## Applied Category Theory Seminar

14 December, 2018

We’re going to have a seminar on applied category theory here at U. C. Riverside! My students have been thinking hard about category theory for a few years, but they’ve decided it’s time to get deeper into applications. Christian Williams, in particular, seems to have caught my zeal for trying to develop new math to help save the planet.

We’ll try to videotape the talks to make it easier for you to follow along. I’ll also start discussions here and/or on the Azimuth Forum. It’ll work best if you read the papers we’re talking about and then join these discussions. Ask questions, and answer any questions you can!

Here’s how the schedule of talks is shaping up so far. I’ll add more information as it becomes available, either here or on a webpage devoted to the task.

#### January 8, 2019: John Baez – Mathematics in the 21st century

I’ll give an updated synthesized version of these earlier talks of mine, so check out these slides and the links:

#### January 15, 2019: Jonathan Lorand – Problems in symplectic linear algebra

Lorand is visiting U. C. Riverside to work with me on applications of symplectic geometry to chemistry. Here is the abstract of his talk:

In this talk we will look at various examples of classification problems in symplectic linear algebra: conjugacy classes in the symplectic group and its Lie algebra, linear lagrangian relations up to conjugation, tuples of (co)isotropic subspaces. I will explain how many such problems can be encoded using the theory of symplectic poset representations, and will discuss some general results of this theory. Finally, I will recast this discussion from a broader category-theoretic perspective.

#### January 22, 2019: Christina Vasilakopoulou – Wiring diagrams

Vasilakopoulou, a visiting professor here, previously worked with David Spivak. So, we really want to figure out how two frameworks for dealing with networks relate: Brendan Fong’s ‘decorated cospans’, and Spivak’s ‘monoidal category of wiring diagrams’. Since Fong is now working with Spivak they’ve probably figured it out already! But anyway, Vasilakopoulou will give a talk on systems as algebras for the wiring diagram monoidal category. It will be based on this paper:

• Patrick Schultz, David I. Spivak and Christina Vasilakopoulou, Dynamical systems and sheaves.

but she will focus more on the algebraic description (and conditions for deterministic/total systems) rather than the sheaf theoretic aspect of the input types. This work builds on earlier papers such as these:

• Dmitry Vagner, David I. Spivak and Eugene Lerman, Algebras of open dynamical systems on the operad of wiring diagrams.

#### January 29, 2019: Daniel Cicala – Dynamical systems on networks

Cicala will discuss a topic from this paper:

• Mason A. Porter and James P. Gleeson, Dynamical systems on networks: a tutorial.

His leading choice is a model for social contagion (e.g. opinions) which is discussed in more detail here:

• Duncan J. Watts, A simple model of global cascades on random networks.

## Second Symposium on Compositional Structures

7 December, 2018

I’ve been asleep at the switch; this announcement is probably too late for anyone outside the UK. But still, it’s great to see how applied category theory is taking off! And this conference is part of a series, so if you miss this one you can still go to the next.

Second Symposium on Compositional Structures (SYCO2), 17-18 December 2018, University of Strathclyde, Glasgow.

#### Accepted presentations

http://events.cs.bham.ac.uk/syco/2/accepted.html

#### Registration

Please register asap so that catering can be arranged. Late registrants
might go hungry.

#### Invited speakers

• Corina Cirstea, University of Southampton – Quantitative Coalgebras for
Optimal Synthesis

• Martha Lewis, University of Amsterdam – Compositionality in Semantic Spaces

#### Description

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. The first SYCO was held at the School of Computer Science, University of Birmingham, 20-21 September, 2018, attracting 70 participants.

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.

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.

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.

This new series aims to bring together the communities behind many previous successful events which have taken place over the last decade, including “Categories, Logic and Physics”, “Categories, Logic and Physics (Scotland)”, “Higher-Dimensional Rewriting and Applications”, “String Diagrams in Computation, Logic and Physics”, “Applied Category Theory”, “Simons Workshop on Compositionality”, and the “Peripatetic Seminar in Sheaves and Logic”.

SYCO will be a regular fixture in the academic calendar, running regularly throughout the year, and becoming over time a recognized venue for presentation and discussion of results in an informal and friendly atmosphere. To help create this community, and to avoid the need to make difficult choices between strong submissions, 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. This would be done based largely on submission order, giving an incentive for early submission, but would also take into account other requirements, such as ensuring a broad scientific programme. Deferred submissions would be accepted for presentation at any future SYCO meeting without the need for peer review. This will allow us to ensure that speakers have enough time to present their ideas, without creating an unnecessarily competitive reviewing process. Meetings would be held sufficiently frequently to avoid a backlog of deferred papers.

