into three sub-relations: the extension of the$P$and$L$-operations on major and minor chords to the augmented ones (these are transformations), and the remaining relation$U$which connects the major and minor chords to the augmented chords (and which is not a transformation). Binary relations on the same set can be composed, so we get a monoid$\mathbf{M}$generated by the three relations$P,L$and$U$. The action of$\mathbf{M}$on the$28$chords no longer gives us an ordinary presheaf (because$U$is not a transformation), but a relational presheaf as in the paper On the use of relational presheaves in transformational music theory by Alexandre Popoff. That is, the action defines a contravariant functor$\mathbf{M} \rightarrow \mathbf{Rel}$where$\mathbf{Rel}$is the category (actually a$2$-category) of sets, but with binary relations as morphisms (that is,$Hom(X,Y)$is all subsets of$X \times Y$), and the natural notion of composition of such relations. The$2$-morphism between relations is that of inclusion. To compute with monoids generated by binary relations in GAP one needs to download, compile and load the package semigroups, and to represent the binary relations as partitioned binary relations as in the paper by Martin and Mazorchuk. This is a bit more complicated than working with ordinary transformations:  P:=PBR([[-13],[-14],[-15],[-16],[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-9],[-10],[-11],[-12],[-25],[-26],[-27],[-28]],[[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[25],[26],[27],[28]]); L:=PBR([[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-13],[-14],[-15],[-16],[-9],[-10],[-11],[-12],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-25],[-26],[-27],[-28]],[[17],[18],[19],[20],[21],[22],[23],[24],[13],[14],[15],[16],[9],[10],[11],[12],[1],[2],[3],[4],[5],[6],[7],[8],[25],[26],[27],[28]]); U:=PBR([[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-17,-21,-13,-4,-8,-12],[-5,-1,-9,-18,-14,-22],[-2,-6,-10,-15,-23,-19],[-24,-16,-20,-11,-3,-7]],[[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[25],[25],[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[17,21,13,4,8,12],[5,1,9,18,14,22],[2,6,10,15,23,19],[24,16,20,11,3,7]]);  But then, GAP quickly tells us that$\mathbf{M}$is a monoid consisting of$40$elements.  gap> M:=Semigroup([P,L,U]); gap> Size(M); 40  The Semigroups-package can also compute Green’s relations and tells us that there are seven such$R$-classes, four consisting of$6$elements, two of four, and one of eight elements. These are also visible in the Cayley graph, exactly as last time. Or, if you prefer the cleaner picture of the Cayley graph from the paper Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis by Popoff, Andreatta and Ehresmann. This then allows us to compute the Heyting algebra of the subobject classifier, and all the Grothendieck topologies, at least for the ordinary presheaf topos of$\mathbf{M}$-sets, not for the relational presheaves we need here. We can consider the same binary relation on the larger set of triads when we add the suspended triads. These are the ordered triples in$\mathbb{Z}_{12}$of the form$(n,n+5,n+7)$, as in the rightmost triangle below. There are twelve suspended chords (numbered from$29$to$40$), so we now have a binary relation$T$on a set of$40$triads. The relation$T$is too coarse, and the art is to subdivide$T$is disjoint sub-relations which are musically significant, between major and minor triads, between major/minor and augmented triads, and so on. For each such partition we can then consider the monoids generated by these sub-relations. In his paper, Popoff suggest relevant sub-relations$P,L,T_U,T_V$and$T_U \cup T_V$of$T$which in our numbering of the$40$chords can be represented by these PBR’s (assuming I made no mistakes…ADDED march 24th: I did make a mistake in the definition of L, see comment by Alexandre Popoff, below the corect L):  