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September 12, 2017

Applied Category Theory 2018

Posted by John Baez

We’re having a conference on applied category theory!

The plenary speakers will be:

  • Samson Abramsky (Oxford)
  • John Baez (UC Riverside)
  • Kathryn Hess (EPFL)
  • Mehrnoosh Sadrzadeh (Queen Mary)
  • David Spivak (MIT)

There will be a lot more to say as this progresses, but for now let me just quote from the conference website.

Applied Category Theory (ACT 2018) is a five-day workshop on applied category theory running from April 30 to May 4 at the Lorentz Center in Leiden, the Netherlands.

Towards an integrative science: in this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one scientific discipline can be reused in another. The aim of the workshop is to (1) explore the use of category theory within and across different disciplines, (2) create a more cohesive and collaborative ACT community, especially among early-stage researchers, and (3) accelerate research by outlining common goals and open problems for the field.

While the workshop will host talks on a wide range of applications of category theory, there will be three special tracks on exciting new developments in the field:

  1. Dynamical systems and networks
  2. Systems biology
  3. Cognition and AI
  4. Causality

Accompanying the workshop will be an Adjoint Research School for early-career researchers. This will comprise a 16 week online seminar, followed by a 4 day research meeting at the Lorentz Center in the week prior to ACT 2018. Applications to the school will open prior to October 1, and are due November 1. Admissions will be notified by November 15.

The organizers

Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford)

We welcome any feedback! Please send comments to this link.

About Applied Category Theory

Category theory is a branch of mathematics originally developed to transport ideas from one branch of mathematics to another, e.g. from topology to algebra. Applied category theory refers to efforts to transport the ideas of category theory from mathematics to other disciplines in science, engineering, and industry.

This site originated from discussions at the Computational Category Theory Workshop at NIST on Sept. 28-29, 2015. It serves to collect and disseminate research, resources, and tools for the development of applied category theory, and hosts a blog for those involved in its study.

The Proposal: Towards an Integrative Science

Category theory was developed in the 1940s to translate ideas from one field of mathematics, e.g. topology, to another field of mathematics, e.g. algebra. More recently, category theory has become an unexpectedly useful and economical tool for modeling a range of different disciplines, including programming language theory [10], quantum mechanics [2], systems biology [12], complex networks [5], database theory [7], and dynamical systems [14].

A category consists of a collection of objects together with a collection of maps between those objects, satisfying certain rules. Topologists and geometers use category theory to describe the passage from one mathematical structure to another, while category theorists are also interested in categories for their own sake. In computer science and physics, many types of categories (e.g. topoi or monoidal categories) are used to give a formal semantics of domain-specific phenomena (e.g. automata [3], or regular languages [11], or quantum protocols [2]). In the applied category theory community, a long-articulated vision understands categories as mathematical workspaces for the experimental sciences, similar to how they are used in topology and geometry [13]. This has proved true in certain fields, including computer science and mathematical physics, and we believe that these results can be extended in an exciting direction: we believe that category theory has the potential to bridge specific different fields, and moreover that developments in such fields (e.g. automata) can be transferred successfully into other fields (e.g. systems biology) through category theory. Already, for example, the categorical modeling of quantum processes has helped solve an important open problem in natural language processing [9].

In this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one discipline can be reused in another. Tangibly and in the short-term, we will bring together people from different disciplines in order to write an expository survey paper that grounds the varied research in applied category theory and lays out the parameters of the research program.

In formulating this research program, we are motivated by recent successes where category theory was used to model a wide range of phenomena across many disciplines, e.g. open dynamical systems (including open Markov processes and open chemical reaction networks), entropy and relative entropy [6], and descriptions of computer hardware [8]. Several talks will address some of these new developments. But we are also motivated by an open problem in applied category theory, one which was observed at the most recent workshop in applied category theory (Dagstuhl, Germany, in 2015): “a weakness of semantics/CT is that the definitions play a key role. Having the right definitions makes the theorems trivial, which is the opposite of hard subjects where they have combinatorial proofs of theorems (and simple definitions). […] In general, the audience agrees that people see category theorists only as reconstructing the things they knew already, and that is a disadvantage, because we do not give them a good reason to care enough” [1, pg. 61].

