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Stream: theory: category theory

Topic: Categorifying energetic sets

John Baez (Jul 23 2024 at 13:34):

There's a concept of 'energetic set' which generalizes the concept of finite set, and any energetic set has a 'partition function' which generalizes the cardinality of finite sets. I'm looking for useful categorified, or homotopified, or derived versions of these simple concepts. Can you help? First let me explain the basics.

An energetic set is a set $X$ equipped with a function $E : X \to \mathbb{R}$. Thanks to the physics applications let's call $E$ the energy.

There's a category of energetic sets: it's just the slice category $\mathsf{Set}/\mathbb{R}$. Let's focus on the full subcategory of finite energetic sets to avoid worrying about convergence of sum. So, henceforth my energetic sets will be finite by default.

To an energetic set $E \colon X \to \mathbb{R}$ we can assign a real-valued function on the real line called its partition function, defined by

$Z(\beta) = \sum_{i \in X} e^{-\beta E(i)}$

The minus sign is a convention taken from physics.

Here's how the partition function of an energetic set generalizes the cardinality of a finite set:

Theorem. Two (finite) energetic sets are isomorphic if and only if their partition functions are equal.

Theorem. When $\beta = 0$, the partition function of an energetic set $Z: Z \to \mathbb{R}$ equals the cardinality $|X|$ of its underlying set.

Theorem. There is a functor from finite sets to energetic sets sending each finite set $X$ to that finite set equipped with the energy function $E = 0$. The partition function of the resulting energetic set equals the cardinality $|X|$.

Theorem. There is a way to make the category of (finite) energetic sets into a rig category where the addition $\oplus$ is coproduct and the tensor product $\otimes$ of energetic sets $E: X \to \mathbb{R},$ $E': X' \to \mathbb{R}$ is given by taking the cartesian product $X \times X'$ and equipping it with the energy function

$(E \otimes E')(x,y) = E(x) + E'(y)$

Theorem. The partition function of energetic sets $A$ and $B$ obeys

$Z(A \oplus B) = Z(A) + Z(B)$

and

$Z(A \otimes B) = Z(A) Z(B)$

for any (finite) energetic sets $X$ and $Y$.

John Baez (Jul 23 2024 at 13:40):

All this is explained in more detail starting on page 71 of my book (page 78 of the pdf file).

John Baez (Jul 23 2024 at 13:41):

So, can we get some of this to work if we replace finite sets by something more general like groupoids or categories or homotopy types, and replace the energy function by... something more general?

Vincent Moreau (Jul 23 2024 at 14:19):

I am puzzled by the theorem about the cardinality. In the book I see that $\beta$ is meant to be $1/kT$. Did you meant that $Z(0)$ is the cardinality $|X|$?

John Baez (Jul 23 2024 at 15:30):

You're right, the partition function reduces to the cardinality at infinite temperature, which is when $\beta = 0$. I'll correct that mistake of mine here. Thanks!

(In the book I explain the point of this math in terms of statistical mechanics.)

Chris Grossack (they/them) (Jul 23 2024 at 15:50):

In the generic case where all the $E(i)$ are distinct, isn't $Z$ a fourier transform? In this case we can identify $X$ with the image of $E$ and we get

$Z(\beta) = \int_{\mathbb{R}} \delta_{X} e^{- \beta x}$

So maybe by analogy we want to consider some kind of categorified fourier transform?

Eric M Downes (Jul 23 2024 at 15:54):

If one wants to proceed in small steps, the Fourier transform has a well studied generalization to abelian groups.

So for an energetic set these become ordered abelian groups (pullback from $\mathbb{R}$)

Eric M Downes (Jul 23 2024 at 15:56):

This would seem to work well with Persei Diaconis book, which I have if you need access, and Terry Tao has some notes on this as well.

Eric M Downes (Jul 23 2024 at 16:27):

Just guessing, based on physical intuition, the symmetry breaking that assigns different probabilities to group elements that should otherwise be equivalent comes from $E:G\to\mathbb{R}$. For any given $E$ there should be an associated Markov model equivalence class on the Cayley graph of $G$ having $e^{-\beta E(g)}/Z$ as a steady-state distribution; so the directed edges in the Cayley graph of $G$ are labelled with values $k\in\mathbb{R}_+$ with for a given pair $E(g)-E(h)\propto k_{gh^{-1}}/k_{hg^{-1}}$.