P:=PBR([[-13],[-14],[-15],[-16],[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-9],[-10],[-11],[-12],[-25],[-26],[-27],[-28],[-36],[-37],[-38],[-39],[-40],[-29],[-30],[-31],[-32],[-33],[-34],[-35]],[[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[25],[26],[27],[28],[34],[35],[36],[37],[38],[39],[40],[29],[30],[31],[32],[33]]); L:=PBR([[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-13],[-14],[-15],[-16],[-9],[ -10],[-11],[-12],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-25],[-26],[-27],[-28],[-29], [-30],[-31],[-32],[-33],[-34],[-35],[-36],[-37],[-38],[-39],[-40]],[[17], [18], [19], [ 20],[21],[22],[23],[24],[13],[14],[15],[16],[9],[10],[11],[12],[1],[2],[3],[4],[5], [6], [7],[8],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40] ]); TU:=PBR([[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-4,-8,-12,-13,-17,-21],[-1,-5,-9,-14,-18,-22],[-2,-6,-10,-15,-19,-23],[-3,-7,-11,-16,-20,-24],[],[],[],[],[],[],[],[],[],[],[],[]],[[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[25],[25],[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[4,8,12,13,17,21],[1,5,9,14,18,22],[2,6,10,15,19,23],[3,7,11,16,20,24],[],[],[],[],[],[],[],[],[],[],[],[]]); TV:=PBR([[-29],[-30],[-31],[-32],[-33],[-34],[-35],[-36],[-37],[-38],[-39],[-40],[-36],[-37],[-38],[-39],[-40],[-29],[-30],[-31],[-32],[-33],[-34],[-35],[],[],[],[],[-1,-18],[-2,-19],[-3,-20],[-4,-21],[-5,-22],[-6,-23],[-7,-24],[-8,-13],[-9,-14],[-10,-15],[-11,-16],[-12,-17]],[[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40],[36],[37],[38],[39],[40],[29],[30],[31],[32],[33],[34],[35],[],[],[],[],[1,18],[2,19],[3,20],[4,21],[5,22],[6,23],[7,24],[8,13],[9,14],[10,15],[11,16],[12,17]]); TUV:=PBR([[-26,-29],[-27,-30],[-28,-31],[-25,-32],[-26,-33],[-27,-34],[-28,-35],[-25,-36],[-26,-37],[-27,-38],[-28,-39],[-25,-40],[-25,-36],[-26,-37],[-27,-38],[-28,-39],[-25,-40],[-26,-29],[-27,-30],[-28,-31],[-25,-32],[-26,-33],[-27,-34],[-28,-35],[-4,-8,-12,-13,-17,-21],[-1,-5,-9,-14,-18,-22],[-2,-6,-10,-15,-19,-23],[-3,-7,-11,-16,-20,-24],[-1,-18],[-2,-19],[-3,-20],[-4,-21],[-5,-22],[-6,-23],[-7,-24],[-8,-13],[-9,-14],[-10,-15],[-11,-16],[-12,-17]],[[26,29],[27,30],[28,31],[25,32],[26,33],[27,34],[28,35],[25,36],[26,37],[27,38],[28,39],[25,40],[25,36],[26,37],[27,38],[28,39],[25,40],[26,29],[27,30],[28,31],[25,32],[26,33],[27,34],[28,35],[4,8,12,13,17,21],[1,5,9,14,18,22],[2,6,10,15,19,23],[3,7,11,16,20,24],[1,18],[2,19],[3,20],[4,21],[5,22],[6,23],[7,24],[8,13],[9,14],[10,15],[11,16],[12,17]]);  The resulting monoids are huge:  gap> G:=Semigroup([P,L,TU,TV]); gap> Size(G); 473293 gap> H:=Semigroup([P,L,TUV]); gap> Size(H); 994624  In Popoff’s paper these monoids have sizes respectively$473,293$and$994,624$. Strangely, the offset is in both cases$144=12^2$. (Added march 24: with the correct L I get the same sizes as in Popoff’s paper). Perhaps we should try to transform such relational presheaves to ordinary presheaves. One approach is to use the Grothendieck construction and associate to a set with such a relational monoid action a directed graph, coloured by the elements of the monoid. That is, an object in the presheaf topos of the category $\xymatrix{C & E \ar[l]^c \ar@/^2ex/[r]^s \ar@/_2ex/[r]_t & V}$ and then we should consider the slice topos over the one-vertex bouquet graph with one loop for each element in the monoid. If you want to have more details on the musical side of things, for example if you want to know what the opening twelve chords of “Take a Bow” by Muse have to do with the Cube Dance graph, here are some more papers: A categorical generalization of Klumpenhouwer networks, A. Popoff, M. Andreatta and A. Ehresmann. From K-nets to PK-nets: a categorical approach, A. Popoff, M. Andreatta and A. Ehresmann. From a Categorical Point of View: K-Nets as Limit Denotators, G. Mazzola and M. Andreatta. Last time, we’ve viewed major and minor triads (chords) as