Published: 3rd November 2022
DOI: 10.4204/EPTCS.372
ISSN: 2075-2180


Proceedings of the Fourth International Conference on
Applied Category Theory
Cambridge, United Kingdom, 12-16th July 2021

Edited by: Kohei Kishida

Kohei Kishida
Functorial Manifold Learning
Dan Shiebler
Learners' Languages
David I. Spivak
Jacobians and Gradients for Cartesian Differential Categories
Jean-Simon Pacaud Lemay
A Categorical Construction of the Real Unit Interval
John van de Wetering
A Categorical Semantics for Bounded Petri Nets
Fabrizio Romano Genovese, Fosco Loregian and Daniele Palombi
The Word Problem for Braided Monoidal Categories is Unknot-Hard
Antonin Delpeuch and Jamie Vicary
Constructing Initial Algebras Using Inflationary Iteration
Andrew M. Pitts and S. C. Steenkamp
Situated Transition Systems
Chad Nester
Native Type Theory
Christian Williams and Michael Stay
Polynomial Life: the Structure of Adaptive Systems
Toby St Clere Smithe
Coequalisers under the Lens
Matthew Di Meglio
Limits and Colimits in a Category of Lenses
Emma Chollet, Bryce Clarke, Michael Johnson, Maurine Songa, Vincent Wang and Gioele Zardini
String Diagrammatic Electrical Circuit Theory
Guillaume Boisseau and Paweł Sobociński
Operadic Modeling of Dynamical Systems: Mathematics and Computation
Sophie Libkind, Andrew Baas, Evan Patterson and James Fairbanks
Exponential Modalities and Complementarity (extended abstract)
Robin Cockett and Priyaa Varshinee Srinivasan
Translating Extensive Form Games to Open Games with Agency
Matteo Capucci, Neil Ghani, Jérémy Ledent and Fredrik Nordvall Forsberg
Towards Foundations of Categorical Cybernetics
Matteo Capucci, Bruno Gavranović, Jules Hedges and Eigil Fjeldgren Rischel
The Sierpinski Carpet as a Final Coalgebra
Victoria Noquez and Lawrence S. Moss
The Cost of Compositionality: A High-Performance Implementation of String Diagram Composition
Paul Wilson and Fabio Zanasi
Temporal Landscapes: A Graphical Logic of Behavior
Brendan Fong, Alberto Speranzon and David I. Spivak
Quantaloidal Approach to Constraint Satisfaction
Soichiro Fujii, Yuni Iwamasa and Kei Kimura
A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces
Sean Tull
Tracelet Hopf Algebras and Decomposition Spaces (Extended Abstract)
Nicolas Behr and Joachim Kock
A Graphical Calculus for Lagrangian Relations
Cole Comfort and Aleks Kissinger
Composing Conversational Negation
Razin A. Shaikh, Lia Yeh, Benjamin Rodatz and Bob Coecke
Grounding Game Semantics in Categorical Algebra
Jérémie Koenig


The Fourth International Conference on Applied Category Theory took place at the Computer Laboratory of the University of Cambridge on 12–16 July 2021, following the previous meetings at Leiden (2018), Oxford (2019), and MIT (2020, fully online). It was preceded by the Adjoint School 2021 (5–9 July), a collaborative research event in which junior researchers worked on cutting-edge topics under the mentorship of experts. The conference comprised fifty-one contributed talks, and also hosted a poster session, an industry showcase session, and a session in which junior researchers who had attended the Adjoint School presented the results of their research at the school.

ACT 2021 was a hybrid event, with physical attendees present in Cambridge and other participants taking part online. The local organizers did a spectacular job of bringing the in-person and online modalities together, making an active exchange of ideas possible while enabling people to join in from across the world. All the talks were recorded and the videos have been posted online, links to which can be found on the conference website (

Continuing the trend in the previous meetings of ACT, the contributions to ACT 2021 ranged from pure to applied and represented a great variety of categorical techniques and application topics. Many talks discussed diagrammatic and graphical calculi, as anyone familiar with applied category theory would expect. Also, as one might predict from the precedent of the 2020 meeting, lenses were heavily studied and applied to game theory, cybernetics, and machine learning. Differential categories and categorical probability theory figured significantly as well, and were applied to machine learning. Indeed, machine learning was probably the most prominent subject of application in the conference. Natural language semantics and processing was another popular subject, as it has been in the ACT conferences. It should also be stressed, however, that the 2021 meeting saw subjects that the previous meetings had not, such as cryptography and finite model theory.

This proceedings volume contains about half of the papers that were presented as talks at ACT 2021. This selection is a reflection of the authors' choice as to whether to publish their papers in this volume or elsewhere. I hope that readers of this volume will enjoy the state of the art in applied category theory through this representative selection of papers as well as the aforementioned online videos of the talks. I am very much looking forward to seeing all of you at the future meetings of ACT.

Kohei Kishida
Chair of the ACT 2021 Programme Committee