## Symposium on Compositional Structures 9

9 July, 2022

The Symposium on Compositional Structures is a nice informal conference series that happens more than once a year. You can now submit talks for this one.

Ninth Symposium on Compositional Structures (SYCO 9), Como, Italy, 8-9 September 2022. Deadline to submit a talk: Monday 1 August 2022.

Apparently you can attend online but to give a talk you have to go there. Here are some details:

The Symposium on Compositional Structures (SYCO) is an interdisciplinary series of meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. Previous SYCO events have been held in Birmingham, Strathclyde, Oxford, Chapman, Leicester and Tallinn.

We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering friendly discussion, disseminating new ideas, and spreading knowledge between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students. Submissions is easy, with no formatting or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere. You could submit work-in-progress, or a recently completed paper, or even a PhD or Masters thesis.

While no list of topics could be exhaustive, SYCO welcomes submissions with a compositional focus related to any of the following areas, in particular from the perspective of category theory:

• logical methods in computer science, including classical and quantum programming, type theory, concurrency, natural language processing and machine learning;
• graphical calculi, including string diagrams, Petri nets and reaction networks;
• languages and frameworks, including process algebras, proof nets, type theory and game semantics;
• abstract algebra and pure category theory, including monoidal category theory, higher category theory, operads, polygraphs, and relationships to homotopy theory;
• quantum algebra, including quantum computation and representation theory;
• tools and techniques, including rewriting, formal proofs and proof assistants, and game theory;
• industrial applications, including case studies and real-world problem descriptions.

### Important dates

All deadlines are 23:59 Anywhere on Earth.

Author notification: Monday 8 August 2022
Symposium dates: Thursday 8 and Friday 9 September 2022

### Submission instructions

Submissions are by EasyChair, via the SYCO 9 submission page:

Submission is easy, with no format requirements or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere. Think creatively: you could submit a recent paper, or notes on work in progress, or even a recent Masters or PhD thesis.

In the event that more good-quality submissions are received than can be accommodated in the timetable, the programme committee may choose to defer some submissions to a future meeting, rather than reject them. Deferred submissions can be re-submitted to any future SYCO meeting, where they will not need peer review, and where they will be prioritised for inclusion in the programme. Meetings will be held sufficiently frequently to avoid a backlog of deferred papers.

If you have a submission which was deferred from a previous SYCO meeting, it will not automatically be considered for SYCO 9; you still need to submit it again through EasyChair. When submitting, append the words “DEFERRED FROM SYCO X” to the title of your paper, replacing “X” with the appropriate meeting number. There is no need to attach any documents.

### Programme committee

The PC chair is John van de Wetering, Radboud University. The Programme Committee will be announced soon.

### Steering committee

Ross Duncan, University of Strathclyde
Chris Heunen, University of Edinburgh
Dominic Horsman, University of Oxford
Aleks Kissinger, University of Oxford
Samuel Mimram, École Polytechnique
Simona Paoli, University of Aberdeen
Pawel Sobocinski, Tallinn University of Technology
Jamie Vicary, University of Cambridge

## Compositional Modeling with Decorated Cospans

27 June, 2022

It’s finally here: software that uses category theory to let you build models of dynamical systems! We’re going to train epidemiologists to use this to model the spread of disease. My first talk on this will be on Wednesday June 29th. You’re invited!

Compositional modeling with decorated cospans, Graph Transformation Theory and Practice (GReTA) seminar, 19:00 UTC, Wednesday 29 June 2022.

You can attend live on Zoom if you click here. You can also watch it live on YouTube, or later recorded, here:

Abstract. Decorated cospans are a general framework for composing open networks and mapping them to dynamical systems. We explain this framework and illustrate it with the example of stock and flow diagrams. These diagrams are widely used in epidemiology to model the dynamics of populations. Although tools already exist for building these diagrams and simulating the systems they describe, we have created a new software package called StockFlow which uses decorated cospans to overcome some limitations of existing software. Our approach cleanly separates the syntax of stock and flow diagrams from the semantics they can be assigned. We have implemented a semantics where stock and flow diagrams are mapped to ordinary differential equations, although others are possible. We illustrate this with code in StockFlow that implements a simplified version of a COVID-19 model used in Canada. This is joint work with Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson.