#### PROGRAMME COMMITTEE

Ross Duncan, University of Strathclyde
Fabrizio Romano Genovese, Statebox and University of Oxford
Jules Hedges, University of Oxford
Chris Heunen, University of Edinburgh
Dominic Horsman, University of Grenoble
Aleks Kissinger, Radboud University Nijmegen
Eliana Lorch, University of Oxford
Guy McCusker, University of Bath
Samuel Mimram, École Polytechnique
Koko Muroya, RIMS, Kyoto University & University of Birmingham
Paulo Oliva, Queen Mary
Nina Otter, UCLA
Simona Paoli, University of Leicester
Robin Piedeleu, University of Oxford and UCL
Julian Rathke, University of Southampton
Bernhard Reus, Univeristy of Sussex
David Reutter, University of Oxford
Pawel Sobocinski, University of Southampton (chair)
Jamie Vicary, University of Birmingham and University of Oxford (co-chair)

## Geometric Quantization (Part 1)

1 December, 2018

I can’t help thinking about geometric quantization. I feel it holds some lessons about the relation between classical and quantum mechanics that we haven’t fully absorbed yet. I want to play my cards fairly close to my chest, because there are some interesting ideas I haven’t fully explored yet… but still, there are also plenty of ‘well-known’ clues that I can afford to explain.

The first one is this. As beginners, we start by thinking of geometric quantization as a procedure for taking a symplectic manifold and constructing a Hilbert space: that is, taking a space of classical states and contructing the corresponding space of quantum states. We soon learn that this procedure requires additional data as its input: a symplectic manifold is not enough. We learn that it works much better to start with a Kähler manifold equipped with a holomorphic hermitian line bundle with a connection whose curvature is the imaginary part of the Kähler structure. Then the space of holomorphic sections of that line bundle gives the Hilbert space we seek.

That’s quite a mouthful—but it makes for such a nice story that I’d love to write a bunch of blog articles explaining it with lots of examples. Unfortunately I don’t have time, so try these:

• Matthias Blau, Symplectic geometry and geometric quantization.

• A. Echeverria-Enriquez, M.C. Munoz-Lecanda, N. Roman-Roy, C. Victoria-Monge, Mathematical foundations of geometric quantization.

But there’s a flip side to this story which indicates that something big and mysterious is going on. Geometric quantization is not just a procedure for converting a space of classical states into a space of quantum states. It also reveals that a space of quantum states can be seen as a space of classical states!

To reach this realization, we must admit that quantum states are not really vectors in a Hilbert space $H$; from a certain point of view they are really 1-dimensonal subspaces of a Hilbert space, so the set of quantum states I’m talking about is the projective space $PH.$ But this projective space, at least when it’s finite-dimensional, turns out to be the simplest example of that complicated thing I mentioned: a Kähler manifold equipped with a holomorphic hermitian line bundle whose curvature is the imaginary part of the Kähler structure!

So a space of quantum states is an example of a space of classical states—equipped with precisely all the complicated extra structure that lets us geometrically quantize it!

At this point, if you don’t already know the answer, you should be asking: and what do we get when we geometrically quantize it?

The answer is exciting only in that it’s surprisingly dull: when we geometrically quantize $PH,$ we get back the Hilbert space $H.$

You may have heard of ‘second quantization’, where we take a quantum system, treat it as classical, and quantize it again. In the usual story of second quantization, the new quantum system we get is more complicated than the original one… and we can repeat this procedure again and again, and keep getting more interesting things:

• John Baez, Nth quantization.

The story I’m telling now is different. I’m saying that when we take a quantum system with Hilbert space $H,$ we can think of it as a classical system whose symplectic manifold of states is $PH,$ but then we can geometrically quantize this and get $H$ back.

The two stories are not in contradiction, because they rely on two different notions of what it means to ‘think of a quantum system as classical’. In today’s story that means getting a symplectic manifold $PH$ from a Hilbert space $H.$ In the other story we use the fact that $H$ itself is a symplectic manifold!