If one wanted to check against physics at this point one could also look at two finite groups $G,H$ where the dynamics on $G\times H$ could be deterministic (where there is uncertainty over initial conditions) or random by reversible and in either case possibly coupled, but $|G|\ll |H|$, so $H$ serves as a kind of “thermodynamic bath” when there is uncertainty over the initial configuration or some such.

It would also be natural and possible to talk about the entropy generated along a trajectory vs its inverse because the groups provide the critical time-reversal piece from Hamiltonian dynamics that allows us to discuss non-equilibrium thermodynamic systems. (See work by Jarzynski, Crooks, England or ask me.)

John Baez (Jul 23 2024 at 16:30):

Chris Grossack (they/them) said:

In the generic case where all the $E(i)$ are distinct, isn't $Z$ a fourier transform?

It's called a Laplace transform. A Laplace transform is like a Fourier transform missing the $\sqrt{-1}$ in the exponential. Regardless of whether the energies $E(i)$ are distinct, $Z$ is the Laplace transform of a sum of delta functions, one at each point $E(i) \in \mathbb{R}$.

For some purposes the difference between Fourier transforms and Laplace transforms is pretty small, while for others it's pretty big. Laplace transforms are to statistical mechanics (which is what I'm thinking about now) as Fourier transforms are to quantum mechanics. The quantity $\beta$ in my Laplace transform is basically inverse temperature $1/T$, which plays a role in statistical mechanics analogous to time in quantum mechanics.

So, I had a different network of ideas in my head right now than if I'd written down

$Z(t) = \sum_{i \in X} e^{-\sqrt{-1} E(i) t}$

But I can switch if it helps.

John Baez (Jul 23 2024 at 16:33):

There are various flavors of categorified Fourier transform, like the [[Fourier-Mukai transform]] going from the derived category of coherent sheaves on an abelian variety to the derived category of coherent sheaves on its dual. Here we are thinking of a coherent sheaf as a categorified version of a function, and an abelian variety as being like an abelian group. ([[Pontryagin duality]] already generalized Fourier duality from $\mathbb{R}$ to locally compact abelian groups.)

John Baez (Jul 23 2024 at 16:49):

But I don't actually see how the Fourier-Mukai transform would help me.

Eric M Downes (Jul 23 2024 at 18:04):

(somehow deleted my message while trying to edit it...)
John Baez said:

But I don't actually see how the Fourier-Mukai transform would help me.

Briefly

1. To see if what you are already doing has been done in quantum mechanics; any partition function is a [[Wick rotation]] applied to the eigendecomposition solution of some Schrödinger equation.
2. It gives you access to physics models in which trajectory reversibility and entropy both have precise meaning within the same system. Consider if there is a functor $\mathcal{G}:{\sf Set\to Grp}$ such that for every $E:X\to\mathbb{R}$ there is a unqiue $E':\mathcal{G}X\to\mathbb{R}$ commuting with the forgetful functor taking a group to its underlying set. Many classic statmech systems are realizable as markov models on the cayley graph of some group, where the transition rates label the edges; these systems may serve as hypothesis and extension generating examples for what other things a categorization might do, because the fourier transform on groups is so well understood.

Other stuff could be said, about how $\beta<0$ isnt as unphysical as it seems, etc. and I'll retype if anyone would like me to.

David Egolf (Jul 23 2024 at 18:15):

The notation you've used leads me to wonder if we can nicely promote $Z$ to a functor! I haven't been able to figure out a nice way to achieve this though, yet. I wonder if there is some nice notion of morphism between partition functions.

In the direction of trying to promote $Z$ to a functor, the other idea that occurs to me is to think about the "most informative concise observation" of $\mathsf{FinSet}/\mathbb{R}$, as illustrated in this picture:
picture

The idea is to look for some functor $Z:\mathsf{FinSet}/\mathbb{R} \to A$ that is initial among functors that preserve the rig category structure and also satisfy some more properties analogous to those that the original version of $Z$ satisfies. (Such a functor may or may not exist.) For example, we might require each such functor $F$ to satsify this condition:

• $F(E) = F(E') \iff E \cong E'$.