My talk is at a seminar on graph rewriting, so I’ll explain how the math applies to graphs before turning to ‘stock-flow diagrams’, like this one here:

Stock-flow diagrams are used to create models in epidemiology. There’s a functor mapping them to dynamical systems.

But the key idea in our work is ‘compositional modeling’. This lets different teams build different models and then later assemble them into a larger model. The most popular existing software for stock-flow diagrams does not allow this. Category theory to the rescue!

This work would be impossible without the right team! Brendan Fong developed decorated cospans and then started the Topos Institute. My coauthors Evan Patterson and Sophie Libkind work there, and they know how to program using category theory.

Evan started a seminar on epidemiological modeling – and my old grad school pal Nate Osgood showed up, along with his grad student Xiaoyan Li! Nate is a computer scientist who now runs the main COVID model for the government of Canada.

So, all together we have serious expertise in category theory, computer science, and epidemiology. Any two parts alone would not be enough for this project.

And I’m not even listing all the people whose work was essential. For example, Kenny Courser and Christina Vasilakopoulou helped modernize the theory of decorated cospans in a way we need here. James Fairbanks, Evan and others designed AlgebraicJulia, the software environment that our package StockFlow relies on. And so on!

Moral: to apply category theory to real-world problems, you need a team.

And we’re just getting started!

## Hoàng Xuân Sính

20 June, 2022

During the Vietnam war, Grothendieck taught math to the Hanoi University mathematics department staff, out in the countryside. Hoàng Xuân Sính took notes and later did a PhD with him — by correspondence! She mailed him her hand-written thesis. She is the woman in this picture:

As you might guess, there’s a very interesting story behind this. I’ve looked into it, but what I found raises even more questions. Hoàng Xuân Sính’s life really deserves a good biography.

Hoàng Xuân Sính was born in 1933 in a village called Cót, one of seven children of a fabric merchant. Her mother died when she was eight years old, and she was raised by a stepmother.

She spent a lot of time sewing and designing clothes. However, this image of a magazine from 1981 shows a picture of her holding a book—and on the website where I found this, a caption in Vietnamese says “Cót village girl is passionate about math”.

She would have been 18 at the time. In this year she completed a bachelor’s degree in Hanoi, studying English and French, and then traveled to Paris for a second baccalaureate in mathematics. She stayed in France to study for the agrégation (the competitive examination for civil service) at the University of Toulouse, which she completed in 1959, before returning to Vietnam and teaching mathematics at the Hanoi National University of Education.

Grothendieck visited North Vietnam in late 1967, during the Vietnam War, and spent a month teaching mathematics to the Hanoi University mathematics department staff, including Hoàng Xuân Sính, who took the notes for the lectures. Because of the war, Grothendieck’s lectures were held away from Hanoi, first in the nearby countryside and later in Đại Từ.

After Grothendieck returned to France, he continued to teach Hoàng Xuân Sính in an exchange of letters. According to the web page of the university she founded, Thang Long University, Hoàng Xuân Sính remembers two main impressions from her contacts with Alexander Grothendieck:

1) A good teacher is a teacher who turns something difficult into something easy.

2) We should always avoid anything that is fictitious, live in accordance to our own feelings and value simple people.

She finished her thesis in 1972. Around Christmas that year, the United States dropped over 20,000 tons of bombs on North Vietnam, mainly Hanoi. So, it’s not surprising that she only defended her thesis three yearslater, when the North had almost won. But she mentions another reason. She later wrote:

I was a doctorate student during wartime. Back then, I was teaching at Hanoi Pedagogical University, there was not a mode to take leave to study for the doctorate. I taught during the day and worked on my thesis during the night under the kerosene lamp light. I wrote in French under my distant teacher’s guidance. When I got the approval from France to come over to defend, there were disagreeable talks about not letting me because they was afraid I wasn’t coming back. The most supportive person during the time was Lady Ha Thi Que—President of the Vietnamese Women Coalescent organization. Madame Que was a guerilla, without the conditions to get much education, but gave very convincing reasons to support me. She said, firstly, I was 40 years old, it is very difficult to get a job abroad at 40 years old, and without a job, how can I live? Second, my child is at home, no woman would ever leave her child… so comrades, let’s not be worried, let her go. I finished my thesis in 1972, and 3 years later with the help and struggle of the women’s organization, I was able to travel over to defend in 1975….

