Applied Category Theory 2019 happened last week! It was very exciting: about 120 people attended, and they’re pushing forward to apply category theory in many different directions. The topics ranged from ultra-abstract to ultra-concrete, sometimes in the same talk.

The talks are listed above — click for a more readable version. Below you can read what Jules Hedges and I wrote about all those talks:

• Jules Hedges, Applied Category Theory 2019.

I tend to give terse summaries of the talks, with links to the original papers or slides. Jules tends to give his impressions of their overall significance. They’re nicely complementary.

You can also see videos of some talks, created by Jelle Herold with help from Fabrizio Genovese:

• Giovanni de Felice, Functorial question answering.

• Antonin Delpeuch, Autonomization of monoidal categories.

• Colin Zwanziger, Natural model semantics for comonadic and adjoint modal type theory.

• Nicholas Behr, Tracelets and tracelet analysis Of compositional rewriting systems.

• Dan Marsden, No-go theorems for distributive laws.

• Christian Williams, Enriched Lawvere theories for operational semantics.

• Walter Tholen, Approximate composition.

• Erwan Beurier, Interfacing biology, category theory & mathematical statistics.

• Stelios Tsampas, Categorical contextual reasoning.

• Fabrizio Genovese, idris-ct: A library to do category theory in Idris.

• Michael Johnson, Machine learning and bidirectional transformations.

• Bruno Gavranović, Learning functors using gradient descent

• Zinovy Diskin, Supervised learning as change propagation with delta lenses.

• Bryce Clarke, Internal lenses as functors and cofunctors.

• Ryan Wisnewsky, Conexus AI.

• Ross Duncan, Cambridge Quantum Computing.

• Beurier Erwan, Memoryless systems generate the class of all discrete systems.

• Blake Pollard, Compositional models for power systems.

• Martti Karvonen, A comonadic view of simulation and quantum resources.

• Quanlong Wang, ZX-Rules for 2-qubit Clifford+T quantum circuits, and beyond.

• James Fairbank, A Compositional framework for scientific model augmentation.

• Titoan Carette, Completeness of graphical languages for mixed state quantum mechanics.

• Antonin Delpeuch, A complete language for faceted dataflow languages.

• John van der Wetering, An effect-theoretic reconstruction of quantum mechanics.

• Vladimir Zamdzhiev, Inductive datatypes for quantum programming.

• Octavio Malherbe, A categorical construction for the computational definition of vector spaces.

• Vladimir Zamdzhiev, Mixed linear and non-linear recursive types.

]]>• Applied Category Theory 2019, 15–19 July, 2019, Lecture Theatre B of the Department of Computer Science, 10 Keble Road, Oxford.

You can now see the program here, or below. Hope to see you soon!

]]>Here is his talk:

• Kenny Courser, Structured cospans.

In July 11th I’m going to talk about structured cospans at the big annual category theory conference, CT2019:

• John Baez, Structured cospans.

I borrowed more than just the title from Kenny’s talk… but since I’m an old guy, they’re giving me time to say more stuff. For full details, try Kenny’s thesis:

• Kenny Courser, *The Mathematics of Open Systems from a Double Categorical Perspective*.

This thesis is not quite in its final form, so I won’t try to explain it all now. But it’s full of great stuff, so I hope you look at it! If you have any questions or corrections please let us know.

We’ve been working on this project for a couple of years, so there’s a lot to say… but right now let me just tell you what a ‘structured cospan’ is.

Suppose you have any functor Then a **structured cospan** is a diagram like this:

For example if is the functor from sets to graphs sending each set to the graph with that set of vertices and no edges, a structured cospan looks like this:

It’s a graph with two sets getting mapped into its set of vertices. I call this an **open graph**. Or if is the functor from sets to Petri nets sending each set to the Petri having that set of places and nothing else, a structured cospan looks like this:

You can read a lot more about this example here:

• John Baez, Open Petri nets, *Azimuth*, 15 August 2018.