We might need to put more conditions on these functors to potentially get something interesting from this approach. Maybe we need something relating to cardinality? But I'm not sure what exactly this would look like!

David Egolf (Jul 23 2024 at 18:52):

I notice that $Z$ sends two isomorphic energetic sets to two equal partition functions. And it sends two non-isomorphic energetic sets to two non-equal partition functions. For this reason, I wonder if we can form a category of partition functions by taking a skeleton of $\mathsf{Set}/\mathbb{R}$. Might we be able to realize $Z$ as a functor, using this?

(I'm getting tired though, so I'll stop here.)

John Baez (Jul 23 2024 at 19:30):

For any category $C$ there's a functor from its core (the groupoid of objects of $C$ and isomorphisms) to the discrete category whose objects are isomorphism classes of objects of $C$. This functor sends any object to its isomorphism class.

When $C = \mathsf{Set}$ this functor is called cardinality. When $C$ is the category of energetic sets this functor is $Z$ (up to isomorphism).

Matteo Capucci (he/him) (Jul 24 2024 at 06:52):

One naive way is to make X a category and E a functor into energetic sets again. I don't really know how to define Z in that case without going in unexplored quantitative category theory land...

Peva Blanchard (Jul 24 2024 at 07:54):

The partition function as cardinality reminds me of Tom Leinser's work on magnitude (see e.g. the introduction here)

If I go on with the analogy. Every energetic set $(X,E)$ comes with a similarity matrix

$$A_{ij} = \frac{e^{\beta \cdot (E_j - E_i)}}{Z(\beta)}$$

Then the vector $u = (e^{-\beta E_j})_{j \in X}$ is a weigthing

$$\sum_j A_{ij} u_j = 1$$

and $Z(\beta)$ is the corresponding magnitude.

Tom Leinster generalizes this to enriched category: $A_{ij}$ becomes a hom object (in some enriching category $\mathcal{V}$), and $u$ an enriched presheaf (or op-presheaf).

John Baez (Jul 24 2024 at 08:53):

Interesting idea! Let me just check:

$\displaystyle{ \sum_j A_{ij} u_j = \sum_j \frac{e^{\beta(E_j - E_i)}}{Z(\beta)} e^{-\beta E_j} = \sum_j \frac{e^{-\beta E_i}}{Z(\beta)} \ne 1 }$

John Baez (Jul 24 2024 at 08:56):

Did I screw up or do you need to fix your formulas somehow?

Peva Blanchard (Jul 24 2024 at 09:50):

Oh you're right, my formula for $A_{ij}$ is wrong!

Actually, I don't see any natural formula for $A_{ij}$ in terms of the energetic set. Given an arbitrary stochastic matrix $\mathbb{P}(j|i)$, we can build a matrix $A_{ij} = \mathbb{P}(j | i) \cdot e^{\beta E_j}$. And $Z(\beta)$ is the magnitude of $A$.

I don't know if it provides additional insights though.

Peva Blanchard (Jul 24 2024 at 14:56):

There is another analogy (I hope I'm not wrong this time).

The partition function can be rewritten

$$Z(\beta) = \sum_{x \in X} e^{ -\beta E_x} = \int \omega(\epsilon)d\epsilon \cdot e^{- \beta \epsilon}$$

where $\omega(\epsilon) d\epsilon$ counts the number of points $x \in X$ with $E_x = \epsilon$.

In your paper Groupoidification made easy, there is a generating function $Z \in \mathbb{R}^{Iso(B)}$ associated with every groupoid morphism $E : X \to B$. We can formally write

$$Z(\beta) = \sum_{\epsilon \in Iso(B)} \lvert E^{-1}(\epsilon) \rvert \cdot e^{-\beta\epsilon}$$

where $\lvert E^{-1}(\epsilon)\rvert$ is the groupoid cardinality of the essential preimage $E^{-1}(\epsilon)$ of $\epsilon$.

$$\lvert E^{-1}(\epsilon)\rvert = \sum_{[x] \in Iso(E^{-1}(\epsilon))} \frac{1}{\lvert Aut~x \rvert}$$

And $e^{-\beta\epsilon}$ is a formal symbol that represents the function in $\mathbb{R}^{Iso(B)}$ that outputs $1$ on $\epsilon$ and $0$ everywhere else.

John Baez (Jul 24 2024 at 15:02):

I don't think I want a formal symbol $e^{-\beta \epsilon}$. I want a function on the real line: an exponential function. I'm doing a Laplace transform.

That is, whatever abstract framework we've got, I hope it has the usual partition function $Z$ as a special case.

David Egolf (Jul 24 2024 at 16:11):

The mention of Leinster's work above reminds me that he wrote a book called "Entropy and Diversity: The Axiomatic Approach". Chapter 12 is entitled "The categorical origins of entropy", which might be relevant. (I haven't read much of this book yet - so I don't know how relevant it is).

John Baez (Jul 24 2024 at 17:30):

Tom Leinster is one of my best friends and I know all that stuff: we wrote a paper on that.

John Baez (Jul 24 2024 at 17:31):

(He teaches at the University of Edinburgh and got me over here for a 2-year Leverhulme Fellowship; I liked it there so much that I got a job here.)

John Baez (Jul 24 2024 at 17:32):

Nonetheless your suggestion is a useful one in this way: you're making me realize I should ask Tom my question here.

John Baez (Jul 24 2024 at 17:33):

He's just not here now: he's down in Barcelona.

David Egolf (Jul 24 2024 at 18:09):

I've been thinking about this puzzle with an extended analogy to medical imaging :sweat_smile:. I don't know if this will be useful, but it's kind of fun, so I'd thought I'd share it here:

David Egolf (Jul 24 2024 at 18:11):

We think of our finite set $X$ as a set of "possible reconstructions of a point". Each element of $X$ corresponds to some particular "brightness" of that point (e.g. this "brightness" could correspond to the estimated amplitude of an ultrasound wave that bounced off this point).

David Egolf (Jul 24 2024 at 18:13):

The energy $E(x)$ of some $x \in X$ is some measure of how much we "like" the particular reconstruction $x$. This could be based on some measure of consistency between observed data and some theoretically predicted data (that we would expect to be observed if $x$ was present).

(As $E(x)$ goes up, how much we like $x$ decreases).

David Egolf (Jul 24 2024 at 18:16):

We then want to put a probability distribution on all the different reconstructions available to us. To do this, we introduce a parameter $\beta$. Using $\beta$, the probability we assign to the reconstruction $x \in X$ is proportional to $e^{-\beta E(x)}$.

Intuitively, different values of $\beta$ correspond to different ways of using the information about how much we "like" each reconstruction $x$, which is encoded by $E(x)$. When $\beta$ is zero, we completely ignore our $E(x)$ preference for each $x$. Presumably we take $E$ increasingly more into account as $\beta$ becomes larger.

David Egolf (Jul 24 2024 at 18:24):

I am wondering if this perspective on $X$ can help with upgrading $X$ to a non-discrete category. In the case above, each element of $X$ is to be thought of as a particular reconstruction of a single (fixed) part of an image. So, perhaps we could consider a category where objects correspond to reconstructions of various parts of an image!

David Egolf (Jul 24 2024 at 18:25):

Using this analogy, we might consider categories $X$ where each object of $X$ corresponds to a particular state of some part of a (fixed) system.

David Egolf (Jul 24 2024 at 18:31):

Returning to the imaging analogy, if the objects of some category are reconstructions for parts of an image, then what could morphisms be? The first idea that occurs to me is this:

• let $I \sub \mathbb{R}^2$ be the "canvas" for our image
• then a partial reconstruction is a function $f:U \to \mathbb{R}$ such that $U \subseteq I$
• a morphism from $f:U \to \mathbb{R}$ to $f':U' \to \mathbb{R}$ is then a "witness of compatibility", namely an open set $V \subseteq U \cap U'$ such that $f|_V = f'_V$

I have not checked that this actually forms a category, yet.

David Egolf (Jul 24 2024 at 18:34):

Hmm. This probably doesn't form a category - composition seems like it would be problematic.

David Egolf (Jul 24 2024 at 18:36):

A simpler category can maybe still be obtained though:

• each object is a function $f:U \to \mathbb{R}$ for $U$ some subset of $I$
• we have a single morphism from $f:U \to \mathbb{R}$ to $f':U' \to \mathbb{R}$ if $U \subseteq U'$, and $f$ and $f'$ agree on $U$

I think this is a category, and in particular a subcategory of $\mathsf{Set}/\mathbb{R}$.