I should explain the relation of these two stories, but that would be a big digression from today’s intended blog article: indeed I’m already regretting having drifted off course. I only brought up this other story to heighten the mystery I’m talking about now: namely, that when we geometrically quantize the space $PH,$ we get $H$ back.

The math is not mysterious here; it’s the physical meaning of the math that’s mysterious. The math seems to be telling us that contrary to what they say in school, quantum systems are special classical systems, with the special property that when you quantize them nothing new happens!

This idea is not mine; it goes back at least to Kibble, the guy who with Higgs invented the method whereby the Higgs boson does its work:

• Tom W. B. Kibble, Geometrization of quantum mechanics, Comm. Math. Phys. 65 (1979), 189–201.

This led to a slow, quiet line of research that continues to this day. I find this particular paper especially clear and helpful:

• Abhay Ashtekar, Troy A. Schilling, Geometrical formulation of quantum mechanics, in On Einstein’s Path, Springer, Berlin, 1999, pp. 23–65.

so if you’re wondering what the hell I’m talking about, this is probably the best place to start. To whet your appetite, here’s the abstract:

Abstract. States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a Kähler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different appearance. In particular, states are now represented by points of a symplectic manifold (which happens to have, in addition, a compatible Riemannian metric), observables are represented by certain real-valued functions on this space and the Schrödinger evolution is captured by the symplectic flow generated by a Hamiltonian function. There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. Features—such as uncertainties and state vector reductions—which are specific to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric—a structure which is absent in classical mechanics. The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semi-classical considerations and the WKB approximation. More importantly, it suggests generalizations of quantum mechanics. The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. The geometrical reformulation provides a unified framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found.

Personally I’m not interested in the generalizations of quantum mechanics: I’m more interested in what this circle of ideas means for quantum mechanics.

One rather cynical thought is this: when we start our studies with geometric quantization, we naively hope to extract a space of quantum states from a space of classical states, e.g. a symplectic manifold. But we then discover that to do this in a systematic way, we need to equip our symplectic manifold with lots of bells and whistles. Should it really be a surprise that when we’re done, the bells and whistles we need are exactly what a space of quantum states has?

I think this indeed dissolves some of the mystery. It’s a bit like the parable of ‘stone soup’: you can make a tasty soup out of just a stone… if you season it with some vegetables, some herbs, some salt and such.

However, perhaps because by nature I’m an optimist, I also think there are interesting things to be learned from the tight relation between quantum and classical mechanics that appears in geometric quantization. And I hope to talk more about those in future articles.

## Category Theory Course

13 October, 2018

I’m teaching a course on category theory at U.C. Riverside, and since my website is still suffering from reduced functionality I’ll put the course notes here for now. I taught an introductory course on category theory in 2016, but this one is a bit more advanced.

The hand-written notes here are by Christian Williams. They are probably best seen as a reminder to myself as to what I’d like to include in a short book someday.

Lecture 1: What is pure mathematics all about? The importance of free structures.

Lecture 2: The natural numbers as a free structure. Adjoint functors.

Lecture 3: Adjoint functors in terms of unit and counit.

Lecture 4: 2-Categories. Adjunctions.

Lecture 5: 2-Categories and string diagrams. Composing adjunctions.

Lecture 6: The ‘main spine’ of mathematics. Getting a monad from an adjunction.

Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

Lecture 8: The walking monad, the augmented simplex category and the simplex category.

Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

Lecture 10: The Dold-Thom theorem: the category of simplicial abelian groups is equivalent to the category of chain complexes of abelian groups. The homology of a chain complex.

Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

Lecture 8: The walking monad, the
augmented simplex category and the simplex category.

Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

Lecture 10: Chain complexes from simplicial abelian groups. The homology of a chain complex.

Lecture 12: The bar construction: getting a simplicial objects from an adjunction. The bar construction for G-sets, previewed.

Lecture 13: The adjunction between G-sets and sets.

Lecture 14: The bar construction for groups.

Lecture 15: The simplicial set $\mathbb{E}G$ obtained by applying the bar construction to the one-point $G$-set, its geometric realization $EG = |\mathbb{E}G|,$ and the free simplicial abelian group $\mathbb{Z}[\mathbb{E}G].$

Lecture 16: The chain complex $C(G)$ coming from the simplicial abelian group $\mathbb{Z}[\mathbb{E}G],$ its homology, and the definition of group cohomology $H^n(G,A)$ with coefficients in a $G$-module.