She went to France to defend her thesis at Paris Diderot University (also called Paris 7). Her thesis committee included not only Alexander Grothendieck but also Henri Cartan, Laurent Schwartz, Michel Zisman, and Jacques Deny.

Her thesis defense lasted two and a half hours. And soon thereafter she defended a second thesis, entitled “The embedding of a one-dimensional complex in a two-dimensional differential manifold”. I don’t know who, if anyone, directed this second thesis.

She later became the first woman mathematics professor in Vietnam — and the second came 35 years later.

In 1988, she started the first private university in Vietnam, Thang Long University in Hanoi. For a while she was not only the head, but also the janitor, bringing water to the school and sweeping floors. Later she said “When I look back at it, I thought it was the most romantic idea I’ve had.”

In 2003 she was awarded France’s Ordre des Palmes Académiques. She is still alive! I hope someone has interviewed her, or does it now. Her stories must be very interesting.

Her thesis classified Gr-categories, which are now called ‘2-groups’ for short. A 2-group is the categorified version of a group: it’s a monoidal category where every object and morphism is invertible. (An object $x$ is invertible if there’s an object $y$ with $x \otimes y \cong y \otimes x \cong I$, where $I$ is the unit for the tensor product.)

From a 2-group you can get two groups:

• the group $G$ of isomorphism classes of objects, and

• the group $A$ of automorphisms of the unit object $I$.

The group $A$ is abelian, and $G$ acts on $A$. But there’s one more thing! The associator can be used to get a map

$a \colon G^3 \to A$

The pentagon identity for the associator implies that this map obeys an equation. And this equation is familiar in the subject of group cohomology: it says $a$ is a ‘3-cocycle’ on the group $G$ with coefficients in $H$.

Even better, she showed that cohomologous 3-cocycles give equivalent 2-groups. (Equivalent as monoidal categories, that is.)

So, we can classify 2-groups using cohomology! The most exciting, least obvious part of this is the cohomology class $[a] \in H^3(G,A)$. This is often called the ‘Sính invariant’, though I believe Hoàng is her surname, not Sính.

This connection between 2-groups and cohomology is no coincidence. It’s best understood using a bit more topology.

Any connected space with a basepoint, say $X$, has a fundamental group. But it also has a fundamental 2-group! This 2-group has $G = \pi_1(X)$ and $A = \pi_2(X)$. And if all the higher homotopy groups of $X$ vanish, this 2-group knows everything about the
homotopy type of $X$, at least if $X$ is reasonably nice, like a CW complex.

So, Hoàng Xuân Sính’s thesis sheds light on ‘homotopy 2-types’: that is, homotopy types of nice spaces with $\pi_n(X) = 0$ for $n > 2$. They are just 2-groups!

Thus, her thesis illuminated one of the simplest — yet still important — special cases of Grothendieck’s ‘homotopy hypothesis’, namely that homotopy n-types correspond to n-groupoids.

You can see Hoàng Xuân Sính’s thesis along with a handwritten
summary in English here:

That website also has three nice photos of Grothendieck in Vietnam. I showed a colorized version of one at the top of this article, and here is another, with Hoàng Xuân Sính at far left:

Here are the original uncolorized versions of all three:

## Tutorial on Categorical Semantics of Entropy

28 May, 2022

Here are two talks on the categorical semantics of entropy, given on Wednesday May 11th 2022 at CUNY. First one there’s one by me and then starting around 1:31:00 there’s one by Tai-Danae Bradley:

My talk is called “Shannon entropy from category theory”:

Shannon entropy is a powerful concept. But what properties single out Shannon entropy as special? Instead of focusing on the entropy of a probability measure on a finite set, it can help to focus on the “information loss”, or change in entropy, associated with a measure-preserving function. Shannon entropy then gives the only concept of information loss that is functorial, convex-linear and continuous. This is joint work with Tom Leinster and Tobias Fritz.