It illustrates many ideas from the general theory of structured cospans: for example, what we *do* with them.

You may have heard of a similar idea: ‘decorated cospans’, invented by Brendan Fong. You may wonder what’s the difference!

Kenny’s talk explains the difference pretty well. Basically, decorated cospans that look isomorphic may not be technically isomorphic. For example, if we have an open graph like this:

and its set of edges is this is *not* isomorphic to the identical-looking open graph whose set of edges is That’s right: the names of the edges matter!

This is an annoying glitch in the formalism. As Kenny’s talk explains, structured cospans don’t suffer from this problem.

My talk at CT2019 explains another way to fix this problem: using a new improved concept of decorated cospan! This new improved concept gives results that match those coming from structured cospan in many cases. Proving this uses some nice theorems proved by Kenny Courser, Christina Vasilakopoulou and also Daniel Cicala.

But I think structured cospans are simpler than decorated cospans. They get the job done more easily in most cases, though they don’t handle *everything* that decorated cospans do.

I’ll be saying more about structured cospans as time goes on. The basic theorem, in case you’re curious but don’t want to look at my talk, is this:

**Theorem.** Let be a category with finite coproducts, a category with finite colimits, and a functor preserving finite coproducts. Then there is a symmetric monoidal category where:

• an object is an object of

• a morphism is an isomorphism class of structured cospans:

Here two structured cospans are **isomorphic** if there is a commutative diagram of this form:

If you don’t want to work with *isomorphism classes* of structured cospans, you can use a symmetric monoidal bicategory where the 1-morphisms are actual structured cospans. But following ideas of Mike Shulman, it’s easier to work with a symmetric monoidal double category. So:

**Theorem.** Let be a category with finite coproducts, a category with finite colimits, and a functor preserving finite coproducts. Then there is a symmetric monoidal double category where:

• an object is an object of

• a vertical 1-morphism is a morphism of

• a horizontal 1-cell is a structured cospan

• a 2-morphism is a commutative diagram

]]>• Nancy M. Haegel *et al.*, Terawatt-scale photovoltaics: transform global energy, *Science* **364** (2019), 836–838.

Important topic! Here’s the abstract:

Solar energy has the potential to play a central role in the future global energy system because of the scale of the solar resource, its predictability, and its ubiquitous nature. Global installed solar photovoltaic (PV) capacity exceeded 500 GW at the end of 2018, and an estimated additional 500 GW of PV capacity is projected to be installed by 2022–2023, bringing us into the era of TW-scale PV. Given the speed of change in the PV industry, both in terms of continued dramatic cost decreases and manufacturing-scale increases, the growth toward TW-scale PV has caught many observers, including many of us (1), by surprise. Two years ago, we focused on the challenges of achieving 3 to 10 TW of PV by 2030. Here, we envision a future with ∼10 TW of PV by 2030 and 30 to 70 TW by 2050, providing a majority of global energy. PV would be not just a key contributor to electricity generation but also a central contributor to all segments of the global energy system. We discuss ramifications and challenges for complementary technologies (e.g., energy storage, power to gas/liquid fuels/chemicals, grid integration, and multiple sector electrification) and summarize what is needed in research in PV performance, reliability, manufacturing, and recycling.

Of course, increased energy storage is needed to take advantage of solar power. Let’s see what they say about that:

]]>

Energy storageAt high penetration, increased PV installation is synergistic with increased storage. Tesla recently installed a 100-MW battery in South Australia and in the first 6 months recovered 14% of the capital cost. California is also setting aggressive targets for storage. The price of lithium-ion batteries has decreased by more than 80% in the past 8 years, and improvements are expected to continue through a combination of technological advances and increased manufacturing capacity. To achieve the U.S. Department of Energy target price of U.S. $150/kWh for automotive batteries capable of charging within 15 minutes, research should explore materials with higher energy density to further reduce costs, focusing on nickel-rich, critical-materials–free cathodes and advanced anodes for lithium-ion systems. With further research and cost reduction, flow batteries and sodium-ion and multivalent-ion or conversion systems could also hold the promise of long-term competitors to lithium ion.