inscribed triangles in a regular$12$-gon. If we move clockwise along the$12$-gon, starting from the endpoint of the longest edge (the root of the chord, here the$0$-vertex) the edges skip$3,2$and$4$vertices (for a major chord, here on the left the major$0$-chord) or$2,3$and$4$vertices (for a minor chord, here on the right the minor$0$-chord). The symmetries of the$12$-gon, the dihedral group$D_{12}$, act on the$24$major- and minor-chords transitively, preserving the type for rotations, and interchanging majors with minors for reflections. Mathematical Music Theoreticians (MaMuTh-ers for short) call this the$T/I$-group, and view the rotations of the$12$-gon as transpositions$T_k : x \mapsto x+k~\text{mod}~12$, and the reflections as involutions$I_k : x \mapsto -x+k~\text{mod}~12$. Note that the elements of the$T/I$-group act on the vertices of the$12$-gon, from which the action on the chord-triangles follows. There is another action on the$24$major and minor chords, mapping a chord-triangle to its image under a reflection in one of its three sides. Note that in this case the reflection$I_k$used will depend on the root of the chord, so this action on the chords does not come from an action on the vertices of the$12$-gon. There are three such operations: (pictures are taken from Alexandre Popoff’s blog, with the ‘funny names’ removed) The$P$-operation is reflection in the longest side of the chord-triangle. As the longest side is preserved,$P$interchanges the major and minor chord with the same root. The$L$-operation is refection in the shortest side. This operation interchanges a major$k$-chord with a minor$k+4~\text{mod}~12$-chord. Finally, the$R$-operation is reflection in the middle side. This operation interchanges a major$k$-chord with a minor$k+9~\text{mod}~12$-chord. From this it is already clear that the group generated by$P$,$L$and$R$acts transitively on the$24$major and minor chords, but what is this$PLR$-group? If we label the major chords by their root-vertex$1,2,\dots,12$(GAP doesn’t like zeroes), and the corresponding minor chords$13,14,\dots,24$, then these operations give these permutations on the$24$chords:  P:=(1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) L:=(1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16) R:=(1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)  Then GAP gives us that the$PLR$-group is again isomorphic to$D_{12}$:  gap> G:=Group(P,L,R);; gap> Size(G); 24 gap> IsDihedralGroup(G); true  In fact, if we view both the$T/I$-group and the$PLR$-group as subgroups of the symmetric group$Sym(24)$via their actions on the$24$major and minor chords, these groups are each other centralizers! That is, the$T/I$-group and$PLR$-group are dual to each other. For more on this, there’s a beautiful paper by Alissa Crans, Thomas Fiore and Ramon Satyendra: Musical Actions of Dihedral Groups. What does this new MaMuTh info learns us more about our Elephant, the Topos of Triads, studied by Thomas Noll? Last time we’ve seen the eight element triadic monoid$T$of all affine maps preserving the three tones$\{ 0,4,7 \}$of the major$0$-chord, computed the subobject classified$\Omega$of the corresponding topos of presheaves, and determined all its six Grothendieck topologies, among which were these three: Why did we label these Grothendieck topologies (and corresponding elements of$\Omega$) by$P$,$L$and$R$? We’ve seen that the sheafification of the presheaf$\{ 0,4,7 \}$in the triadic topos under the Grothendieck topology$j_P$gave us the sheaf$\{ 0,3,4,7 \}$, and these are the tones of the major$0$-chord together with those of the minor$0$-chord, that is the two chords in the$\langle P \rangle$-orbit of the major$0$-chord. The group$\langle P \rangle$is the cyclic group$C_2$. For the sheafication with respect to$j_L$we found the$T$-set$\{ 0,3,4,7,8,11 \}$which are the tones of the major and minor$0$-,$4$-, and$8$-chords. Again, these