In this workshop, we wish to articulate a natural response to the above: instead of treating the reconstruction as a weakness, we should treat the use of categorical concepts as a natural part of transferring and integrating knowledge across disciplines. The restructuring employed in applied category theory cuts through jargon, helping to elucidate common themes across disciplines. Indeed, the drive for a common language and comparison of similar structures in algebra and topology is what led to the development category theory in the first place, and recent hints show that this approach is not only useful between mathematical disciplines, but between scientific ones as well. For example, the ‘Rosetta Stone’ of Baez and Stay demonstrates how symmetric monoidal closed categories capture the common structure between logic, computation, and physics [4].

[1] Samson Abramsky, John C. Baez, Fabio Gadducci, and Viktor Winschel. Categorical methods at the crossroads. Report from Dagstuhl Perspectives Workshop 14182, 2014.

[2] Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. In Handbook of Quantum Logic and Quantum Structures. Elsevier, Amsterdam, 2009.

[3] Michael A. Arbib and Ernest G. Manes. A categorist’s view of automata and systems. In Ernest G. Manes, editor, Category Theory Applied to Computation and Control. Springer, Berlin, 2005.

[4] John C. Baez and Mike Stay. Physics, topology, logic and computation: a Rosetta stone. In Bob Coecke, editor, New Structures for Physics. Springer, Berlin, 2011.

[5] John C. Baez and Brendan Fong. A compositional framework for passive linear networks. arXiv e-prints, 2015.

[6] John C. Baez, Tobias Fritz, and Tom Leinster. A characterization of entropy in terms of information loss. Entropy, 13(11):1945-1957, 2011.

[7] Michael Fleming, Ryan Gunther, and Robert Rosebrugh. A database of categories. Journal of Symbolic Computing, 35(2):127-135, 2003.

[8] Dan R. Ghica and Achim Jung. Categorical semantics of digital circuits. In Ruzica Piskac and Muralidhar Talupur, editors, Proceedings of the 16th Conference on Formal Methods in Computer-Aided Design. Springer, Berlin, 2016.

[9] Dimitri Kartsaklis, Mehrnoosh Sadrzadeh, Stephen Pulman, and Bob Coecke. Reasoning about meaning in natural language with compact closed categories and Frobenius algebras. In Logic and Algebraic Structures in Quantum Computing and Information. Cambridge University Press, Cambridge, 2013.

[10] Eugenio Moggi. Notions of computation and monads. Information and Computation, 93(1):55-92, 1991.

[11] Nicholas Pippenger. Regular languages and Stone duality. Theory of Computing Systems 30(2):121-134, 1997.

[12] Robert Rosen. The representation of biological systems from the standpoint of the theory of categories. Bulletin of Mathematical Biophysics, 20(4):317-341, 1958.

[13] David I. Spivak. Category Theory for Scientists. MIT Press, Cambridge MA, 2014.

[14] David I. Spivak, Christina Vasilakopoulou, and Patrick Schultz. Dynamical systems and sheaves. arXiv e-prints, 2016.

Posted at September 12, 2017 2:09 PM UTC

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Re: Applied Category Theory

Does the conference start on 29 or 30 April? My invitation from Brendan said the former, but the website and your post say the latter. 29 is Sun, 30 is Mon. I’m forming the conjecture that the 29th is merely for the drinking of beer.

Posted by: Tom Leinster on September 13, 2017 6:36 PM | Permalink | Reply to this

Re: Applied Category Theory

I just copied the information from the website. You could ask Brendan when to show up… or wait for a more detailed schedule to be created. I’ll be showing up on April 23rd to teach students the joys of applied category theory. These may include beer.