This talk will open with a basic introduction to operads and their representations, with the main example being the operad of probabilities. I’ll then give a light sketch of how this framework leads to a small, but interesting, connection between information theory, abstract algebra, and topology, namely a correspondence between Shannon entropy and derivations of the operad of probabilities.

• John Baez, Tobias Fritz and Tom Leinster, A characterization of entropy in terms of information loss, 2011.

• Tom Leinster, An operadic introduction to entropy, 2011.

• John Baez and Tobias Fritz, A Bayesian characterization of relative entropy, 2014.

• Tom Leinster, A short characterization of relative entropy, 2017.

• Nicolas Gagné and Prakash Panangaden,
A categorical characterization of relative entropy on standard Borel spaces, 2017.

• Tom Leinster, Entropy and Diversity: the Axiomatic Approach, 2020.

• Arthur Parzygnat, A functorial characterization of von Neumann entropy, 2020.

• Arthur Parzygnat, Towards a functorial description of quantum relative entropy, 2021.

## Shannon Entropy from Category Theory

22 April, 2022

I’m giving a talk at Categorical Semantics of Entropy on Wednesday May 11th, 2022. You can watch it live on Zoom if you register, or recorded later. Here’s the idea:

Shannon entropy is a powerful concept. But what properties single out Shannon entropy as special? Instead of focusing on the entropy of a probability measure on a finite set, it can help to focus on the “information loss”, or change in entropy, associated with a measure-preserving function. Shannon entropy then gives the only concept of information loss that is functorial, convex-linear and continuous. This is joint work with Tom Leinster and Tobias Fritz.

You can see the slides now, here. I talk a bit about all these papers:

• John Baez, Tobias Fritz and Tom Leinster, A characterization of entropy in terms of information loss, 2011.

• Tom Leinster, An operadic introduction to entropy, 2011.

• John Baez and Tobias Fritz, A Bayesian characterization of relative entropy, 2014.

• Tom Leinster, A short characterization of relative entropy, 2017.

• Nicolas Gagné and Prakash Panangaden, A categorical characterization of relative entropy on standard Borel spaces, 2017.

• Tom Leinster, Entropy and Diversity: the Axiomatic Approach, 2020.

• Arthur Parzygnat, A functorial characterization of von Neumann entropy, 2020.

• Arthur Parzygnat, Towards a functorial description of quantum relative entropy, 2021.

## Categorical Semantics of Entropy

19 April, 2022

There will be a workshop on the categorical semantics of entropy at the CUNY Grad Center in Manhattan on Friday May 13th, organized by John Terilla. I was kindly invited to give an online tutorial beforehand on May 11, which I will give remotely to save carbon. Tai-Danae Bradley will also be giving a tutorial that day in person:

Tutorial: Categorical Semantics of Entropy, Wednesday 11 May 2022, 13:00–16:30 Eastern Time, Room 5209 at the CUNY Graduate Center and via Zoom. Organized by John Terilla. To attend, register here.

12:00-1:00 Eastern Daylight Time — Lunch in Room 5209.

1:00-2:30 — Shannon entropy from category theory, John Baez, University of California Riverside; Centre for Quantum Technologies (Singapore); Topos Institute.

Shannon entropy is a powerful concept. But what properties single out Shannon entropy as special? Instead of focusing on the entropy of a probability measure on a finite set, it can help to focus on the “information loss”, or change in entropy, associated with a measure-preserving function. Shannon entropy then gives the only concept of information loss that is functorial, convex-linear and continuous. This is joint work with Tom Leinster and Tobias Fritz.

2:30-3:00 — Coffee break.

3:00-4:30 — Operads and entropy, Tai-Danae Bradley, The Master’s University; Sandbox AQ.

