## September 15, 2024

### The Space of Physical Frameworks (Part 4)

#### Posted by John Baez

In Part 1, I explained my hopes that classical statistical mechanics reduces to thermodynamics in the limit where Boltzmann’s constant $k$ approaches zero. In Part 2, I explained exactly what I mean by ‘thermodynamics’. I also showed how, in this framework, a quantity called ‘negative free entropy’ arises as the Legendre transform of entropy.

In Part 3, I showed how a Legendre transform can arise as a limit of something like a Laplace transform.

Today I’ll put all the puzzle pieces together. I’ll explain exactly what I mean by ‘classical statistical mechanics’, and how negative free entropy is defined in *this* framework. Its definition involves a Laplace transform. Finally, using the result from Part 3, I’ll show that as $k \to 0$, negative free entropy in classical statistical mechanics approaches the negative free entropy we’ve already seen in thermodynamics!

## September 9, 2024

### The Space of Physical Frameworks (Part 3)

#### Posted by John Baez

In Part 1, I explained how statistical mechanics is connected to a rig whose operations depend on a real parameter $\beta$ and approach the ‘tropical rig’, with operations $\min$ and $+$, as $\beta \to +\infty$. I explained my hope that if we take equations from classical statistical mechanics, expressed in terms of this $\beta$-dependent rig, and let $\beta \to +\infty$, we get equations in thermodynamics. That’s what I’m slowly trying to show.

As a warmup, last time I explained a bit of thermodynamics. We saw that some crucial formulas involve Legendre transforms, where you take a function $f \colon \mathbb{R} \to [-\infty,\infty]$ and define a new one $\tilde{f} \colon \mathbb{R} \to [-\infty,\infty]$ by

$\tilde{f}(s) = \inf_{x \in \mathbb{R}} (s x - f(x))$

I’d like the Legendre transform to be something like a limit of the Laplace transform, where you take a function $f$ and define a new one $\hat{f}$ by

$\hat{f}(s) = \int_{-\infty}^\infty e^{-s x} f(x) \, d x$

Why do I care? As we’ll see later, classical statistical mechanics features a crucial formula that involves a Laplace transform. So it would be great if we could find some parameter $\beta$ in that formula, take the limit $\beta \to +\infty$, and get a corresponding equation in thermodynamics that involves a Legendre transform!

As a warmup, let’s look at the purely mathematical question of how to get the Legendre transform as a limit of the Laplace transform — or more precisely, something *like* the Laplace transform. Once we understand that, we can tackle the physics in a later post.

## September 7, 2024

### The Space of Physical Frameworks (Part 2)

#### Posted by John Baez

I’m trying to work out how classical statistical mechanics can reduce to thermodynamics in a certain limit. I sketched out the game plan in Part 1 but there are a lot of details to hammer out. While I’m doing this, let me stall for time by explaining more precisely what I mean by ‘thermodynamics’. Thermodynamics is a big subject, but I mean something more precise and limited in scope.

## August 31, 2024

### The Space of Physical Frameworks (Part 1)

#### Posted by John Baez

Besides learning about individual physical theories, students learn different *frameworks* in which physical theories are formulated. I’m talking about things like this:

- classical statics
- classical mechanics
- quantum mechanics
- thermodynamics
- classical statistical mechanics
- quantum statistical mechanics

A physical framework often depends on some physical constants that we can imagine varying, and in some limit one framework may reduce to another. This suggests that we should study a ‘moduli space’ or ‘moduli stack’ of physical frameworks. To do this formally, in full generality, we’d need to define what counts as a ‘framework’, and what means for two frameworks to be equivalent. I’m not ready to try that yet. So instead, I want to study an example: a 1-parameter family of physical frameworks that includes classical statistical mechanics — and, I hope, also thermodynamics!

## August 25, 2024

### Stirling’s Formula from Statistical Mechanics

#### Posted by John Baez

Physicists like to study all sorts of simplified situations, but here’s one I haven’t seen them discuss. I call it an ‘energy particle’. It’s an imaginary thing with no qualities except energy, which can be any number $\ge 0$.

I hate it when on Star Trek someone says “I’m detecting an energy field” — as if energy could exist without any specific form. That makes no sense! Yet here I am, talking about energy particles.

Earlier on the *n*-Café, I once outlined a simple proof of Stirling’s formula using Laplace’s method. When I started thinking about statistical mechanics, I got interested in an alternative proof using the Central Limit Theorem, mentioned in a comment by Mark Meckes. Now I want to dramatize that proof using energy particles.

## August 16, 2024

### Bernoulli Numbers and the Harmonic Oscillator

#### Posted by John Baez

I keep wanting to understand Bernoulli numbers more deeply, and people keep telling me stuff that’s fancy when I want to understand things *simply*. But let me try again.

## August 15, 2024

### Galois Theory

#### Posted by Tom Leinster

I’ve just arXived my notes for Edinburgh’s undergraduate Galois theory course, which I taught from 2021 to 2023.

I first shared the notes on my website some time ago. But it took me a while to arXiv them, because I wanted to simultaneously make public most of the other course materials. I now have, which means the following are now available to all:

Notes forming a complete, self-contained account of the part of Galois theory that we covered.

About 40 short explanatory videos.

A large collection of problems.

Nearly 500 multiple choice questions.

## August 14, 2024

### Confluence in Graph Rewriting

#### Posted by John Baez

*guest post by Anna Matsui and Innocent Ob*

In this blog post we discuss the main (‘algebraic’) ideas of term rewriting and how they can be applied to term graph and
arbitrary graph (i.e. string diagrams) rewriting. Our group was
interested in the application of critical pairs for string diagram
rewriting. We read Detlef Plump’s *Critical Pairs in Term Graph Rewriting* and given the task before us wanted to understand the following question: if Plump needed a new decision procedure for critical pairs in graph rewriting, how would we know what we might need for string diagram rewriting? In the this blog post we start at the root (pun intended) and work our way up from term rewriting to rewriting on arbitrary graphs. The goal is to understand, for example, why we can’t just apply, without modification,
the ideas in Plump’s paper to rewriting in the *λ*-calculus. This post is aimed at an audience unfamiliar with term rewriting and presents foundational ideas and results to set the stage for (old and more) recent categorical applications.

## August 13, 2024

### Introduction to Categorical Probability

#### Posted by Tom Leinster

*Guest post by Utku Boduroğlu, Drew McNeely, and Nico Wittrock*

*When is it appropriate to completely reinvent the wheel?*
To an outsider, that seems to happen a lot in category theory, and probability theory isn’t spared from this treatment.
We’ve had a useful language to describe probability since the 17th century, and it works. Why change it up now?

This may be a tempting question when reading about categorical probability, but we might argue that this isn’t completely reinventing traditional probability from the ground up.
Instead, we’re developing a set of tools that allows us to work with traditional probability in a really powerful and intuitive way, and these same tools also allow us define new kinds of uncertainty where traditional probability is limited.
In this blog post, we’ll work with examples of both traditional finite probability and nondeterminism, or *possibility* to show how they can be unified in a single categorical language.

## August 12, 2024

### Skew Monoidal Categories Through Triangulations and Examples

#### Posted by Tom Leinster

*Guest post by Thea Li and Pablo S. Ocal*

The study of monoidal categories and their applications is an essential part of the research and applications of category theory. However, on occasion the coherence conditions of these categories turns out to be too strong for what one might want (or one might simply be curious as to what happens when their axioms are not all met), leading to the development of various weakenings of these notions. In our case at hand, we consider a weakening in which neither the associator nor the unitors are invertible, which is known as a skew monoidal category. A particularity of these categories is that Mac Lane’s coherence theorem does not apply, whereas there may be parallel arrows formed by structural maps that are not equal.

## August 10, 2024

### Prismatic Category Theory

#### Posted by Tom Leinster

*Guest post by C.B. Aberlé and Rubén Maldonado*

Fibrations are a fundamental concept of category theory and categorical logic that have become increasingly relevant to the world of applied category theory thanks to their prominent use in applications such as the categorical semantics of dependent type theory, the study of dynamical systems, etc. With the increasing occurrence of *higher*-categorical structures in these applications, there is an evident need for both pure and applied category theorists to develop and refine higher-categorical analogues of the ordinary theory of fibrations.

This blog post aims to sketch the basis of a general framework for the study of higher-categorical fibrations. We begin with a recap of the fundamental building-blocks of the theory of Grothendieck fibrations, including Cartesian arrows and the Grothendieck construction. In doing so, we develop a simple graphical framework for studying properties of functors. An analysis of this framework then reveals that it can be interpreted in the internal language of the topos of *simplicial sets*, which paves the way for studying fibrations on higher-categorical structures such as double categories.

## August 8, 2024

### Dimensional Analysis in Algebra and Geometry

#### Posted by John Baez

Let’s take a look at how various mathematicians over the ages have dealt with the idea that quantities have ‘dimensions’, in the sense of dimensional analysis.

## July 20, 2024

### What Is Entropy?

#### Posted by John Baez

I wrote a little book about entropy; here’s the current draft:

If you see typos and other mistakes, or have trouble understanding things, please let me know!

An alternative title would be *92 Tweets on Entropy*, but people convinced me that title wouldn’t age well: in decade or two few people may remember what ‘tweets’ were.

Here is the foreword, which explains the basic idea.

## July 15, 2024

### Skew-Monoidal Categories: Logical and Graphical Calculi

#### Posted by Emily Riehl

*guest post by Wilf Offord*

One of the earliest and most well-studied definitions in “higher” category theory is that of a monoidal category. These have found ubiquitous applications in pure mathematics, physics, and computer science; from type theory to topological quantum field theory. The machine making them tick is MacLane’s *coherence theorem*: if anything deserves to be called “the fundamental theorem” of monoidal categories, it is this. As such, numerous other proofs have sprung up in recent years, complementing MacLane’s original one. One strategy with a particularly operational flavour uses *rewriting systems*: the morphisms of a free monoidal category are identified with normal forms for some rewriting system, which can take the form of a logical system as in (UVZ20,Oli23), or a diagrammatic calculus as in (WGZ22). In this post, we turn to *skew*-monoidal categories, which no longer satisfy a coherence theorem, but nonetheless can be better understood using rewriting methods.

## July 12, 2024

### Double Limits: A User’s Guide

#### Posted by Emily Riehl

*Guest post by Matt Kukla and Tanjona Ralaivaosaona*

Double limits capture the notion of limits in double categories. In ordinary category theory, a limit is the best way to construct new objects from a given collection of objects related in a certain way. Double limits, extend this idea to the richer structure of double categories. For each of the limits we can think of in an ordinary category, we can ask ourselves: how do these limits look in double categories?