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The MCM2015 conference for Mathematics and Computation in Music is coming in June, and the submission process has taken up quite a lot of my time recently, which explains the relative inactivity of this blog. But everything is finished now, so here is a new post about topos theory and music.

I want to show here a simple example of another topos, applied to the study of time-spans. We are going to introduce a specific monoid $M$ and its action on a particular set $S$, which therefore defines a functor $M \to \textbf{Sets}$. Since the category $\textbf{Sets}^{M}$ is a topos, we are going to calculate its subobject classifier $\Omega$ explicitly and see its use for time-spans. This is a small part in a much larger paper than Moreno Andreatta, Andrée Ehresmann, and my self have submitted to MCM2015, which I will describe in a later post.

A time-span defines a certain duration at a certain point in time. Recall from the previous posts that a time-span, as defined by Lewin, is a couple $(t,\Delta)$, with $t \in \mathbb{R}$, and $\Delta \in \mathbb{R}_{>0}$. The group $G = (\mathbb{R},+) \rtimes (\mathbb{R}_{>0},x)$ acts simply transitively on the set of time-spans, either on the left, by the action of a group element $(u,\delta)$ given by

$(u,\delta).(t,\Delta) = (u+\delta t, \delta \Delta),$

or on the right, by

$(t,\Delta).(u,\delta) = (t+\Delta u, \Delta \delta).$

We can represent the time-spans graphically. Our reference unit time-span will be represented as such:

Given this unit time-span, here are the time-spans (1,1) and (3,1/2):

We are now going to consider a restricted set of dyadic time-spans, namely the set $S=\{(t,2^\delta), t \in \mathbb{Z}[\frac{1}{2}], \delta \in \mathbb{Z}\}$, where $\mathbb{Z}[\frac{1}{2}]$ is the set of the dyadic rationals, i.e. numbers of the form $\frac{p}{2^q}$, where $p$ is an integer, and $q$ is a natural number. This can model time-spans having the duration of a quarter, an eighth, and so on. The time-spans (1,1) and (3,1/2) are examples of dyadic time-spans.

Consider now the monoid $M=\{(u,2^v), u \in \mathbb{Z}[\frac{1}{2}], u \geq 0, v \in \mathbb{Z}\}$ whose multiplication law is given by the following equation.

$(u_1,2^{v_1})*(u_2,2^{v_2})=(u_2+u_1 2^{v_2},2^{v_1+v_2})$

The monoid $M$ is generated by the elements $a=(1,1)$ and $b=(0,1/2)$ and has for presentation $M=\langle a,b \mid a^2b=ba \rangle$. It can be considered as a discrete monoid version of Lewin’s continuous group of time-span transformations. It is also a monoid analog of the Baumslag-Solitar group $BS(1,2)$.

Now, consider the action of an element $(u,2^v) \in M$ on a dyadic time-span $(t,\delta) \in S$ given by

$(u,2^v) \cdot (t,2^\delta)=(t+2^\delta \cdot u,2^{\delta+v}).$

This defines a functor $M \to \textbf{Sets}$ which belongs to the category of functors $\textbf{Sets}^M$. In the example given above, one transforms the time-span (1,1) into (3,1/2) by the action of the monoid element (2,1/2). The transformation of (3,1/2) into (1,1) is not possible since the monoid only acts by positive translations in time.

The category of functors $\textbf{Sets}^M$ is a topos, and we can calculate explicitly its subobject classifier $\Omega$.

For that, we need to calculate the cosieves on the unique object of $M$. It is easy to see that those are of the form $J_p = \{(u,2^\delta) \in M, u \in \mathbb{Z}[\frac{1}{2}], u \geq p, \delta \in \mathbb{Z}\}$, with $p \in \mathbb{Z}[\frac{1}{2}], p \geq 0$. Indeed, you can check that for any element $m \in M$, we have $m \cdot J_p \subset J_p$. We also have the empty cosieve $\emptyset$.

The subobject classifier is therefore $\Omega=\{J_p, p \in \mathbb{Z}[\frac{1}{2}], p \geq 0\} \cup \{x\}$, where $\{x\}$ is a singleton. The first term can be bijectively identified with $\mathbb{Z}[\frac{1}{2}]_{\geq 0}$, so that $\Omega=\mathbb{Z}[\frac{1}{2}]_{\geq 0} \cup \{x\}$.

The action of the monoid $M$ on $\Omega$ is given by $(m, J_p) \to m.J_p = \{f \in M, f.m \in J_p\}$. This can be obtained in a rather straightforward manner by studying the action of the generators $a=(1,1)$ and $b=(0,1/2)$. One thus gets:

• $(1,1) \cdot p = p-1,$ if $p \geq 1$, and 0 otherwise, and
• $(0,1/2) \cdot p = 2p$,

for all $p \in \mathbb{Z}[\frac{1}{2}]$. The singleton $\{x\}$ is a fixed point by the action of $M$. It corresponds to the value False in our topos, whereas 0 corresponds to the value True.

So what can we do with this information ? First, observe that, for a given $k \in \mathbb{Z}[\frac{1}{2}]$, the set $T_k=\{(t,2^\delta), t \in \mathbb{Z}[\frac{1}{2}], t \geq k, \delta \in \mathbb{Z}\}$ equipped with the same action of $M$ is a subobject $A$ of $S$. The characteristic map $\chi_A$ then sends any element $(t,2^\delta) \in S$ to $\dfrac{k-t}{2^\delta}$ if $k \geq t$, or 0 otherwise. In other words, the characteristic map measures the time period $t-k$ from a time-span $(t,2^\delta)$ in units of $2^\delta$.

Let’s look at our small example again, along with the set $T_{9/2}$. We can represent it as such :

The time-span (3,1/2) needs to be translated by three times its duration in order to belong to the set $T_{9/2}$. Thus the characteristic map $\chi_{9/2}$ sends the time-span (3,1/2) to the set $\{3\}$ $\in \Omega$. Similarly, one can check that the map $\chi_{9/2}$ sends the time-span (1,1) to the set $\{7/2\} \in \Omega$.

This topos is rather simple, but I find it nevertheless interesting. First, it gives a precise way to define the distance of a time-span to a given cue in time. Second, it could easily be implemented on a computer and suggests application in automatic music analysis. Last, it shows that monoids can sometimes be more interesting than groups. Indeed the subobject classifier of any functor category $\mathbf{Sets}^{G}$ where $G$ is a group is rather simple: it has only two elements, and the topos is therefore boolean.