This talk will open with a basic introduction to operads and their representations, with the main example being the operad of probabilities. I’ll then give a light sketch of how this framework leads to a small, but interesting, connection between information theory, abstract algebra, and topology, namely a correspondence between Shannon entropy and derivations of the operad of probabilities.

Symposium on Categorical Semantics of Entropy, Friday 13 May 2022, 9:30-3:15 Eastern Daylight Time, Room 5209 at the CUNY Graduate Center and via Zoom. Organized by John Terilla. To attend, register here.

9:30-10:00 Eastern Daylight Time — Coffee and pastries in Room 5209.

10:00-10:45 — Operadic composition of thermodynamic systems, Owen Lynch, Utrecht University.

The maximum entropy principle is a fascinating and productive lens with which to view both thermodynamics and statistical mechanics. In this talk, we present a categorification of the maximum entropy principle, using convex spaces and operads. Along the way, we will discuss a variety of examples of the maximum entropy principle and show how each application can be captured using our framework. This approach shines a new light on old constructions. For instance, we will show how we can derive the canonical ensemble by attaching a probabilistic system to a heat bath. Finally, our approach to this categorification has applications beyond the maximum entropy principle, and we will give an hint of how to adapt this categorification to the formalization of the composition of other systems.

11:00-11:45 — Polynomial functors and Shannon entropy, David Spivak, MIT and the Topos Institute.

The category Poly of polynomial functors in one variable is extremely rich, brimming with categorical gadgets (e.g. eight monoidal products, two closures, limits, colimits, etc.) and applications including dynamical systems, databases, open games, and cellular automata. In this talk I’ll show that objects in Poly can be understood as empirical distributions. In part using the standard derivative of polynomials, we obtain a functor to Set × Setop which encodes an invariant of a distribution as a pair of sets. This invariant is well-behaved in the sense that it is a distributive monoidal functor: it acts on both distributions and maps between them, and it preserves both the sum and the tensor product of distributions. The Shannon entropy of the original distribution is then calculated directly from the invariant, i.e. only in terms of the cardinalities of these two sets. Given the many applications of polynomial functors and of Shannon entropy, having this link between them has potential to create useful synergies, e.g. to notions of entropic causality or entropic learning in dynamical systems.

12:00-1:30 — Lunch in Room 5209

1:30-2:15 — Higher entropy, Tom Mainiero, Rutgers New High Energy Theory Center.

Is the frowzy state of your desk no longer as thrilling as it once was? Are numerical measures of information no longer able to satisfy your needs? There is a cure! In this talk we’ll learn about: the secret topological lives of multipartite measures and quantum states; how a homological probe of this geometry reveals correlated random variables; the sly decategorified involvement of Shannon, Tsallis, Réyni, and von Neumann in this larger geometric conspiracy; and the story of how Gelfand, Neumark, and Segal’s construction of von Neumann algebra representations can help us uncover this informatic ruse. So come to this talk, spice up your entropic life, and bring new meaning to your relationship with disarray.

2:30-3:15 — On characterizing classical and quantum entropy, Arthur Parzygnat, Institut des Hautes Études Scientifiques.

In 2011, Baez, Fritz, and Leinster proved that the Shannon entropy can be characterized as a functor by a few simple postulates. In 2014, Baez and Fritz extended this theorem to provide a Bayesian characterization of the classical relative entropy, also known as the Kullback–Leibler divergence. In 2017, Gagné and Panangaden extended the latter result to include standard Borel spaces. In 2020, I generalized the first result on Shannon entropy so that it includes the von Neumann (quantum) entropy. In 2021, I provided partial results indicating that the Umegaki relative entropy may also have a Bayesian characterization. My results in the quantum setting are special applications of the recent theory of quantum Bayesian inference, which is a non-commutative extension of classical Bayesian statistics based on category theory. In this talk, I will give an overview of these developments and their possible applications in quantum information theory.

Wine and cheese reception to follow, Room 5209.

## Compositional Thermostatics (Part 4)

8 March, 2022

guest post by Owen Lynch

This is the fourth and final part of a blog series on this paper:

• John Baez, Owen Lynch and Joe Moeller, Compositional thermostatics.