Lecture 17: Extensions of groups. The Jordan-Hölder theorem. How an extension of a group $G$ by an abelian group $A$ gives an action of $G$ on $A$ and a 2-cocycle $c \colon G^2 \to A.$

Lecture 18: Classifying abelian extensions of groups. Direct products, semidirect products, central extensions and general abelian extensions. The groups of order 8 as abelian extensions.

Lecture 19: Group cohomology. The chain complex for the cohomology of $G$ with coefficients in $A$, starting from the bar construction, and leading to the 2-cocycles used in classifying abelian extensions. The classification of extensions of $G$ by $A$ in terms of $H^2(G,A).$

Lecture 20: Examples of group cohomology: nilpotent groups and the fracture theorem. Higher-dimensional algebra and homotopification: the nerve of a category and the nerve of a topological space. $\mathbb{E}G$ as the nerve of the translation groupoid $G/\!/G.$ $BG = EG/G$ as the walking space with fundamental group $G.$

Lecture 21: Homotopification and higher algebra. Internalizing concepts in categories with finite products. Pushing forward internalized structures using functors that preserve finite products. Why the ‘discrete category on a set’ functor $\mathrm{Disc} \colon \mathrm{Set} \to \mathrm{Cat},$ the ‘nerve of a category’ functor $\mathrm{N} \colon \mathrm{Cat} \to \mathrm{Set}^{\Delta^{\mathrm{op}}},$ and the ‘geometric realization of a simplicial set’ functor $|\cdot| \colon \mathrm{Set}^{\Delta^{\mathrm{op}}} \to \mathrm{Top}$ preserve products.

Lecture 22: Monoidal categories. Strict monoidal categories as monoids in $\mathrm{Cat}$ or one-object 2-categories. The periodic table of strict $n$-categories. General ‘weak’ monoidal categories.

Lecture 23: 2-Groups. The periodic table of weak $n$-categories. The stabilization hypothesis. The homotopy hypothesis. Classifying 2-groups with $G$ as the group of objects and $A$ as the abelian group of automorphisms of the unit object in terms of $H^3(G,A).$ The Eckmann–Hilton argument.

## Lebesgue Universal Covering Problem (Part 3)

7 October, 2018

Back in 2015, I reported some progress on this difficult problem in plane geometry. I’m happy to report some more.

First, remember the story. A subset of the plane has diameter 1 if the distance between any two points in this set is ≤ 1. A universal covering is a convex subset of the plane that can cover a translated, reflected and/or rotated version of every subset of the plane with diameter 1. In 1914, the famous mathematician Henri Lebesgue sent a letter to a fellow named Pál, challenging him to find the universal covering with the least area.

Pál worked on this problem, and 6 years later he published a paper on it. He found a very nice universal covering: a regular hexagon in which one can inscribe a circle of diameter 1. This has area

0.86602540…

But he also found a universal covering with less area, by removing two triangles from this hexagon—for example, the triangles C1C2C3 and E1E2E3 here:

The resulting universal covering has area

0.84529946…

In 1936, Sprague went on to prove that more area could be removed from another corner of Pál’s original hexagon, giving a universal covering of area

0.8441377708435…

In 1992, Hansen took these reductions even further by removing two more pieces from Pál’s hexagon. Each piece is a thin sliver bounded by two straight lines and an arc. The first piece is tiny. The second is downright microscopic!

Hansen claimed the areas of these regions were 4 · 10-11 and 6 · 10-18. This turned out to be wrong. The actual areas are 3.7507 · 10-11 and 8.4460 · 10-21. The resulting universal covering had an area of

0.844137708416…

This tiny improvement over Sprague’s work led Klee and Wagon to write:

it does seem safe to guess that progress on [this problem], which has been painfully slow in the past, may be even more painfully slow in the future.

However, in 2015 Philip Gibbs found a way to remove about a million times more area than Hansen’s larger region: a whopping 2.233 · 10-5. This gave a universal covering with area

0.844115376859…

Karine Bagdasaryan and I helped Gibbs write up a rigorous proof of this result, and we published it here:

• John Baez, Karine Bagdasaryan and Philip Gibbs, The Lebesgue universal covering problem, Journal of Computational Geometry 6 (2015), 288–299.

Greg Egan played an instrumental role as well, catching various computational errors.