An additional approach to battery-based storage is pumped-storage hydropower (pumped hydro). Recent research indicates that there is a substantial technical potential for untapped off-river (closed-loop) pumped hydro and other forms of gravity storage in many parts of the world (9, 10). Pumped hydro has the advantage of being able to provide short-term responsiveness and diurnal-scale storage potentially at low cost.

The biggest challenge may be to meet energy requirements during the winter at high latitudes. However, wind power tends to be more abundant in many of these locations, whereas most of the world’s population lives closer to the equator. Economic development as well as population growth may be dominated by countries within 35° of the equator in the coming decades.

The American Mathematical Society is having their Fall Western meeting here at U. C. Riverside during the weekend of November 9th and 10th, 2019. Joe Moeller and I are organizing a session on Applied Category Theory! We already have some great speakers lined up:

• Tai-Danae Bradley

• Vin de Silva

• Brendan Fong

• Nina Otter

• Evan Patterson

• Blake Pollard

• Prakash Panangaden

• David Spivak

• Brad Theilman

• Dmitry Vagner

• Zhenghan Wang

Alas, we have no funds for travel and lodging. If you’re interested in giving a talk, please submit an abstract here:

• General information about abstracts, American Mathematical Society.

More precisely, please read the information there and then click on the link on that page to submit an abstract. It should then magically fly through the aether to me! Abstracts are due September 3rd, but the sooner you submit one, the greater the chance that we’ll have space.

For the program of the whole conference, go here:

• Fall Western Sectional Meeting, U. C. Riverside, Riverside, California, 9–10 November 2019.

I will also be running a special meeting on diversity and excellence in mathematics on Friday November 8th. There will be a banquet that evening, and at some point I’ll figure out how tickets for that will work.

We had a special session like this in 2017, and it’s fun to think about how things have evolved since then.

David Spivak had already written *Category Theory for the Sciences*, but more recently he’s written another book on applied category theory, *Seven Sketches*, with Brendan Fong. He already had a company, but now he’s helping run Conexus, which plans to award grants of up to $1.5 million to startups that use category theory (in exchange for equity). Proposals are due June 30th, by the way!

I guess Brendan Fong was already working with David Spivak at MIT in the fall of 2017, but since then they’ve written *Seven Sketches* and developed a graphical calculus for logic in regular categories. He’s also worked on a functorial approach to machine learning—and now he’s using category theory to unify learners and lenses.

Blake Pollard had just finished his Ph.D. work at U.C. Riverside back in 2018. He will now talk about his work with Spencer Breiner and Eswaran Subrahmanian at the National Institute of Standards and Technology, using category theory to help develop the “smart grid”—the decentralized power grid we need now. Above he’s talking to Brendan Fong at the Centre for Quantum Technologies, in Singapore. I think that’s where they first met.

Nina Otter was a grad student at Oxford in 2017, but now she’s at UCLA and the University of Leipzig. She worked with Ulrike Tillmann and Heather Harrington on stratifying multiparameter persistent homology, and is now working on a categorical formulation of positional and role analysis in social networks. Like Brendan, she’s on the executive board of the applied category theory journal *Compositionality*.

I first met Tai-Danae Bradley at ACT2018. Now she will talk about her work at Tunnel Technologies, a startup run by her advisor John Terilla. They model sequences—of letters from an alphabet, for instance—using quantum states and tensor networks.

Vin de Silva works on topological data analysis using persistent cohomology so he’ll probably talk about that. He’s studied the “interleaving distance” between persistence modules, using category theory to treat it and the Gromov-Hausdorff metric in the same setting. He came to the last meeting and it will be good to have him back.