are exactly the six chords in the$\langle P,L \rangle$-orbit of the major$0$-chord. The group$\langle P,L \rangle$is isomorphic to$Sym(3)$. The$j_R$-topology gave us the$T$-set$\{ 0,1,3,4,6,7,9,10 \}$which are the tones of the major and minor$0$-,$3$-,$6$-, and$9$-chords, and lo and behold, these are the eight chords in the$\langle P,R \rangle$-orbit of the major$0$-chord. The group$\langle P,R \rangle$is the dihedral group$D_4$. More on this can be found in the paper Commuting Groups and the Topos of Triads by Thomas Fiore and Thomas Noll. The operations$P$,$L$and$R$on major and minor chords are reflexions in one side of the chord-triangle, so they preserve two of the three tones. There’s a distinction between the$P$and$L$operations and$R$when it comes to how the third tone changes. Under$P$and$L$the third tone changes by one halftone (because the corresponding sides skip an even number of vertices), whereas under$R$the third tone changes by two halftones (a full tone), see the pictures above. The$\langle P,L \rangle = Sym(3)$subgroup divides the$24$chords in four orbits of six chords each, three major chords and their corresponding minor chords. These orbits consist of the •$0$-,$4$-, and$8$-chords (see before) •$1$-,$5$-, and$9$-chords •$2$-,$6$-, and$10$-chords •$3$-,$7$-, and$11$-chords and we can view each of these orbits as a cycle tracing six of the eight vertices of a cube with one pair of antipodal points removed. These four ‘almost’ cubes are the NE-, SE-, SW-, and NW-regions of the Cube Dance Graph, from the paper Parsimonious Graphs by Jack Douthett and Peter Steinbach. To translate the funny names to our numbers, use this dictionary (major chords are given by a capital letter): The four extra chords (at the N, E, S, and P places) are augmented triads. They correspond to the triads$(0,4,8),~(1,5,9),~(2,6,10)$and$(3,7,11)$. That is, two triads are connected by an edge in the Cube Dance graph if they share two tones and differ by an halftone in the third tone. This graph screams for a group or monoid acting on it. Some of the edges we’ve already identified as the action of$P$and$L$on the$24$major and minor triads. Because the triangle of an augmented triad is equilateral, we see that they are preserved under$P$and$L$. But what about the edges connecting the regular triads to the augmented ones? If we view each edge as two directed arrows assigned to the same operation, we cannot do this with a transformation because the operation sends each augmented triad to six regular triads. Alexandre Popoff, Moreno Andreatta and Andree Ehresmann suggest in their paper Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis that one might use a monoid generated by relations, and they show that there is such a monoid with$40$elements acting on the Cube Dance graph. Popoff claims that usual presheaf toposes, that is contravariant functors to$\mathbf{Sets}$are not enough to study transformational music theory. He suggest to use instead functors to$\mathbf{Rel}$, that is Sets with as the morphisms binary relations, and their compositions. Another Elephant enters the room… (to be continued) Here, MaMuTh stands for Mathematical Music Theory which analyses the pitch, timing, and structure of works of music. The Elephant is the nickname for the ‘bible’ of topos theory, Sketches of an Elephant: A Topos Theory Compendium, a two (three?) volume book, written by Peter Johnstone. How can we get as quickly as possible from the MaMuth to the Elephant, musical illiterates such as myself? What Mamuth-ers call a pitch class (sounds that are a whole number of octaves apart), is for us a residue modulo$12$, as an octave is usually divided into twelve (half)tones. We’ll just denote them by numbers from$0$to$11$, or view them as the vertices of a regular$12$-gon, and forget the funny names given to them, as there are several such encodings, and we don’t know a$G$from a$D\#$. Our regular$12$-gon has exactly$24$symmetries. Twelve rotations, which they call transpositions, given by the affine transformations $T_k~:~x \mapsto x+k~\text{mod}~12$ and twelve reflexions, which they call involutions, given by $I_k~:~x \mapsto -x+k~\text{mod}~12$ What for us is the dihedral group$D_{12}$(all symmetries of the$12$-gon), is for them the$T/I$-group (for transpositions/involutions). Let’s move from individual notes (or pitch classes) to chords (or triads), that is, three notes played together. Not all triples of notes sound nice when played together, that’s why the most commonly played chords are among the major and minor triads. A major triad is an ordered triple of elements from$\mathbb{Z}_{12}$of the form $(n,n+4~\text{mod}~12,n+7~\text{mod}~12)$ and a minor triad is an ordered triple of the form $(n,n+3~\text{mod}~12,n+7~\text{mod}~12)$ where the first entry$n$is called the root of the triad (or chord) and its funny name is then also the name of that chord. For us, it is best to view a triad as an inscribed triangle in our regular$12$-gon. The triangles of major and minor triads have edges of different lengths, a small one, a middle, and a large one. Starting from the root, and moving clockwise, we encounter in a major chord-triangle first the middle edge, then the small edge, and finally the large edge. For a minor chord-triangle, we have first the small edge, then the middle one, and finally the large edge. On the left, two major triads, one with root$0$, the other with root$6$. On the right, two minor triads, also with roots$0$and$6$. (Btw. if you are interested in the full musical story, I strongly recommend the alpof blog by Alexandre Popoff, from which the above picture is taken.) Clearly, there are$12$major triads (one for each root), and$12$minor triads. From the shape of the triad-triangles it is also clear that rotations (transpositions) send major triads to major triads (and minors to minors), and that reflexions (involutions) interchange major with minor triads. That is, the dihedral group$D_{12}$(or if you prefer the$T/I$-group) acts on the set of$24$major and minor triads, and this action is transitive (an element stabilising a triad-triangle must preserve its type (so is a rotation) and its root (so must be the identity)). Can we hear the action of the very special group element$T_6$(the unique non-trivial central element of$D_{12}$) on the chords? This action is not only the transposition by three full tones, but also a point-reflexion with respect to the center of the$12$-gon (see two examples in the picture above). This point reflexion can be compositionally meaningful to refer to two very different upside-down worlds. In It’s$T_6$-day, Alexandre Popoff gives several examples. Here’s one of them, the Ark theme in Indiana Jones – Raiders of the Lost Ark. “The$T_6$transformation is heard throughout the map room scene (in particular at 2:47 in the video): that the ark is a dreadful object from a very different world is well rendered by the$T_6$transposition, with its inherent tritone and point reflection.” Let’s move on in the direction of the Elephant. We saw that the only affine map of the form$x \mapsto \pm x + k$fixing say the major$0$-triad$(0,4,7)$is the identity map. But, we can ask for the collection of all affine maps$x \mapsto a x + b$fixing this major$0$-triad set-wise, that is, such that $\{ b, 4a+b~\text{mod}~12, 7a+b~\text{mod}~2 \} \subseteq \{ 0,4,7 \}$ A quick case-by-case analysis shows that there are just eight such maps: the identity and the constant maps $x \mapsto x,~x \mapsto 0,~x \mapsto 4, ~x \mapsto 7$ and the four maps $\underbrace{x \mapsto 3x+7}_a,~\underbrace{x \mapsto 8x+4}_b,~x \mapsto 9x+4,~x \mapsto 4x$ Compositions of such maps again preserve the set$\{ 0,4,7 \}$so they form a monoid, and a quick inspection with GAP learns that$a$and$b$generate this monoid.  gap> a:=Transformation([10,1,4,7,10,1,4,7,10,1,4,7]);; gap> b:=Transformation([12,8,4,12,8,4,12,8,4,12,8,4]);; gap> gens:=[a,b];; gap> T:=Monoid(gens); gap> Size(T); 8  The monoid$T$is the triadic monoid of Thomas Noll’s paper The topos of triads. The monoid$T$can be seen as a one-object category (with endomorphisms the elements of$T$). The corresponding presheaf topos is then the category of all sets equipped with a right$T$-action. Actually, Noll considers just one such presheaf (and its sub-presheaves) namely$\mathcal{F}=\mathbb{Z}_{12}$with the action of$T$by affine maps described before. He is interested in the sheafifications of these presheaves with respect to Grothendieck topologies, so we have to describe those. For any monoid category, the subobject classifier$\Omega$is the set of all right ideals in the monoid. Using the GAP sgpviz package we can draw its Cayley graph (red coloured vertices are idempotents in the monoid, the blue vertex is the identity map).  gap> DrawCayleyGraph(T);  The elements of$T$(vertices) which can be connected by oriented paths (in both ways) in the Cayley graph, such as here$\{ 2,4 \}$,$\{ 3,7 \}$and$\{ 5,6,8 \}$, will generate the same right ideal in$T$, so distinct right ideals are determined by unidirectional arrows, such as from$1$to$2$and$3$or from$\{ 2,4 \}$to$5$, or from$\{ 3,7 \}$to$6$. This gives us that$\Omega$consists of the following six elements: •$0 = \emptyset$•$C = \{ 5,6,8 \} = a.T \wedge b.T$•$L = \{ 2,4,5,6,8 \}=a.T$•$R = \{ 3,7,5,6,8 \}=b.T$•$P = \{ 2,3,4,5,6,7,8 \}=a.T \vee b.T$•$1 = T$As a subobject classifier$\Omega$is itself a presheaf, so wat is the action of the triad monoid$T$on it? For all$A \in \Omega$, and$s \in T$the action is given by$A.s = \{ t \in T | s.t \in A \}$and it can be read off from the Cayley-graph.$\Omega$is a Heyting algebra of which the inclusions, and logical operations can be summarised in the picture below, using the Hexboards and Heytings-post. In this case, Grothendieck topologies coincide with Lawvere-Tierney topologies, which come from closure operators$j~:~\Omega \rightarrow \Omega$which are order-increasing, idempotent, and compatible with the$T$-action and with the$\wedge$, that is, • if$A \leq B$, then$j(A) \leq j(B)$•$j(j(A)) = j(A)$•$j(A).t=j(A.t)$•$j(A \wedge B) = j(A) \wedge j(B)$Colouring all cells with the same$j$-value alike, and remaining cells$A$with$j(A)=A$coloured yellow, we have six such closure operations$j$, that is, Grothendieck topologies. The triadic monoid$T$acts via affine transformations on the set of pitch classes$\mathbb{Z}_{12}$and we’ve defined it such that it preserves the notes$\{ 0,4,7 \}$of the major$(0,4,7)$-chord, that is,$\{ 0,4,7 \}$is a subobject of$\mathbb{Z}_{12}$in the topos of$T$-sets. The point of the subobject classifier$\Omega$is that morphisms to it classify subobjects, so there must be a$T$-equivariant map$\chi$making the diagram commute (vertical arrows are the natural inclusions) $\xymatrix{\{ 0,4,7 \} \ar[r] \ar[d] & 1 \ar[d] \\ \mathbb{Z}_{12} \ar[r]^{\chi} & \Omega}$ What does the morphism$\chi$do on the other pitch classes? Well, it send an element$k \in \mathbb{Z}_{12} = \{ 1,2,\dots,12=0 \}$to •$1$iff$k \in \{ 0,4,7 \}$•$P$iff$a(k)$and$b(k)$are in$\{ 0,4,7 \}$•$L$iff$a(k) \in \{ 0,4,7 \}$but$b(k)$is not •$R$iff$b(k) \in \{ 0,4,7 \}$but$a(k)$is not •$C$iff neither$a(k)$nor$b(k)$is in$\{ 0,4,7 \}$Remember that$a$and$b$are the transformations (images of$(1,2,\dots,12)$)  a:=Transformation([10,1,4,7,10,1,4,7,10,1,4,7]);; b:=Transformation([12,8,4,12,8,4,12,8,4,12,8,4]);;  so we see that •$0,1,4$are mapped to$1$•$3$is mapped to$P$•$8,11$are mapped to$L$•$1,6,9,10$are mapped to$R$•$2,5$are mapped to$C$Finally, we can compute the sheafification of the sub-presheaf$\{ 0,4,7 \}$of$\mathbb{Z}$with respect to a Grothendieck topology$j$: it consists of the set of those$k \in \mathbb{Z}_{12}$such