In Part 1, we went over our definition of thermostatic system: it’s a convex space $X$ of states and a concave function $S \colon X \to [-\infty, \infty]$ saying the entropy of each state. We also gave examples of thermostatic systems.

In Part 2, we talked about what it means to compose thermostatic systems. It amounts to constrained maximization of the total entropy.

In Part 3 we laid down a categorical framework for composing systems when there are choices that have to be made for how the systems are composed. This framework has been around for a long time: operads and operad algebras.

In this post we will bring together all of these parts in a big synthesis to create an operad of all the ways of composing thermostatic systems, along with an operad algebra of thermostatic systems!

Recall that in order to compose thermostatic systems $(X_1, S_1), \ldots, (X_n, S_n),$ we need to use a ‘parameterized constraint’, a convex subset

$R \subseteq X_1 \times \cdots \times X_n \times Y$,

where $Y$ is some other convex set. We end up with a thermostatic system on $Y$, with $S \colon Y \to [-\infty,\infty]$ defined by

$S(y) = \sup_{(x_1,\ldots,x_n,y) \in R} S_1(x_1) + \cdots + S_n(x_n)$

In order to model this using operads and operad algebras, we will make an operad $\mathcal{CR}$ which has convex sets as its types, and convex relations as its morphisms. Then we will make an operad algebra that assigns to any convex set $X$ the set of concave functions

$S \colon X \to [-\infty,\infty]$

This operad algebra will describe how, given a relation $R \subseteq X_1 \times \cdots \times X_n \times Y$, we can ‘push forward’ entropy functions on $X_1,\ldots,X_n$ to form an entropy function on $Y$.

The operad $\mathcal{CR}$ is built using a construction from Part 3 that takes a symmetric monoidal category and produces an operad. The symmetric monoidal category that we start with is $\mathsf{ConvRel},$ which has convex sets as its objects and convex relations as its morphisms. This symmetric monoidal category has $\mathsf{Conv}$ (the category of convex sets and convex-linear functions) as a subcategory with all the same objects, and $\mathsf{ConvRel}$ inherits a symmetric monoidal structure from the bigger category $\mathsf{Conv}.$

Following the construction from Part 3, we see that we get an operad

$\mathcal{CR} = \mathrm{Op}(\mathsf{ConvRel})$

exactly as described before: namely it has convex sets as types, and

$\mathcal{CR}(X_1,\ldots,X_n;Y) = \mathsf{ConvRel}(X_1 \times \cdots \times X_n, Y)$

Next we want to make an operad algebra on $\mathcal{CR}$. To do this we use a lax symmetric monoidal functor $\mathrm{Ent}$ from $\mathsf{ConvRel}$ to $\mathsf{Set},$ defined as follows. On objects, $\mathrm{Ent}$ sends any convex set $X$ to the set of entropy functions on it:

$\mathrm{Ent}(X) = \{ S \colon X \to [-\infty,\infty] \mid S \: \text{is concave} \}$

On morphisms, $\mathrm{Ent}$ sends any convex relation to to the map that “pushes forward” an entropy function along that relation:

$\mathrm{Ent}(R \subseteq X \times Y) = (y \mapsto \sup_{(x,y) \in R} S(x))$

And finally, the all-important laxator $\epsilon$ produces an entropy function on $X_1 \times X_2$ by summing an entropy function on $X_1$ and an entropy function on $X_2$:

$\epsilon_{X_1,X_2} = ((S_1,S_2) \mapsto S_1 + S_2)$

The proof that all this indeed defines a lax symmetric monoidal functor can be found in our paper. The main point is that once we have proven this really is a lax symmetric monoidal functor, we can invoke the machinery of lax symmetric monoidal functors and operad algebras to prove that we get an operad algebra! This is very convenient, because proving that we have an operad algebra directly would be somewhat tedious.

We have now reached the technical high point of the paper, which is showing that this operad algebra exists and thus formalizing what it means to compose thermostatic systems. All that remains to do now is to show off a bunch of examples of composition, so that you can see how all this categorical machinery works in practice. In our paper we give many examples, but here let’s consider just one.