Evan Patterson is a statistics grad student at Stanford. He’s worked on knowledge representation in bicategories of relations, and on teaching machines to understand data science code by the semantic enrichment of dataflow graphs. He too came to the last meeting.

Dmitry Vagner was also at the last meeting, where he spoke about his work with Spivak on open dynamical systems and the operad of wiring diagrams. He is now working on mathematically defining and implementing (in Idris) wiring diagrams for symmetric monoidal categories.

Prakash Panangaden has long been a leader in applied category theory, focused on semantics and logic for probabilistic systems and languages, machine learning, and quantum information theory.

Brad Theilman is a grad student in computational neuroscience at U.C. San Diego. I first met him at ACT2018. He’s using algebraic topology to design new techniques for quantifying the spatiotemporal structure of neural activity in the auditory regions of the brain of the European starling. (I bet you didn’t see those last two words coming!)

Last but not least, Zhenghan Wang works on condensed matter physics and modular tensor categories at U.C. Santa Barbara. At Microsoft’s Station Q, he is using this research to help design topological quantum computers.

In short: a lot has been happening in applied category theory, so it will be good to get together and talk about it!

]]>
*There seems to be a murky abyss lurking at the bottom of mathematics. While in many ways we cannot hope to reach solid ground, mathematicians have built impressive ladders that let us explore the depths of this abyss and marvel at the limits and at the power of mathematical reasoning at the same time.*

This is a quote from Matthew Katz and Jan Reimann’s book *An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics*. I’ve been been talking to my old friend Michael Weiss about nonstandard models of Peano arithmetic on his blog. We just got into a bit of Ramsey theory. But you might like the whole series of conversations, which are precisely about this murky abyss.

Here it is so far:

• Part 1: I say I’m trying to understand ‘recursively saturated’ models of Peano arithmetic, and Michael dumps a lot of information on me. The posts get easier to read after this one!

• Part 2: I explain my dream: to show that the concept of ‘standard model’ of Peano arithmetic is more nebulous than many seem to think. We agree to go through Ali Enayat’s paper Standard models of arithmetic.

• Part 3: We talk about the concept of ‘standard model’, and the ideas of some ultrafinitists: Alexander Yessenin-Volpin and Edward Nelson.

• Part 4: Michael mentions “the theory of true arithmetic”, and I ask what that means. We decide that a short dive into the philosophy of mathematics may be required.

• Part 5: Michael explains his philosophies of mathematics, and how they affect his attitude toward the natural numbers and the universe of sets.

• Part 6: After explaining my distaste for the Punch-and-Judy approach to the philosophy of mathematics (of which Michael is thankfully not guilty), I point out a strange fact: our views on the infinite cast shadows on our study of the natural numbers. For example: large cardinal axioms help us name larger *finite* numbers.

• Part 7: We discuss Enayat’s concept of “a T-standard model of PA”, where T is some axiom system for set theory. I describe my crazy thought: maybe *your* standard natural numbers are nonstandard for *me*. We conclude with a brief digression into Hermetic philosophy: “as above, so below”.

• Part 8: We discuss the tight relation between PA and ZFC with the axiom of infinity replaced by its negation. We then chat about Ramsey theory as a warmup for the Paris–Harrington Theorem.

• Part 9: Michael sketches the proof of the Paris–Harrington Theorem, which says that a certain rather simple theorem about combinatorics can be stated in PA, and proved in ZFC, but not proved in PA. The proof he sketches builds a nonstandard model in which this theorem does not hold!

• Part 10: Michael and I talk about “ordinal analysis”: a way of assigning ordinals to theories of arithmetic, that measures how strong they are.

• Part 11: Michael begins explaining Enayat’s paper Standard models of arithmetic. I pull him into explaining “Craig’s trick” and “Rosser’s trick”, two famous tricks in mathematical logic.

]]>There’s another conference involving applied category theory at Chapman University!