Posted by: John Baez on September 14, 2017 1:03 AM | Permalink | Reply to this

Re: Applied Category Theory

Oops, sorry Tom! Your conjecture is correct. The conference programme starts on Monday, April 30th. The beer starts, well, whenever.

Posted by: Brendan Fong on September 15, 2017 3:56 AM | Permalink | Reply to this

Re: Applied Category Theory

I think I can manage! Actually, there’s something I’d really like to do if I have a spare day or two in Leiden, which is to visit the natural history museum, and in particular the associated biodiversity centre.

The reason is that about a year ago, I took an excellent MOOC run from the Leiden biodiversity centre, called Evolution Today, via the MOOC provider Coursera. It was so pleasant and civilized and gentle, and at the same time I learned a great deal from it. So I’d like to go the museum and relive those happy weeks of learning about the genetic code and intersexual vs. intrasexual selection and DNA transcription and the weird social structure of bees, which explains why it’s in worker bees’ genetic interest to help someone other than themselves lay eggs. And then, do some delicious category theory.

Posted by: Tom Leinster on September 15, 2017 11:29 PM | Permalink | Reply to this

Re: Applied Category Theory

As someone in industry I would say that the “weakness [in that] definitions play a key role” is a feature of CT, not a bug. In fact that is the entire point of the subject to researchers in my position: frequently one is confronted with messy or ill-defined structures thrown together by engineers and the key task is to understand what they were really trying to do, and then to do it properly.

For example, it’s pretty clear that deterministic network calculus (a sort of queueing theory that gives deterministic bounds on latency and backlog of communications) could benefit from some categorical elbow grease. In fact that perspective, which I would say came from Brendan and Spencer Breiner, turned out to anticipate (alas, slightly after the fact) a groundbreaking algorithm (see section 4.3 of As another example, it turns out that a certain type of statistical dimensionality reduction algorithm called tree-preserving embedding is (after a trivial modification) functorial in the sense of Carlsson and Memoli, and moreover that it underlies all such functorial dimensionality reduction techniques–but the authors didn’t realize this.

Posted by: Steve Huntsman on September 15, 2017 1:42 PM | Permalink | Reply to this

Re: Applied Category Theory

Hi! I hope to see you in Riverside in November!

That quote infuriates me for a number of reasons:

“a weakness of semantics/CT is that the definitions play a key role. Having the right definitions makes the theorems trivial, which is the opposite of hard subjects where they have combinatorial proofs of theorems (and simple definitions).”

First, the idea that “definition play a key role” in some branches of mathematics—but apparently not others?—looks prima facie absurd. Pure math starts with definitions and then proves theorems: the definitions always play a key role.

Second, if having the definitions makes some theorems trivial, that’s a good thing. There’s no benefit in having theorems be harder to prove.

Third, there are hard theorems even in category theory. There are always hard theorems to be proved, in any area of math, if you go looking for them. The right definitions can make some theorems trivial, but that just means you can go further before you run into the problems that are so hard you can’t solve them. Even if you want to climb Mt. Everest, you still might as well scrape the ice off your front porch: a challenge can be fun, but slipping around on the ice as you walk to your car to drive to the airport to go to Nepal to climb Mt. Everest—that’s a challenge you don’t really need.

Fourth, most definitions in category theory are not so complicated. What they mainly are is abstract: it takes time to wrap your head around them.

Posted by: John Baez on September 15, 2017 3:00 PM | Permalink | Reply to this

Re: Applied Category Theory

That’s exactly the point of category theory: you dig “down to the engine of the things” already at the level of definitions. In most other areas of math, instead, definitions and theorems are like a clean plastic panel, but each time you have to prove something you have to remove the panel and mess with the tangle of cables behind.

So yeah, definitions look abstract, but that’s just because that’s how math is “inside”.

Posted by: Paolo on September 18, 2017 10:22 AM | Permalink | Reply to this

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