Consider the following setup with two ideal gases connected by a movable divider.

The state space of each individual ideal gas is $\mathbb{R}^3_{> 0}$, with coordinates $(U,V,N)$ representing energy, volume, and number of particles respectively. Let $(U_1, V_1, N_1)$ be the coordinates for the left-hand gas, and $(U_2, V_2, N_2)$ be the coordinates for the right-hand gas. Then as the two gases move to thermodynamic equilibrium, the conserved quantities are $U_1 + U_2$, $V_1 + V_2$, $N_1$ and $N_2$. We picture this with the following diagram.

Ports on the inner circles represent variables for the ideal gases, and ports on the outer circle represent variables for the composed system. Wires represent relations between those variables. Thus, the entire diagram represents an operation in $\mathcal{CR}$, given by

$U_1 + U_2 = U^e$
$V_1 + V_2 = V^e$
$N_1 = N_1^e$
$N_2 = N_2^e$

We can then use the operad algebra to take entropy functions $S_1,S_2 \colon \mathbb{R}^3_{> 0} \to [-\infty, \infty]$ on the two inner systems (the two ideal gases), and get an entropy function $S^e \colon \mathbb{R}^4_{> 0} \to [-\infty,\infty]$ on the outer system.

As a consequence of this entropy maximization procedure, the inner state $(U_1,V_1,N_1), (U_2,V_2,N_2)$ are such that the temperature and pressure equilibriate between the two ideal gases. This is because constrained maximization with the constraint $U_1 + U_2 = U^e$ leads to the following equations at a maximizer:

$\displaystyle{ \frac{1}{T_1} = \frac{\partial S_1}{\partial U_1} = \frac{\partial S_2}{\partial U_2} = \frac{1}{T_2} }$

(where $T_1$ and $T_2$ are the respective temperatures), and

$\displaystyle{ \frac{p_1}{T_1} = \frac{\partial S_1}{\partial V_1} = \frac{\partial S_2}{\partial V_2} = \frac{p_2}{T_2} }$

(where $p_1$ and $p_2$ are the respective pressures).

Thus we arrive at the expected conclusion, which is that temperature and pressure equalize when we maximize entropy under constraints on the total energy and volume.

And that concludes this series of blog posts! For more examples of thermostatic composition, I invite you to read our paper, which has some “thermostatic systems” that one does not normally see thought of in this way, such as heat baths and probabilistic systems! And if you find this stuff interesting, don’t hesitate to reach out to me! Just drop a comment here or email me at the address in the paper.

See all four parts of this series:

Part 1: thermostatic systems and convex sets.

Part 2: composing thermostatic systems.

Part 3: operads and their algebras.

Part 4: the operad for composing thermostatic systems.

## Topos Institute Research Associates

5 March, 2022

Come spend the summer at the Topos Institute! For early-career researchers, we’re excited to open up applications for our summer research associate (RA) program.

Summer RAs are an important part of life at Topos — they help explore new directions relevant to Topos projects, and they bring new ideas, energy, and expertise to our research groups. This year we’ll welcome a new cohort of RAs to our offices in Berkeley, CA, with the program running from June to August.

RAs will be responsible for performing an in-depth research or teaching project, mentored by a Topos faculty mentor or advisor. This year possible mentors may include Conal Elliott, Valeria de Paiva, Evan Patterson, Dana Scott, David Spivak, and others. RAs will work closely with their mentor to define and pursue a research project.

Along the way, RAs will also participate in our weekly lunches and seminars, blog about their time here, and produce papers, books, software, or policy, according to the parameters of their project.

Applications are now open. The position is full-time (~40 hours per week) and paid hourly starting at \$30/hour. Unfortunately, Topos is not able to sponsor US work visas for participants.

Please apply by Sunday March 27th. Offers of positions will be made in early April.