• Quantum Physics and Logic 2019, June 9-14, 2019, Chapman University, Beckman Hall 404. Organized by Matthew Leifer, Lorenzo Catani, Justin Dressel, and Drew Moshier.

The QPL series started out being about quantum programming languages, but it later broadened its scope while keeping the same acronym. This conference series now covers quite a range of topics, including the category-theoretic study of physical systems. My students Kenny Courser, Jade Master and Joe Moeller will be speaking there, and I’ll talk about Kenny’s new work on structured cospans as a tool for studying open systems.

• John Baez (UC Riverside), Structured cospans.

• Anna Pappa (University College London), Classical computing via quantum means.

• Joel Wallman (University of Waterloo), TBA.

• Ana Belen Sainz (Perimeter Institute), Bell nonlocality: correlations from principles.

• Quanlong Wang (University of Oxford) and KangFeng Ng (Radboud University), Completeness of the ZX calculus.

]]>• Joe Moeller and Christina Vasilakopoulou, Monoidal Grothendieck construction.

The monoidal Grothendieck construction plays an important role in our team’s work on network theory, in at least two ways. First, we use it to get a symmetric monoidal category, and then an operad, from any network model. Second, we use it to turn any decorated cospan category into a ‘structured cospan category’. I haven’t said anything about structured cospans yet, but they are an alternative approach to open systems, developed by my grad student Kenny Courser, that I’m very excited about. Stay tuned!

The Grothendieck construction turns a functor

into a category equipped with a functor

The construction is quite simple but there’s a lot of ideas and terminology connected to it: for example a functor is called an indexed category since it assigns a category to each object of while the functor is of a special sort called a fibration.

I think the easiest way to learn more about the Grothendieck construction and this new monoidal version may be Joe’s talk:

• Joe Moeller, Monoidal Grothendieck construction, SYCO4, Chapman University, 22 May 2019.

Abstract.We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this global monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a fibrewise one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal structures on a functor to Cat bijectively correspond to lifts of the functor to MonCat. Finally, we give some indicative examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.

To dig deeper, try this talk Christina gave at the big annual category theory conference last year:

• Christina Vasilakopoulou, Monoidal Grothendieck construction, CT2018, University of Azores, 10 July 2018.

Then read Joe and Christina’s paper!

Here is the Grothendieck construction in a nutshell:

]]>• John Baez and Christian Williams, Enriched Lawvere theories for operational semantics.

Abstract.Enriched Lawvere theories are a generalization of Lawvere theories that allow us to describe the operational semantics of formal systems. For example, a graph-enriched Lawvere theory describes structures that have a graph of operations of each arity, where the vertices are operations and the edges are rewrites between operations. Enriched theories can be used to equip systems with operational semantics, and maps between enriching categories can serve to translate between different forms of operational and denotational semantics. The Grothendieck construction lets us study all models of all enriched theories in all contexts in a single category. We illustrate these ideas with the SKI-combinator calculus, a variable-free version of the lambda calculus, and with Milner’s calculus of communicating processes.

When Mike Stay came to U.C. Riverside to work with me about ten years ago, he knew about computation and I knew about category theory, and we started trying to talk to each other. I’d heard that categories and computer science were deeply connected: for example, people like to say that the lambda-calculus is all about cartesian closed categories. But we soon realized something funny was going on here.

Computer science is deeply concerned with *processes of computation*, and category theory uses morphisms to describe processes… but when cartesian closed categories are applied to the lambda calculus, their morphisms do *not* describe processes of computation. In fact, the process of computation is effectively ignored!

We decided that to fix this we could use 2-categories where

• objects are types. For example, there could be a type of integers, `INT`

. There could be a type of pairs of integers, `INT × INT`

. There could also be a boring type `1`

, which represents something there’s just one of.

