I serve as chief executive at Topos Institute.
I am interested in the mathematics of open systems, particularly through a category theoretic lens. An introduction to my work can be found in my book An Invitation to Applied Category Theory, my DPhil thesis, or on John Baez's blog. I am also interested in the social process by which science and technology is produced, and how it impacts our lives.
I believe that mathematicians and computer scientists play a critical role in shaping today's world, and that the values our research communities uphold deserve careful attention. As part of this, I believe in actively working to improve inclusivity in category theory and mathematics at large.
I am a steering committee member of The Adjoint School, an annual research school for researchers new to the applied category theory community. I was previously an executive editor of the diamond open-access journal Compositionality from its founding in 2017 until 2023, and organised the MIT Categories Seminar from 2018 until 2020.
An Invitation to Applied Category Theory
My book (with David Spivak) is available in print or on the arXiv:
- B. Fong, D. Spivak, An Invitation to Applied Category Theory: Seven Sketches in Compositionality, Cambridge University Press 2019.
The e-book version linked above is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. © Brendan Fong and David Spivak 2019. A print version is available here.
A course based on this book was taught at MIT in January 2019; videos are available here.
Programming with Categories
In January 2020, Bartosz Milewski, David Spivak and I taught a course at MIT on programming with categories. Videos are available here; an accompanying textbook is in draft.
Papers
Here are some papers I have written, with thanks to many wonderful collaborators:
Category theory
- Regular calculi I: graphical regular logic (with t. clingman and D. Spivak), arXiv:2109.14123.
- Regular and relational categories: revisiting "Cartesian bicategories I" (with D. Spivak), arXiv:1909.00069.
- Supplying bells and whistles in symmetric monoidal categories (with D. Spivak), arXiv:1908.02633.
- Graphical regular logic (with D. Spivak), arXiv:1812.05765.
- A recipe for black box functors (with M. Sarazola), Theory and Applications of Categories, Vol. 35, 2020, No. 26, pp 979-1011, arXiv:1812.03601.
- Hypergraph categories (with D. Spivak), Journal of Pure and Applied Algebra, Vol. 223, No. 11, pp.4746--4777, 2019, arXiv:1806.08304.
- Universal constructions for (co)relations: categories, monoidal categories, props (with F. Zanasi), Logical Methods in Computer Science, Vol. 14, Issue 3, 2018, arXiv:1710.03894.
- Decorated corelations, Theory and Applications of Categories, Vol. 33, 2018, No. 22, pp. 608--643, arXiv:1703.09888.
- A universal construction for (co)relations (with F. Zanasi), CALCO 2017, arXiv:1703.08247.
- Corelations are the prop for extraspecial commutative Frobenius monoids (with B. Coya), Theory and Applications of Categories, Vol. 32, 2017, No. 11, pp. 380-395, arXiv:1601.02307.
- Decorated cospans, Theory and Applications of Categories, Vol. 30, 2015, No. 33, pp. 1096--1120, arXiv:1502.00872.
Logic and computation
- Temporal landscapes: a graphical temporal logic (with A. Speranzon, D. Spivak), arXiv:1904.01081.
- String diagrams for regular logic (with D. Spivak), Proceedings of the 2nd International Conference on Applied Category Theory (ACT 2019), arXiv:2009.06836.
- Backprop as functor: a compositional perspective on supervised learning (with D. Spivak, R. Tuyeras), Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2019), arXiv:1711.10455.
- Lenses and learners (with M. Johnson) Proceedings of the Eighth International Workshop on Bidirectional Transformations (Bx 2019), arXiv:1903.03671.
- Additive monotones for resource theories of parallel combinable processes (with H. Nava Kopp), Electronic Proceedings in Theoretical Computer Science (QPL 2015), 195 (2015), pp. 170--178, arXiv:1505.02651.
- A note on interleaved quantum teleportation Technical Report CS-RR-14-09, University of Oxford, 2014.
Networks and systems
- Behavioral mereology (with D. Myers, D. Spivak), Proceedings of the Third International Conference on Applied Category Theory (ACT 2020) arXiv:1811.00420.
- A compositional framework for passive linear networks (with J. C. Baez), Theory and Applications of Categories, Vol. 33, No. 22, pp. 608--643, 2018, arXiv:1504.05625.
- A categorical approach to open and interconnected systems (with P. Sobocinski, P. Rapisarda), Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2016), arXiv:1510.05076.
- A compositional framework for Markov processes (with J. C. Baez, B. Pollard), Journal of Mathematical Physics, 57, 2016, 033301, arXiv:1508.06448.
- Quantum techniques for studying equilibrium in chemical reaction networks (with J. C. Baez), Journal of Complex Networks (2014), doi: 10.1093/comnet/cnu013, arXiv:1305.4988.
- A Noether theorem for Markov processes (with J. C. Baez), Journal of Mathematical Physics 54 (2013), 013301, arXiv:1203.2035.
Tools for mathematics
- Extracting mathematical concepts from text (with J. Collard, V. de Paiva, and E. Subrahmanian), Proceedings of the Eighth Workshop on Noisy User-generated Text (W-NUT 2022), arXiv:2208.13830.
Theses
- The algebra of open and interconnected systems, DPhil thesis, University of Oxford, 2016, arXiv:1609.05382.
- Causal theories: a categorical perspective on Bayesian networks, Master's thesis, University of Oxford, 2012, arXiv:1301.6201.
- On families of elliptic curves, Honours thesis, Australian National University and National University of Singapore, 2010.
Other writing
My articles for the Topos Institute blog can be found here.