## Applied Category Theory 2022

25 February, 2022

The Fifth International Conference on Applied Category Theory, ACT2022, will take place at the University of Strathclyde from 18 to 22 July 2022, preceded by the Adjoint School 2022 from 11 to 15 July. This conference follows previous events at Cambridge (UK), Cambridge (MA), Oxford and Leiden.

Applied category theory is important to a growing community of researchers who study computer science, logic, engineering, physics, biology, chemistry, social sciences, linguistics and other subjects using category-theoretic tools. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, strengthen the applied category theory community, disseminate the latest results, and facilitate further development of the field.

### Submissions

We accept submissions in English of original research papers, talks about work accepted/submitted/published elsewhere, and demonstrations of relevant software. Accepted original research papers will be published in a proceedings volume. The keynote addresses will be chosen from the accepted papers. The conference will include an industry showcase event and community meeting. We particularly encourage people from underrepresented groups to submit their work and the organizers are committed to non-discrimination, equity, and inclusion.

### Submission formats

Extended Abstracts should be submitted describing the contribution and providing a basis for determining the topics and quality of the anticipated presentation (1-2 pages). These submissions will be adjudicated for inclusion as a talk at the conference. Such work should include references to any longer papers, preprints, or manuscripts providing additional details.

Conference Papers should present original, high-quality work in the style of a computer science conference paper (up to 14 pages, not counting the bibliography; detailed proofs may be included in an appendix for the convenience of the reviewers). Such submissions should not be an abridged version of an existing journal article (see item 1) although pre-submission Arxiv preprints are permitted. These submissions will be adjudicated for both a talk and publication in the conference proceedings.

Software Demonstrations should be submitted in the format of an Extended Abstract (1-2 pages) giving the program committee enough information to assess the content of the demonstration. We are particularly interested in software that makes category theory research easier, or uses category theoretic ideas to improve software in other domains.

Extended abstracts and conference papers should be prepared with LaTeX. For conference papers please use the EPTCS style files available at

http://style.eptcs.org

https://easychair.org/conferences/?conf=act2022

### Important dates

The following dates are all in 2022, and Anywhere On Earth.

• Submission Deadline: Monday 9 May
• Author Notification: Tuesday 7 June
• Camera-ready version due: Tuesday 28 June
• Adjoint School: Monday 11 to Friday 15 July
• Main Conference: Monday 18 to Friday 22 July

### Conference format

We hope to run the conference as a hybrid event with talks recorded or streamed for remote participation. However, due to the state of the pandemic, the possibility of in-person attendance is not yet confirmed. Please be mindful of changing conditions when booking travel or hotel accommodations.

### Program committee

• Jade Master, University of Strathclyde (Co-chair)
• Martha Lewis, University of Bristol (Co-chair)

The full program committee will be announced soon.

### Organizing committee

• Jules Hedges, University of Strathclyde
• Jade Master, University of Strathclyde
• Fredrik Nordvall Forsberg, University of Strathclyde
• James Fairbanks, University of Florida

### Steering committee

• John Baez, University of California, Riverside
• Bob Coecke, Cambridge Quantum
• Dorette Pronk, Dalhousie University
• David Spivak, Topos Institute

## Categories: the Mathematics of Connection

17 February, 2022

I gave this talk at Mathematics of Collective Intelligence, a workshop organized by Jacob Foster at UCLA’s Institute of Pure and Applied Mathematics, or IPAM for short. There have been a lot of great talks here, all available online.

Perhaps the main interesting thing about this talk is that I sketch some work happening at the Topos Institute where we are using techniques from category theory to design epidemiological models:

Categories: the mathematics of connection

Abstract. As we move from the paradigm of modeling one single self-contained system at a time to modeling ‘open systems’ which interact with their — perhaps unmodeled — environment, category theory becomes a useful tool. It gives a mathematical language to describe the interface between an open system and its environment, the process of composing open systems along their interfaces, and how the behavior of a composite system relates to the behaviors of its parts. It is far from a silver bullet: at present, every successful application of category theory to open systems takes hard work. But I believe we are starting to see real progress.

You can see my slides or watch a video of my talk on the IPAM website or here:

For some other related talks, see:

To read more about my work on categories and open systems, go here: