What: Applied Category Theory 2018, a workshop on applied category theory

When: Monday, April 30 – Friday, May 4

Where: the Lorentz Center in Leiden, the Netherlands

Who: researchers and practitioners of applied category theory

The workshop is heavily slanted toward discussion and collaboration. A limited number of speakers will give overview talks intended to instigate discussion, including

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

###### 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]).1 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].

###### Bibliography

- [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. 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.

###### Prior and Related Workshops

- Compositionality at Berkeley
- CCT 2015 at NIST – Computational Category Theory Workshop
- Categorical Methods at the Crossroads 2015 at Dagstuhl
- FMCS 2015, 2016 – Foundational Methods in Computer Science
- More related workshops can be see at the Atlantic Category Theory Group.