Yes, my last post was about ACT2019, but we’re already planning next year’s applied category theory conference and school! I’m happy to say that Brendan Fong and David Spivak have volunteered to run it at MIT on these dates:

• Applied Category Theory School: June 29–July 3, 2020.
• Applied Category Theory Conference: July 6–10, 2020.

The precise dates for the other big category theory conference, CT2020, have not yet been decided. However, it will take place in Genoa sometime in the interval June 18–28, 2020.

There may also be an additional applied theory school in Marrakesh from May 25–29, 2020. More on that later, with any luck!

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:

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.

My grad student Kenny Courser gave a talk at the 4th Symposium on Compositional Structures. He spoke about his work with Christina Vasilakopolou and me. We’ve come up with a theory that can handle a broad class of open systems, from electrical circuits to chemical reaction networks to Markov processes and Petri nets. The idea is to treat open systems as morphisms in a category of a particular kind: a ‘structured cospan category’.

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:

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:

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

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:

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.

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.

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.

Eswaran Subrahmanian will also be giving a talk this time! He works at NIST and Carnegie Mellon; he’s an engineer who specializes in using mathematics to help design smart networks and other complex systems.

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’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.

My grad student Joe Moeller is talking at the 4th Symposium on Compositional Structures this Thursday! He’ll talk about his work with Christina Vasilakopolou, a postdoc here at U.C. Riverside. Together they created a monoidal version of a fundamental construction in category theory: the Grothendieck construction! Here is their paper:

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 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:

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:

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