Thank you, both! I guess I'll mostly be taking a long nap until May 5 then. Since David Spivak will be visiting here May 6–11, I'll be quite distracted during those days, but I'll still try to check in here as frequently as possible.

> I hadn't seen your paper [A resource theory for work and heat](https://arxiv.org/abs/1607.01302). It looks interesting, but it looks more like conventional physics than "symmetric monoidal posets". I wonder how it's connected to your earlier paper.

Roughly, the story is like this: people who study [thermodynamics at the nanoscale](https://arxiv.org/abs/0908.2076)—or *single-shot* thermodynamics as they call it—secretly work with a symmetric monoidal poset. Its elements are states of some physical system where one is above another if there exists a 'thermodynamic transformation' that turns the first into the second. When you take this symmetric monoidal poset and perform the regularization from my [Resource convertibility and ordered commutative monoids](https://arxiv.org/abs/1504.03661) paper on it, then you get macroscopic thermodynamics! Mathematically, the regularization of the symmetric monoidal poset is the positive cone of an ordered vector space. This is the cone that we've studied in the thermodynamics paper: one of its directions is 'amount of substance', and when you slice it a unit amount of substance, then you then get the convex set that we've called the energy-entropy diagram. The energy-entropy diagram is a very convenient tool e.g. for analyzing heat engines with finite (but large) reservoirs. What still astonishes me is that this seems to be new.

When I say 'roughly' in my first sentence, this is because I've made some statements that are morally correct, but technically false. Like in any other information-theoretic context, it's all about being able to implement *approximate* transformations between states that can get arbitrarily close to the target state. I still don't know how to fit this into the framework of symmetric monoidal posets. I've tried out three different approaches which all looked promising at first, but none of them actually works. Another loose end is to find a physically meaningful derivation of what a 'thermodynamic transformation' is—but I should stop talking about this now, unless people want to know more.

***

Matthew, nice to hear from you!

> Is FinStat the opposite category of what Terrence Tao sketches in this 2010 blog post?

No, Terry Tao's category is what I would call 'the category of probability spaces'. If you restrict this to finite sets, then you get a category that we've called **FinProb**—see Definition 1 from our [first paper](https://arxiv.org/abs/1106.1791). A morphism between finite probability spaces \\((X,p)\\) and \\((Y,q)\\) in **FinStat** also consists of a measure-preserving map \\(f : X\to Y\\), but *in addition* you must specify a stochastic map \\(s : Y\to X\\) such that \\(f\circ s = 1_Y\\). There is a functor **FinStat** \\(\rightarrow\\) **FinProb** which forgets \\(s\\).

We've interpreted this definition of **FinStat** in terms of subjective Bayesian probability: imagine that \\((X,p)\\) describes the possible states of the world, and \\((Y,q)\\) describes your possible perceptions of the world. Since every state of the world leads to a unique perception, we have a measure-preserving map \\(f : X\to Y\\). Now if you experience some particular perception \\(y\in Y\\), what do you think that the state of the world is? It clearly must be some element of the set \\(f^{-1}(y)\\). But since you don't know which one, you'll have some probability distribution on \\(f^{-1}(y)\\), and this is exactly \\(s(y)\\).

> I hadn't seen your paper [A resource theory for work and heat](https://arxiv.org/abs/1607.01302). It looks interesting, but it looks more like conventional physics than "symmetric monoidal posets". I wonder how it's connected to your earlier paper.

Roughly, the story is like this: people who study [thermodynamics at the nanoscale](https://arxiv.org/abs/0908.2076)—or *single-shot* thermodynamics as they call it—secretly work with a symmetric monoidal poset. Its elements are states of some physical system where one is above another if there exists a 'thermodynamic transformation' that turns the first into the second. When you take this symmetric monoidal poset and perform the regularization from my [Resource convertibility and ordered commutative monoids](https://arxiv.org/abs/1504.03661) paper on it, then you get macroscopic thermodynamics! Mathematically, the regularization of the symmetric monoidal poset is the positive cone of an ordered vector space. This is the cone that we've studied in the thermodynamics paper: one of its directions is 'amount of substance', and when you slice it a unit amount of substance, then you then get the convex set that we've called the energy-entropy diagram. The energy-entropy diagram is a very convenient tool e.g. for analyzing heat engines with finite (but large) reservoirs. What still astonishes me is that this seems to be new.

When I say 'roughly' in my first sentence, this is because I've made some statements that are morally correct, but technically false. Like in any other information-theoretic context, it's all about being able to implement *approximate* transformations between states that can get arbitrarily close to the target state. I still don't know how to fit this into the framework of symmetric monoidal posets. I've tried out three different approaches which all looked promising at first, but none of them actually works. Another loose end is to find a physically meaningful derivation of what a 'thermodynamic transformation' is—but I should stop talking about this now, unless people want to know more.

***

Matthew, nice to hear from you!

> Is FinStat the opposite category of what Terrence Tao sketches in this 2010 blog post?

No, Terry Tao's category is what I would call 'the category of probability spaces'. If you restrict this to finite sets, then you get a category that we've called **FinProb**—see Definition 1 from our [first paper](https://arxiv.org/abs/1106.1791). A morphism between finite probability spaces \\((X,p)\\) and \\((Y,q)\\) in **FinStat** also consists of a measure-preserving map \\(f : X\to Y\\), but *in addition* you must specify a stochastic map \\(s : Y\to X\\) such that \\(f\circ s = 1_Y\\). There is a functor **FinStat** \\(\rightarrow\\) **FinProb** which forgets \\(s\\).

We've interpreted this definition of **FinStat** in terms of subjective Bayesian probability: imagine that \\((X,p)\\) describes the possible states of the world, and \\((Y,q)\\) describes your possible perceptions of the world. Since every state of the world leads to a unique perception, we have a measure-preserving map \\(f : X\to Y\\). Now if you experience some particular perception \\(y\in Y\\), what do you think that the state of the world is? It clearly must be some element of the set \\(f^{-1}(y)\\). But since you don't know which one, you'll have some probability distribution on \\(f^{-1}(y)\\), and this is exactly \\(s(y)\\).