• morphisms are terms. For example, a morphism `f: 1 → INT`

picks out a specific natural number, like 2 or 3. There could also be a morphism `+: INT × INT → INT`

, called ‘addition’. Combining these, we can get expressions like 2+3.

• 2-morphism are rewrites. For example, there could be a rewrite going from 2+3 to 5.

Later Mike realized that instead of 2-categories, it can be good to use graph-enriched categories: that is, things like categories where instead of a *set* of morphisms from one object to another, we have a *graph*.

In other words: instead of hom-sets, a graph-enriched category has ‘hom-graphs’. The objects of a graph-enriched category can represent types, the vertices of the hom-graphs can represent terms, and the edges of the hom-graphs can represent rewrites.

Mike teamed up with Greg Meredith to write a paper on this:

• Mike Stay and Greg Meredith, Representing operational semantics

with enriched Lawvere theories.

Christian decided to write a paper building on this, and I’ve been helping him out because it’s satisfying to see an old dream finally realized—in a much more detailed, beautiful way than I ever imagined!

The key was to sharpen the issue by considering enriched Lawvere theories. Lawvere theories are an excellent formalism for describing algebraic structures obeying equational laws, but they do not specify how to compute in such a structure, for example taking a complex expression and simplifying it using rewrite rules. *Enriched* Lawvere theories let us study the process of rewriting.

Maybe I should back up a bit. A Lawvere theory is a category with finite products generated by a single object , for ‘type’. Morphisms represent *n*-ary operations, and commutative diagrams specify equations these operations obey. There is a theory for groups, a theory for rings, and so on. We can specify algebraic structures of a given kind in some ‘context’—that is, in some category with finite products—by a product-preserving functor For example, if is the theory of groups and is the category of sets then such a functor describes a group, but if is the category of topological space then such a functor describes a *topological* group.

All this is a simple and elegant form of what computer scientists call *denotational* semantics: roughly, the study of types and terms, and what they signify. However, Lawvere theories know nothing of *operational* semantics: that is, how we actually compute. The objects of our Lawvere are types and the morphisms are terms. But there are no rewrites going between terms, only equations!

This is where enriched Lawvere theories come in. Suppose we fix a cartesian closed category *V*, such as the category of sets, or the category of graphs, or the category of posets, or even the category of categories. Then *V*-enriched category is a thing like a category, but instead of having a *set* of morphisms from any object to any other object, it has an *object of V*. That is, instead of hom-sets it can have hom-graphs, or hom-posets, or hom-categories. If it has hom-categories, then it’s a 2-category—so this setup includes my original dream, but much more!

Our paper explains how to generalize Lawvere theories to this enriched setting, and how to use these enriched Lawvere theories in operational semantics. We rely heavily on previous work, especially by Rory Lucyshyn-Wright, who in turn built on work by John Power and others. But we’re hoping that our paper, which is a bit less high-powered, will be easier for people who are familiar with category theory but not yet enriched categories. The novelty lies less in the math than its applications. Give it a try!

Here is a small piece of a hom-graph in the graph-enriched theory of the SKI combinator calculus, a variable-free version of the lambda calculus invented by Moses Schönfinkel and Haskell Curry back in the 1920s:

Here’s my talk for SYCO4 next week:

Abstract.To describe systems composed of interacting parts, scientists and engineers draw diagrams of networks: flow charts, Petri nets, electrical circuit diagrams, signal-flow graphs, chemical reaction networks, Feynman diagrams and the like. All these different diagrams fit into a common framework: the mathematics of symmetric monoidal categories. Two complementary approaches are presentations of props using generators and relations (which are more algebraic in flavor) and structured cospan categories (which are more geometrical). In this talk we focus on the former. A “prop” is a strict symmetric monoidal category whose objects are tensor powers of a single generating object. We will see that props are a flexible tool for describing many kinds of networks.

You can read a lot more here:

• John Baez, Props in network theory (part 1), *Azimuth*, April 27, 2018.