Recently we lost the great mathematician Bill Lawvere. In what follows I'll start to share with you my attempt to study and understand by means of examples the content of his article "taking categories seriously". The main construction that I'd like to analyze here, making it clear by bathing it in the concreteness of classical dynamics, is the study of periodic behavior of a dynamical system contained in chapter 2.
Why categories: the mathematics of mutation
The point that Lawvere tries to make is the following: category theory captures the essential content of mathematics in that it is the theory of mutability of mathematical objects, and consecutive application of those mutations. Mutability of one object into another allows us to study the former object by means of the latter or vice-versa. We do this all the time in mathematics and category theory is exactly the math of doing this, independently from the specific context (topology, analysis, probability and so on).
If our object of studies can be described as categories then the mutability is expressed by functors, if the structures of interest are functors themselves then we look at the natural transformations between two such functors. Since this mutability is the "essential content of category theory (cit)" Lawvere claims, rightly so, that we should take categories seriously. A theory of abstract mutability should in fact, as it does, guide our intuition about how we should study mathematical object of interest by relating them or mutating one into another.
Lawvere uses a fundamental concrete example to substantiate his claim, that of dynamical systems.
We can begin by taking an arbitrary monoid \(M\) as an object incarnating our concept of time. We don't want to restrict too much, yet, our concept of time like for example by restricting to the classical case \(M=(\mathbb R,+)\) because we don't want to exclude discrete/quantized time concepts, e.g. \(\mathbb N\) or periodic times concepts, eg. \( \mathbb Z/ \theta \mathbb Z\) or the complex unitary circle.
We then look at the ways, all the ways simultaneously, in which our abstract concept of time can be realized in the concrete, by means of functions between sets. Call those representations of the monoid or, equivalently actions of the monoid. That is, we look at all the set based-dynamical systems that are "animated" by that specific abstract concept of time. Those are functors
\[M\to {\bf Set}\]
We can mutate dynamical systems into other dynamical systems in at least three ways, Lawvere explains us.
General spectral analysis 1: the essence of periodicity
I'd like to start by the concept of periodic/spectral analysis of a monoid action. An \(M\)-action is a monoid morphism \(\alpha:M\to {\rm End}(X)\). If \(M=(\mathbb R,+)\) we call it a (continuous time) dynamical system, if \(M=(\mathbb N,+)\) we call it discrete (time) dynamical system.
What it means for a \(M\)-time dynamical system \(X,\alpha\) to be periodical, really? It means that after some time translation \(t+\) all the states of the systems circle back to themselves... they go back to their initial state: in symbols \(\alpha_t={\rm id}_X:X\to X\). This, given the definition of actions as functors, implies, (\forall s\in M\) that \[\alpha_{t+s}=\alpha_s\]
Example 1: (the integer case) We say that an \(\mathbb N\)-action \(\alpha\) is periodic iff \(\alpha^{\circ k}=\alpha^{\circ 0}\) for some \(k\in \mathbb N\). In that case every state \(x\in X\) is a periodic state in that \(\alpha^{\circ k}(x)=x\). For example, idempotent functions are \(2\)-periodic \(\mathbb N\)-actions.
Key Observation 1: (periodic means factorizable) Every time we have a \(k\)-periodic \(\mathbb N\)-action \(\alpha:\mathbb N\to {\bf Set}\) then we can deduce that \(\alpha^{\circ n}=\alpha^{\circ (n \mod\! k) } \). But this is in fact equivalent to the action \(alph\) being factorizable as \[\mathbb N\overset{\mod\! k}{\to} \mathbb Z /k\mathbb Z \overset{\alpha'}{\to} {\bf Set}\]
Lawvere would consider the morphism \(\pi_k:\mathbb N \to \mathbb Z /k\mathbb Z \) rolling up the integers every \(k\) steps, i.e. the act of modding out by \(k\) as a period and would probably say that each action \(\alpha\) that factors as \(\alpha=\alpha' \circ \pi_k\) is a (globally) periodic action of period \(\pi_k\).
Example 2: (real time case) in the last paragraph of page 3, Lawvere brings up the main content of idea of periodic analysis. Consider the canonical winding of the real numbers into the circle group \[\rho:\mathbb R \to S^1\]
This is a monoid map that sends addition of real numbers to addition of angles on the circle: it is essentially modding out the real line by \(2\pi\). But more generally Lawvere defines a period to be any surjective homomorphism \(\theta: \mathbb R\to S^1\).
Definition (period): A surjective monoid morphism \(\theta: \mathbb R\to S^1\) will be called a period of \(\mathbb R\). A map of that kind satisfies \(\theta(0)=0\) and \(\theta(t+s)=\theta(t)+\theta(s)\)
The smart move of Lawvere is that of defining an \(\mathbb R\) dynamical system \(\alpha_t:X\to X\) periodic of period \(\theta\) iff it factors through the map \(\theta\) as follows
\[\mathbb R\overset{\theta}{\to} S^1\overset{\alpha'}{\to} {\bf Set}\]
For some \(S^1\)-time dynamical system \(\alpha '\). Now, if you focus on what \(S^1\)-time dynamical systems are, the smart move should be clear: they are dynamical systems where the time is cyclic, it is running in a circle.
I hope I'll be able to conclude soon this analysis of Lawvere's text, chapter 2. I'll explain step by step the Kan quantifiers and how they are used to extract the spectral information of a dynamical system, and to do this I'll try to work out simple examples. This is highly related to many recent threads on the forum, especially the Fourier-related ones. I want thanks Mick for the remark he made to me many years ago about the importance of periodicity.
Why categories: the mathematics of mutation
The point that Lawvere tries to make is the following: category theory captures the essential content of mathematics in that it is the theory of mutability of mathematical objects, and consecutive application of those mutations. Mutability of one object into another allows us to study the former object by means of the latter or vice-versa. We do this all the time in mathematics and category theory is exactly the math of doing this, independently from the specific context (topology, analysis, probability and so on).
If our object of studies can be described as categories then the mutability is expressed by functors, if the structures of interest are functors themselves then we look at the natural transformations between two such functors. Since this mutability is the "essential content of category theory (cit)" Lawvere claims, rightly so, that we should take categories seriously. A theory of abstract mutability should in fact, as it does, guide our intuition about how we should study mathematical object of interest by relating them or mutating one into another.
Lawvere uses a fundamental concrete example to substantiate his claim, that of dynamical systems.
We can begin by taking an arbitrary monoid \(M\) as an object incarnating our concept of time. We don't want to restrict too much, yet, our concept of time like for example by restricting to the classical case \(M=(\mathbb R,+)\) because we don't want to exclude discrete/quantized time concepts, e.g. \(\mathbb N\) or periodic times concepts, eg. \( \mathbb Z/ \theta \mathbb Z\) or the complex unitary circle.
We then look at the ways, all the ways simultaneously, in which our abstract concept of time can be realized in the concrete, by means of functions between sets. Call those representations of the monoid or, equivalently actions of the monoid. That is, we look at all the set based-dynamical systems that are "animated" by that specific abstract concept of time. Those are functors
\[M\to {\bf Set}\]
We can mutate dynamical systems into other dynamical systems in at least three ways, Lawvere explains us.
- (actions over the same objects) By mutating two actions one into the other. Given \(\alpha:M\to {\bf Set}\) and \(\beta:M\to {\bf Set}\) we consider a natural transformation \(f:\alpha\implies \beta\) to be a function preserving the action, \(M\)-equivariant, like superfunctions and Abel functions.
\[f(\alpha_t(x))=\beta_t(f(x))\]
- (actions on related objects) By mutating the matter we use to concretely realize the action of our time monoid. I.e. assume we can mutate sets into another kind of mathematical object \(F:{\bf Set}\to \mathcal C\), then given an action \(\alpha: M\to {\bf Set}\) we can mutate it into another action \(F\alpha: M\to {\mathcal C}\), this times not set-based but \(\mathcal C\)-based. \[M\to {\bf Set}\overset{F}{\to}\mathcal C\]
- (action of a related time concept) Given a way to mutate our concept of time, i.e. a monoid homomorphism \(\lambda: M\to L\) we obtain the third way of mutating dynamical systems \(\alpha: L\to {\bf Set}\) with time running in the monoid \(L\) into dynamical systems running with time belonging to our original \(M\)
\[M\overset{\lambda}{\to} L\to {\bf Set}\]
This third kind of transformation unlocks to us what Lawvere calls Spectral Analysis of dynamical system because we can take as a special case the mutation of the real numbers into the circle. This mutation is the winding of the real line into the circle \(p:\mathbb R\to S^1\) and as such defines a mutation of \(S^1\)-dynamical systems, i.e. periodic dynamics, into real time-continuous dynamical systems opening up the possibilities of systematizing the study of periodic properties of the latter.
General spectral analysis 1: the essence of periodicity
I'd like to start by the concept of periodic/spectral analysis of a monoid action. An \(M\)-action is a monoid morphism \(\alpha:M\to {\rm End}(X)\). If \(M=(\mathbb R,+)\) we call it a (continuous time) dynamical system, if \(M=(\mathbb N,+)\) we call it discrete (time) dynamical system.
What it means for a \(M\)-time dynamical system \(X,\alpha\) to be periodical, really? It means that after some time translation \(t+\) all the states of the systems circle back to themselves... they go back to their initial state: in symbols \(\alpha_t={\rm id}_X:X\to X\). This, given the definition of actions as functors, implies, (\forall s\in M\) that \[\alpha_{t+s}=\alpha_s\]
Example 1: (the integer case) We say that an \(\mathbb N\)-action \(\alpha\) is periodic iff \(\alpha^{\circ k}=\alpha^{\circ 0}\) for some \(k\in \mathbb N\). In that case every state \(x\in X\) is a periodic state in that \(\alpha^{\circ k}(x)=x\). For example, idempotent functions are \(2\)-periodic \(\mathbb N\)-actions.
Key Observation 1: (periodic means factorizable) Every time we have a \(k\)-periodic \(\mathbb N\)-action \(\alpha:\mathbb N\to {\bf Set}\) then we can deduce that \(\alpha^{\circ n}=\alpha^{\circ (n \mod\! k) } \). But this is in fact equivalent to the action \(alph\) being factorizable as \[\mathbb N\overset{\mod\! k}{\to} \mathbb Z /k\mathbb Z \overset{\alpha'}{\to} {\bf Set}\]
Lawvere would consider the morphism \(\pi_k:\mathbb N \to \mathbb Z /k\mathbb Z \) rolling up the integers every \(k\) steps, i.e. the act of modding out by \(k\) as a period and would probably say that each action \(\alpha\) that factors as \(\alpha=\alpha' \circ \pi_k\) is a (globally) periodic action of period \(\pi_k\).
Example 2: (real time case) in the last paragraph of page 3, Lawvere brings up the main content of idea of periodic analysis. Consider the canonical winding of the real numbers into the circle group \[\rho:\mathbb R \to S^1\]
This is a monoid map that sends addition of real numbers to addition of angles on the circle: it is essentially modding out the real line by \(2\pi\). But more generally Lawvere defines a period to be any surjective homomorphism \(\theta: \mathbb R\to S^1\).
Definition (period): A surjective monoid morphism \(\theta: \mathbb R\to S^1\) will be called a period of \(\mathbb R\). A map of that kind satisfies \(\theta(0)=0\) and \(\theta(t+s)=\theta(t)+\theta(s)\)
The smart move of Lawvere is that of defining an \(\mathbb R\) dynamical system \(\alpha_t:X\to X\) periodic of period \(\theta\) iff it factors through the map \(\theta\) as follows
\[\mathbb R\overset{\theta}{\to} S^1\overset{\alpha'}{\to} {\bf Set}\]
For some \(S^1\)-time dynamical system \(\alpha '\). Now, if you focus on what \(S^1\)-time dynamical systems are, the smart move should be clear: they are dynamical systems where the time is cyclic, it is running in a circle.
I hope I'll be able to conclude soon this analysis of Lawvere's text, chapter 2. I'll explain step by step the Kan quantifiers and how they are used to extract the spectral information of a dynamical system, and to do this I'll try to work out simple examples. This is highly related to many recent threads on the forum, especially the Fourier-related ones. I want thanks Mick for the remark he made to me many years ago about the importance of periodicity.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