that$j(\chi(k)) = 1$. The musically interesting Grothendieck topologies are$j_P, j_L$and$j_R$with corresponding sheaves: • For$j_P$we get the sheaf$\{ 0,3,4,7 \}$which Mamuth-ers call a Major-Minor Mixture as these are the notes of both the major and minor$0$-triads • For$j_L$we get$\{ 0,3,4,7,8,11 \}$which is an example of an Hexatonic scale (six notes), here they are the notes of the major and minor$0,~4$and$8$-triads • For$j_R$we get$\{ 0,1,3,4,6,7,9,10 \}$which is an example of an Octatonic scale (eight notes), here they are the notes of the major and minor$0,~3,~6$and$9$-triads We could have played the same game starting with the three notes of any other major triad. Those in the know will have noticed that so far I’ve avoided another incarnation of the dihedral$D_{12}$group in music, namely the$PLR$-group, which explains the notation for the elements of the subobject classifier$\Omega$, but this post is already way too long. (to be continued…) A couple of days ago, Peter Rowlett posted on The Aperiodical: Introducing hexboard – a LaTeX package for drawing games of Hex. Hex is a strategic game with two players (Red and Blue) taking turns placing a stone of their color onto any empty space. A player wins when they successfully connect their sides together through a chain of adjacent stones. Here’s a short game on a$5 \times 5$board (normal play uses$11\times 11$boards), won by Blue, drawn with the LaTeX-package hexboard. As much as I like mathematical games, I want to use the versability of the hexboard-package for something entirely different: drawing finite Heyting algebras in which it is easy to visualise the logical operations. Every full hexboard is a poset with minimal cell$0$and maximal cell$1$if cell-values increase if we move horizontally to the right or diagonally to the upper-right. With respect to this order,$p \vee q$is the smallest cell bigger than both$p$and$q$, and$p \wedge q$is the largest cell smaller than$p$and$q$. The implication$p \Rightarrow q$is the largest cell$r$such that$r \wedge p \leq q$, and the negation$\neg p$stands for$p \Rightarrow 0$. With these operations, the full hexboard becomes a Heyting algebra. Now the fun part. Every filled area of the hexboard, bordered above and below by a string of strictly increasing cells from$0$to$1$is also a Heyting algebra, with the induced ordering, and with the logical operations defined similarly. Note that this mustn’t be a sub-Heyting algebra as the operations may differ. Here, we have a different value for$p \Rightarrow q$, and$\neg p$is now$0$. If you’re in for an innocent “Where is Wally?”-type puzzle:$W = (\neg \neg p \Rightarrow p)$. Click on the image to get the solution. The downsets in these posets can be viewed as the open sets of a finite topology, so these Heyting algebra structures come from the subobject classifier of a topos. There are more interesting toposes with subobject classifier determined by such hex-Heyting algebras. For example, the Topos of Triads of Thomas Noll in music theory has as its subobject classifier the hex-Heyting algebra (with cell-values as in the paper): Note to self: why not write a couple of posts on this topos? Another example: the category of all directed graphs is the presheaf topos of the two object category ($V$for vertices, and$E$for edges) with (apart from the identities) just two morphisms$s,t : V \rightarrow E$(for start- and end-vertex of a directed edge). The subobject classifier$\Omega$of this topos is determined by the two Heyting algebras$\Omega(E)$and$\Omega(V)\$ below.

These ‘hex-Heyting algebras’ are exactly what Eduardo Ochs calls ‘planar Heyting algebras’.

Eduardo has a very informative research page, containing slides and handouts of talks in which he tries to explain topos theory to “children” (using these planar Heyting algebras) including:

Perhaps now is a good time to revive my old sga4hipsters-project.