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Let’s sum-up what we have seen in the last post. We worked with presheaves, in particular presheaves of sets, and we have seen that such presheaves have the very nice following property:

Given a small category $C$, the functor category $\textbf{Sets}^{C^{\text{op}}}$ is a topos.

Since it’s a topos, it has a subobject classifier $\Omega$, and we can easily construct it.

First, for each object $X$ of $C$, the subobject classifier functor sends $X$ to $\Omega(X)=\{\text{sieves on} X\}$. A sieve $S$ on $X$ in $C$ is a set of morphisms of $C$ with codomain $X$ such that for any morphism $m \in S$ and any morphism $f \in C$ such that the composition exists, $mf$ belongs to $S$.

Then, the image $\Omega(f)$ of a morphism $f$ of $C$ sends a sieve $S$ to the sieve $S.f$ (remember that we are working with $\textbf{Sets}^{C^{\text{op}}}$ so composition is on the right) which is defined by $S.f = \{u \in C, f.u \in S\}$.

Of course, you can also work with the functor category $\textbf{Sets}^{C}$, in which case the subobject classifier is defined as follows.

For each object $X$ of $C$, the subobject classifier functor then sends $X$ to $\Omega(X)=\{\text{cosieves on} X\}$. A cosieve $S$ on $X$ in $C$ is a set of morphisms of $C$ with domain $X$ such that for any morphism $m \in S$ and any morphism $f \in C$ such that the composition exists, $f.m$ belongs to $S$.

Then, the image $\Omega(f)$ of a morphism $f$ of $C$ sends a cosieve $S$ to the cosieve $f.S$ (now composition is on the left) which is defined by $f.S = \{u \in C, u.f \in S\}$.

In the last post, we a particular category $C$ with two objects $M$ and $D$ and one morphism $h_{MD} \colon M \to D$, and we studied its subobject classifier. What if we choose a group $G$ as the category $C$, i.e. a single-object $\bullet$ category with $\text{Hom}(\bullet,\bullet)=G$ ? Can you guess what would be the subobject classifier of the functor category $\textbf{Sets}^{C^{\text{op}}}$ ?

We have to study what are the sieves on $\bullet$, and it will be quick since there aren’t many of them. In fact, two, since a sieve has to be closed under composition: the empty sieve $\emptyset$, which we denote by 0, and the whole set of morphisms of $G$, which we denote by 0. So the subobject classifier $\Omega=\{0,1\}$ looks like the same as that of $\textbf{Sets}$, with the additional action of the group $G$: for any sieve $S$, any group element $g \in G$ sends $S$ to itself.

Consider for example the group $PLR$ that we have introduced in the previous posts. We have seen that this group acts simply transitively on the set $H$ of the 24 major and minor triads. The group $PLR$ has a subgroup $PR$ generated by $P$ and $R$, which is isomorphic to the dihedral group $D_8$. This subgroup acts on the same set of major and minor triads: the group action is free, but this time it is not simply transitive. There are indeed three orbits, given by the three octatonic systems $O_1=\{C_M,C_m,E_{bM},E_{bm},F_{\#M},F_{\#m},A_M,A_m\}$, $O_2=\{B_M,B_m,D_{M},D_{m},F_{M},F_{m},G_{\#M},G_{\#m}\}$, and $O_3=\{B_{bM},B_{bm},C_{\#M},C_{\#m},E_{M},E_{m},G_M,G_m\}$. The $PR$-set $O_1$ being a subobject of the $PR$-set $H$, its characteristic morphism sends it to $1 \in \Omega$, whereas $O_2$ and $O_3$ are sent to $0 \in \Omega$.

If we consider monoids instead of groups (single object categories, where the morphisms are not necessarily invertible), the subobject classifier gets more complicated, and this will bring us to the paper of Noll.

A quick example with monoids

So that everything we said is clear, let’s work first with a simple example of a topos $\textbf{Sets}^{M^{\text{op}}}$ where $M$ is a monoid. We take the monoid $M=$ where $f$ is not invertible. This monoid has three elements, $1, f, f^2$ and you must have easily guessed what is the corresponding subobject classifier. The possible sieves are $\emptyset$, $\{f^2\}$, $\{f, f^2\}$, and $\{1,f,f^2\}$. We will denote these 4 elements by 0, 1/3, 2/3, and 1 respectively. The action of $f$ on $\Omega$ is given by $f(0)=0, f(1/3)=2/3, f(2/3)=1, f(1)=1$.

Below is an example of a $M$-set (a functor $F$ from $M$ to $\mathbf{Set}$):

And here is a subobject of this functor :

The corresponding characteristic morphism sends each element of $F(M)$ to $\Omega$, in this fashion:

Intuitively speaking, the truth values associated with each element of the set measure “how far” the element is from the selected subobject. Let’s now turn to the paper of Noll and the application of topos theory to music.

Previously, we have seen transformations which act on the set $\mathbb{Z}_{12}$ of pitch classes such as the transposition $T(x)=x+1$ and the inversion $I(x)=-x$ (or $I(x)=11x$, since all transformations are understood modulo 12). In general you could consider all affine transformations $h \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ of the form $h(x)=ux+v$. But these transformations are not always invertible (they are if $u=1, 5, 7, 11$), so you cannot use them in the framework of groups and group actions.