At the time Philip was sure he could remove even more area, at the expense of a more complicated proof. Since the proof was already quite complicated, we decided to stick with what we had.

But this week I met Philip at The philosophy and physics of Noether’s theorems, a wonderful workshop in London which deserves a full blog article of its own. It turns out that he has gone further: he claims to have found a vastly better universal covering, with area

0.8440935944…

This is an improvement of 2.178245 × 10-5 over our earlier work—roughly equal to our improvement over Hansen.

You can read his argument here:

• Philip Gibbs, An upper bound for Lebesgue’s universal covering problem, 22 January 2018.

I say ‘claims’ not because I doubt his result—he’s clearly a master at this kind of mathematics!—but because I haven’t checked it and it’s easy to make mistakes, for example mistakes in computing the areas of the shapes removed.

It seems we are closing in on the final result; however, Philip Gibbs believes there is still room for improvement, so I expect it will take at least a decade or two to solve this problem… unless, of course, some mathematicians start working on it full-time, which could speed things up considerably.

## Riverside Math Workshop

6 October, 2018

We’re having a workshop with a bunch of cool math talks at U. C. Riverside, and you can register for it here:

Riverside Mathematics Workshop for Excellence and Diversity, Friday 19 October – Saturday 20 October, 2018. Organized by John Baez, Carl Mautner, José González and Chen Weitao.

This is the first of an annual series of workshops to showcase and celebrate excellence in research by women and other under-represented groups for the purpose of fostering and encouraging growth in the U.C. Riverside mathematical community.

After tea at 3:30 p.m. on Friday there will be two plenary talks, lasting until 5:00. Catherine Searle will talk on “Symmetries of spaces with lower curvature bounds”, and Edray Goins will give a talk called “Clocks, parking garages, and the solvability of the quintic: a friendly introduction to monodromy”. There will then be a banquet in the Alumni Center 6:30 – 8:30 p.m.

On Saturday there will be coffee and a poster session at 8:30 a.m., and then two parallel sessions on pure and applied mathematics, with talks at 9:30, 10:30, 11:30, 1:00 and 2:00. Check out the abstracts here!

(I’m especially interested in Christina Vasilakopoulou’s talk on Frobenius and Hopf monoids in enriched categories, but she’s my postdoc so I’m biased.)

## Applied Category Theory 2019

2 October, 2018

animation by Marius Buliga

I’m helping organize ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019.

More details will come later, but here’s the basic idea. If you’re a grad student interested in this subject, you should apply for the ‘school’. Not yet—we’ll let you know when.

Dear all,

As part of a new growing community in Applied Category Theory, now with a dedicated journal Compositionality, a traveling workshop series SYCO, a forthcoming Cambridge U. Press book series Reasoning with Categories, and several one-off events including at NIST, we launch an annual conference+school series named Applied Category Theory, the coming one being at Oxford, July 15-19 for the conference, and July 22-26 for the school. The dates are chosen such that CT 2019 (Edinburgh) and the ACT 2019 conference (Oxford) will be back-to-back, for those wishing to participate in both.

There already was a successful invitation-only pilot, ACT 2018, last year at the Lorentz Centre in Leiden, also in the format of school+workshop.

For the conference, for those who are familiar with the successful QPL conference series, we will follow a very similar format for the ACT conference. This means that we will accept both new papers which then will be published in a proceedings volume (most likely a Compositionality special Proceedings issue), as well as shorter abstracts of papers published elsewhere. There will be a thorough selection process, as typical in computer science conferences. The idea is that all the best work in applied category theory will be presented at the conference, and that acceptance is something that means something, just like in CS conferences. This is particularly important for young people as it will help them with their careers.

Expect a call for submissions soon, and start preparing your papers now!

The school in ACT 2018 was unique in that small groups of students worked closely with an experienced researcher (these were John Baez, Aleks Kissinger, Martha Lewis and Pawel Sobociński), and each group ended up producing a paper. We will continue with this format or a closely related one, with Jules Hedges and Daniel Cicala as organisers this year. As there were 80 applications last year for 16 slots, we may want to try to find a way to involve more students.

We are fortunate to have a number of private sector companies closely associated in some way or another, who will also participate, with Cambridge Quantum Computing Inc. and StateBox having already made major financial/logistic contributions.

On behalf of the ACT Steering Committee,

John Baez, Bob Coecke, David Spivak, Christina Vasilakopoulou