David Egolf (Jul 24 2024 at 18:42):

Our original case of a set $X$ can be recovered (roughly) as a discrete category. We can do this by considering a category where each object is a distinct function from some fixed $U$ to $\mathbb{R}$. These are "competing reconstructions" on the same part of an image. There will be no morphisms between distinct objects in this setting, because none of these reconstructions are "part of" a different one of these reconstructions.

David Egolf (Jul 24 2024 at 18:46):

By analogy, we might consider a category where:

• objects are states of parts of the entire system
• we have a morphism from one object $a$ to another $b$ if this condition is met: if $a$ specifies a state for some part of our system, then $b$ specifies a state for (at least) that part of the system too, and $a$ and $b$ assign the same state to that part of the system

When each object is associated to the same part of the system, then the resulting category is discrete.

David Egolf (Jul 24 2024 at 18:47):

I don't know if any of this is helpful at all! But it was interesting to me, so I thought I would share.

My hope is that possibly this line of thought could be helpful for upgrading $X$ from a set to a poset.

Nathaniel Virgo (Jul 31 2024 at 02:49):

To @John Baez's original question, I wonder if [[combinatorial species]] might point in a useful direction?

This post contains a basic arithmetic mistake, sorry. It's probably not worth reading.

I know very little about this and I know you know way more, but going from Wikipedia's exposition (which has a combinatorial species being a functor $\rm core({\bf FinSet})\to \rm core({\bf FinSet})$), from a species you can define a generating function

$$f(x) = \sum_{n\in\mathbb{N}}|F(n)|\frac{x^n}{n!}.$$

This can be made more suggestive by defining $E(n) = |F(n)|/n!$ and $\beta = -\log x$ so that we can define

$$\cancel{Z(\beta)\coloneqq f(x) = \sum_{n\in\mathbb{N}} e^{-\beta E(n)}.}$$

(This should have been $\sum_{n\in\mathbb{N}}g(n)e^{-\beta n}$ with $g(n) = |F(n)|/n!$ as David Egolf pointed out. Which still looks a bit like a partition function if you squint, in which the energy is $E(n)=n$ and $g(n)$ is a degeneracy of each state, but this is much less interesting than I thought it was. The rest of the post doesn't really work now.)

This looks like a partition function, which is maybe a trivial observation. It's a bit odd that it's over $\mathbb{N}$ rather than an arbitrary set, and I had to sweep the factors of $1/n!$ under the carpet to get to it.)

But the point is that addition and multiplication of generating functions yields a coproduct and a monoidal product on species, which presumably make them into a rig category. So this is a place where "partition functions" and a rig category come up naturally.

I don't know if there's a way to "fix" this idea so that the sum would be over an arbitrary set instead of $\mathbb{N}$ while keeping the combinatorial flavour, but maybe there's a way?

David Egolf (Jul 31 2024 at 03:38):

I am trying to follow the algebra. $E(n) = |F(n)|/n!$, so our sum becomes $f(x) = \sum_{n \in \mathbb{N}}E(n)x^n$. Then we have $-\beta=\log x$ so that $e^{-\beta}=x$. So I would expect our sum to become $f(x) = \sum_{n \in \mathbb{N}}E(n) e^{-\beta n}$. But this doesn't match what you wrote above, so I suspect I may be missing something!

Nathaniel Virgo (Jul 31 2024 at 05:33):

Oh, you're right. I've got something wrong. Hang on, let me see if I can fix it or if the whole post is nonsense

Peva Blanchard (Jul 31 2024 at 06:04):

FYI, I recently asked about combinatorial species on this thread. One of the things I learned there is that the factor $n!$ is related to some action of the symmetric group $\mathfrak{S}_n$.

While studying other stuff (Feynman diagrams), I learned that the partition function can be related to the exponential generating series of weighted species (or a variant of). I'd need to work the details a bit for our situation here, but I'm relatively confident that there is a reformulation.

(However, I think that John already knows those things very well.)

Peva Blanchard (Jul 31 2024 at 06:49):

I think, we can do it this way.

Let $E : X \to \mathbb{R}$ be the original energetic set.
Then, for every $n \ge 0$, there is a energetic set $Z[n]: X^n \to \mathbb{R}$ defined by

$$Z[n](x_1,\dots,x_n) = \begin{cases} \epsilon^n &\text{if } E(x_1) = \dots = E(x_n) = \epsilon \\ 0 &\text{otherwise} \end{cases}$$

The weighted species $Z$ defined by those $Z[n]$ has the following exponential generating series

$$\begin{align*} Z_{egs}(\beta) &= \sum_{n \ge 0} |Z[n]| \cdot \frac{\beta^n}{n!} \\ &= \sum_{n \ge 0} \Big( \sum_{x \in X^n} Z[n](x_1,\dots,x_n) \Big) \frac{\beta^n}{n!} \\ &= \sum_{n \ge 0} \sum_{\epsilon} \omega(\epsilon) \cdot \frac{(\beta \epsilon)^n}{n!} \\ &= \sum_{\epsilon} \omega(\epsilon) e^{\beta\epsilon} \\ \end{align*}$$

where $\omega(\epsilon)$ counts the number of $x \in X$ with $E(x) = \epsilon$.
Hence, we recognize

$$Z_{egs}(\beta) = \sum_k e^{\beta E_k}$$

(These things are relatively new to me, so I might have made a mistake somewhere)

Nathaniel Virgo (Jul 31 2024 at 06:59):

What's the definition of a weighted species?

Peva Blanchard (Jul 31 2024 at 07:03):

A weighted species is like a species except that the target category is the category of energetic sets (aka weighted sets).

I.e., a weighted species is a functor $F : B \to B/\mathbb{R}$ where $B$ is the groupoid of finite sets and bijection, $B/\mathbb{R}$ is the category of energetic sets with "energy-preserving bijections".

Peva Blanchard (Jul 31 2024 at 07:07):

You can check out chapter 3 of this book.

John Baez (Jul 31 2024 at 08:41):

Nathaniel Virgo said:

To John Baez's original question, I wonder if [[combinatorial species]] might point in a useful direction?

This post contains a basic arithmetic mistake, sorry. It's probably not worth reading.

It was worth reading, because it may be possible to fix the mistake by using the "Dirichlet generating function" of a species instead of the usual "exponential generating function". I've been studying the Dirichlet generating function here:

• John Baez and James Dolan, Dirichlet speces and the Hasse-Weil zeta function.

Suppose $F$ is a species. Then the exponential generating function of $F$ is the formal power series

$\displaystyle{ \sum_{n \ge 0} \frac{|F(n)|}{n!} x^n }\, .$

On the other hand, the Dirichlet series associated to $F$ is

$\displaystyle{ \sum_{n \ge 1} \frac{|F(n)|}{n!} n^{-s} } \, .$

The most famous of all Dirichlet series is the Riemann zeta function:

$\displaystyle{ \zeta(s) = \sum_{n \ge 1} n^{-s}} \, .$

This comes from a species we call the Riemann species, $Z$. A $Z$-structure on a finite set is a way of making it into a semisimple commutative ring - that is, a product of finite fields.

John Baez (Jul 31 2024 at 08:49):

The Riemann zeta function is the partition function of the primon gas, a made-up physical system whose states are positive natural numbers, where the energy of state $n$ is $\ln n$. People influenced by physics think of this system as a gas of prime numbers where the $n$ th state is a state where the gas is made of the prime factors of $n$ - see the link for more.

John Baez (Jul 31 2024 at 08:51):

So this is sort of interesting and deserves to be generalized; in a way James Dolan and I have already done it in the linked paper - but I wasn't thinking in terms of partition functions! So thanks, @Nathaniel Virgo.

John Baez (Jul 31 2024 at 13:07):

@Nathaniel Virgo - thanks to you I now better understand how the Dirichlet series of a species is an example of a partition function, and thus a species is a kind of "energetic groupoid", generalizing an energetic set.

John Baez (Jul 31 2024 at 13:08):

I could explain this but I don't have time now to do a really good job.

The main limitation of this outlook is that it only handles energetic sets, or groupoids, where all the allowed energies are of the form $\ln n$.

John Baez (Jul 31 2024 at 13:11):

I'll just mention this example: the Riemann zeta function is known to be the partition function of the energetic set $\mathbb{N}$ where the energy of $n \in \mathbb{N}$ is $\ln n$. But we can also think of it as the partition function, or Dirichlet series, of a groupoid where the objects are products of finite fields. (This groupoid is the groupoid of elements of the "Riemann species", which I described above.)

So here's the categorification going on: instead of just the set of all natural numbers, we are now working with the groupoid of commutative rings that are products of finite fields. What's the connection? Any such commutative ring has a cardinality which is a natural number.

John Baez (Jul 31 2024 at 13:16):

All of this is going in a very number-theoretic direction, which hadn't been on my mind when I posed my original question in this thread! The number theory and the statistical mechanices were in somewhat separate compartments in my brain, even though I knew about the primon gas.

David Egolf (Jul 31 2024 at 16:37):

I had a thought about the energy function $E: X \to \R$. Such a function associates to each element of $X$ some real number. So, we might view $E$ as specifying an $\mathbb{R}$-weighted "linear combination" of the elements of $X$. That is, at least for finite $X$, we could think of $E$ as an element of the free $\mathbb{R}$-vector space generated by $X$.

So, if someone knows a way to categorify free vector spaces generated by a set, perhaps an "element" (whatever that means) of such a structure could provide a categorified energy function.

John Baez (Jul 31 2024 at 18:41):

Ross Street and George Janelidze have a formalism for dealing with something like monoidal categories, where instead of just being able to form new objects from old by tensoring them like this:

$X \otimes X \otimes Y \otimes X \otimes Z$

or for short

$X^{\otimes 2} \otimes Y \otimes X \otimes Z$

you can form objects like this:

$X^{\otimes 1.739} \otimes Y^{\pi} \otimes Z \otimes X$

John Baez (Jul 31 2024 at 18:43):

In particular you can use this idea to categorify the set $[0,\infty)$. Objects in this category can be thought of as sticks of arbitrary nonnegative length that can be cut into pieces and glued together - but the process of chopping up a stick, rearranging its pieces and gluing it back together is a nontrivial morphism!

John Baez (Jul 31 2024 at 18:48):

I think they call the objects in this category real sets.

Nathaniel Virgo (Aug 01 2024 at 05:41):

John Baez said:

I could explain this but I don't have time now to do a really good job.

The main limitation of this outlook is that it only handles energetic sets, or groupoids, where all the allowed energies are of the form $\ln n$.

It sounds fascinating - I'd be happy to understand it properly if you end up writing it down at some point.

I'm just speculating because I don't really understand it, but maybe the $\log n$ thing makes some intuitive sense if we think of it as entropy rather than energy, in the Boltzmann $\log W$ sense?

John Baez (Aug 01 2024 at 06:42):

It sounds fascinating - I'd be happy to understand it properly if you end up writing it down at some point.

The math is mostly present already in Dirichlet species and the Hasse-Weil zeta function. The new twist would be the physics interpretation of the Dirichlet series of a species as a partition function. This is already 'well known' in the special case of the Riemann species, whose Dirichlet series is the simplest of all, the Riemann zeta function. That is, a bunch of people talk about how the Riemann zeta function is the partition function of the primon gas. So it's 'just' a matter of taking the pushout of these ideas. But I'd never done it before, and I'm not claiming this paragraph counts as a decent explanation of what's going on.

I am claiming that reading the two links may provide you with hours of free entertainment. :upside_down:

John Baez (Aug 01 2024 at 06:55):

I'm just speculating because I don't really understand it, but maybe the $\log n$ thing makes some intuitive sense if we think of it as entropy rather than energy, in the Boltzmann $\log W$ sense?

If we're thinking in terms of a partition function we have to think of $\ln n$ as the energy $E_n$ of the number $n$, because the partition function is always defined as

$\sum_n e^{-\beta E_n}$

and $E_n = \ln n$ makes the partition function equal to the Riemann zeta function:

$\displaystyle{ \sum_n e^{-\beta E_n} = \sum_n e^{- \beta \ln n} = \sum_n \frac{1}{n^\beta} = \zeta(\beta) }$

But there's a very nice interpretation of this: each prime number $p$ has energy $\ln p$, and the energy of an arbitrary positive integer is the sum of the energy of the primes in its prime factorization!

So, we are to imagine a cylinder of gas where the atoms come in different kinds, one for each prime number. If we allow an arbitrary number of atoms of each different kind, the partition function of this system is the Riemann zeta function.