But such transformations can be useful from other point of views: Noll uses a specific set of affine transformations in the framework of topos theory. First, Noll asks the following question: given a specific triad $A=\{p,q,r\}, p,q,r \in \mathbb{Z}_{12}$, what are the affine transformations which stabilize the triad, i.e. $h(x)=ux+v$ such that $\{h(p),h(q),h(r)\} \subset A$ ? In his paper, Noll focuses on two different triads but for the sake of clarity I will only focus on the C major triad $A=\{0,4,7\}$ here (also, Noll uses the circle of fifth instead of the circle of semi-tones so my notation will be a bit different from his).

Noll selects two transformations $f(x)=3x+7$ and $g(x)=8x+4$. The action of $f$ on the pitch classes can be represented as such (I have indicated the triad $A$ in green):

and here is the action of $g$:

As you can see, these transformations are clearly not invertible. But what about the successive composition of these transformations ? We have to consider the monoid $M$ generated by $f$ and $g$. It turns out that this monoid is quite small: it only has 8 elements. Let’s denote by $a$, $b$, and $c$ the constant maps $a(x)=7$, $b(x)=0$, and $c(x)=4$. We can draw the Cayley graph of this monoid:

In his paper, Noll considers covariant functors from $M$ to $\mathbf{Set}$. Therefore, in order to determine the structure of the subobject classifier $\Omega$, we have to find the cosieves of $M$. It is a good exercise to check that there are 6 such cosieves, namely:

• $\mathcal{F}=\emptyset$
• $\mathcal{C}=\{a,b,c\}$
• $\mathcal{L}=\{f,f^2,a,b,c\}$
• $\mathcal{R}=\{g,g^2,a,b,c\}$
• $\mathcal{P}=\{f,f^2,g,g^2,a,b,c\}$
• $\mathcal{T}=\{e,f,f^2,g,g^2,a,b,c\}$

I’m using here Noll’s notation for the cosieves. You can also determine the action of $M$ on this set of cosieves. Now, since the action of the monoid $M$ stabilizes the C-major triad $\{0,4,7\}$, the functor from $M$ to $\mathbf{Set}$ whose image is the set $\{0,4,7\}$ is a subobject of the functor whose image is the set $\mathbb{Z}_{12}$ of pitch-classes. This means that we have a characteristic morphism by which we can look at the truth values in $\Omega$ associated with each pitch-class. Recall that these truth values measure, in some way, “how far” (under the action of $M$) the pitch-class is from the triad $\{0,4,7\}$. If you perform the calculations, you will obtain that

• 0, 4, and 7 (C, G, and E) are mapped to $\mathcal{T}$ (obviously, since they are the elements of the subobject in question)
• 3 (Eb) is mapped to $\mathcal{P}$
• 8, and 11 (G# and B) are mapped to $\mathcal{L}$
• 1, 6, 9, and 10 (C#, F#, A, and Bb) are mapped to $\mathcal{R}$
• 2, and 11 (F and D) are mapped to $\mathcal{C}$

In a sense, this gives a sort of hierarchy of pitch-classes with respect to a given triad, in this case the C-major triad. Of course, you could do it with another triad: determine the affine transformations which stabilize that triad, then determine the subobject classifier and the truth values associated with the pitch-classes.

The monoid introduced by Noll is not the only monoid which stabilizes the C-major triad. For example, one can take the monoid generated by the affine transformations $f'(x)=3x+7$ and $g(x)=4x$. This monoid has only 6 elements, $\{1,f,g,f^2,fg,gf\}$ but has 9 cosieves. Interestingly, the truth values taken by the pitch-classes under the characteristic morphism associated with the C-major triad subobject look very similar to those determined above: 0,4,7 are mapped to the cosieve $M$ (obviously); 3 is mapped to the cosieve $M-\{e\}$; 1, 6, 9, and 10 are mapped to the cosieve $\{g,fg,gf\}$; 8 and 11 are mapped to the cosieve $\{f,f^2,gf,fg\}$; 2 and 5 are mapped to the cosieve $\{fg,gf\}$. There must be something general happening here, when considering all monoids which stabilize the C-major triad.

Noll finally gives an application to the study of Scriabin’s Etude n°3, op.65. Here are the opening bars of this Etude :

As one can see, the left hand generally plays triads of the form [0,2,6] (highlighted in red): the first two notes G and F could be considered part of the triad (G,F,B) (i.e. $\{5,7,11\}$), we then have the triad (F,B,C#), later we find (A,G,C#), and so on. This goes on for most of the Etude. Noll then looks at the monoid which stabilizes such triads and determines the truth values of the pitch-classes in the right hand.

It turns out that the vast majority of these pitch-classes have the truth value $\mathcal{L}$ or $\mathcal{R}$. Some of them have the truth value $\mathcal{P}$ but neither of them have the truth value $\mathcal{C}$. This classification of pitch-classes allows one to immediately determine the relations between notes according to the chosen monoid, and the overall global structure of the piece. I am guessing that the process of determining the truth values according to a particular subobject could be automatized in a computer analysis, but I currently don’t know if someone has looked at it already.

There is more in the paper of Noll, in particular concerning the use of Lawvere-Tierney topologies and their relation in the present case to major-minor, hexatonic, and octatonic sets. I’m saving all of this for a next post. Nevertheless, I hope that the current post has given a fair overview of the value of topos theory as applied to music theory.