ACT 2019

Applied Category Theory 2019

15-19 July 2019
Oxford, United Kingdom

Applied Category Theory Summer School 2019

22-26 July 2019
Oxford, United Kingdom

The Applied Category Theory Conference will take place at the Department of Computer Science of the University of Oxford between Monday 15 and Friday 19 July, 2019.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying systems of all sorts using category-theoretic tools. These systems are found in the natural sciences and social sciences, as well as in computer science, linguistics, and engineering. The background and experience of our members is as varied as the systems being studied. The goal of the ACT2019 School is to help grow this community by pairing ambitious young researchers together with established researchers in order to work on questions, problems, and conjectures in applied category theory.

For information on previouly held events see the ACT website,

The Adjoint Summer School will follow on 22- 26 July 2019.

Important dates

  • Summer School Submission deadline: Wednesday 30 January 2019

All deadline times are Anywhere on Earth.

Call for papers for ACT 2019 Adjoint School

    Who should apply?

    Anyone from anywhere who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language -the definition of monoidal category for example- is encouraged.
    We will consider advanced undergraduates, PhD students, and post-docs. We ask that you commit to the full program as laid out below.

    Instructions for how to apply can be found below the research topic descriptions.


    To apply, please send the following by Wednesday 30 January 2019:

  • Your CV

    A document with:

  • An explanation of any relevant background you have in category theory or any of the specific projects areas
  • The date you completed or expect to complete your Ph.D and a one-sentence summary of its subject matter.
  • Order of project preference
  • To what extent can you commit to coming to Oxford (availability of funding is uncertain at this time)
  • A brief statement (~300 words) on why you are interested in the ACT2019 School.

    Some prompts:

    how can this school contribute to your research goals?

    how can this school help in your career?


For more information, contact Daniel Cicala or Jules Hedges


Programme committee

  • TBC

Local organisation

Invited speakers

Details TBC

Summer School

All of the participants will be divided up into groups corresponding to the projects. A group will consist of several students, a senior researcher, and a TA. Between January and June, we will have a reading course devoted to building the background necessary to meaningfully participate in the projects. Specifically, two weeks are devoted to each paper from the reading list. During this two week period, everybody will read the paper and contribute to discussion in a private online chat forum. There will be a TA serving as a domain expert and moderating this discussion. In the middle of the two week period, the group corresponding to the paper will give a presentation via video conference. At the end of the two week period, this group will compose a blog entry on this background reading that will be posted to the n-category cafe.

After all of the papers have been presented, there will be a two-week visit to Oxford University from 15-26 July 2019. The second week is solely for participants of the ACT2019 School. Groups will work together on research projects, led by the senior researchers.

The first week of this visit is the ACT2019 Conference, where the wider applied category theory community will arrive to share new ideas and results. It is not part of the school, but there is a great deal of overlap and participation is very much encouraged. The school should prepare students to be able to follow the conference presentations to a reasonable degree.

Research mentors and their topics

Below is a list of the senior researchers, each of whom describes a research project that their team will pursue, as well as the background reading that will be studied between now and July 2019.

Miriam Backens

University of Oxford

Simplifying quantum circuits using the ZX-calculus

Simplifying quantum circuits using the ZX-calculus

Description: The ZX-calculus is a graphical calculus based on the category-theoretical formulation of quantum mechanics. A complete set of graphical rewrite rules is known for the ZX-calculus, but not for quantum circuits over any universal gate set. In this project, we aim to develop new strategies for using the ZX-calculus to simplify quantum circuits.

Background reading:
  • Matthes Amy, Jianxin Chen, Neil Ross. A finite presentation of CNOT-Dihedral. Available here.
  • Miriam Backens. The ZX-calculus is complete for stabiliser quantum mechanics. Available here.

  • Tobias Fritz

    Perimeter Institute for Theoretical Physics

    Partial evaluations, the bar construction, and second-order stochastic dominance

    Partial evaluations, the bar construction, and second-order stochastic dominance

    Description: We all know that 2+2+1+1 evaluates to 6. A less familiar notion is that it can *partially evaluate* to 5+1. In this project, we aim to study the compositional structure of partial evaluation in terms of monads and the bar construction and see what this has to do with financial risk via second-order stochastic dominance.

    Background reading:
  • Tobias Fritz, Paolo Perrone. Monads, partial evaluations, and rewriting. Available here.
  • Maria Manuel Clementino, Dirk Hofmann, George Janelidze. The monads of classical algebra are seldom weakly cartesian. Available here.
  • Todd Trimble. On the bar construction. Available here.

  • Pieter Hofstra

    University of Ottowa

    Complexity classes, computation, and Turing categories

    Complexity classes, computation, and Turing categories

    Description: Turing categories form a categorical setting for studying computability without bias towards any particular model of computation. It is not currently clear, however, that Turing categories are useful to study practical aspects of computation such as complexity. This project revolves around the systematic study of step-based computation in the form of stack-machines, the resulting Turing categories, and complexity classes. This will involve a study of the interplay between traced monoidal structure and computation. We will explore the idea of stack machines qua programming languages, investigate the expressive power, and tie this to complexity theory. We will also consider questions such as the following: can we characterize Turing categories arising from stack machines? Is there an initial such category? How does this structure relate to other categorical structures associated with computability?

    Background reading:
  • J.R.B. Cockett, P.J.W. Hofstra. Introduction to Turing categories. APAL, Vol 156, pp 183-209, 2008. Available here.
  • J.R.B. Cockett, P.J.W. Hofstra, P. Hrubes. Total maps of Turing categories. ENTCS (Proc. of MFPS XXX), pp 129-146, 2014. Available here.
  • A. Joyal, R. Street, D. Verity. Traced monoidal categories. Mat. Proc. Cam. Phil. Soc. 3, pp. 447-468, 1996. Available here.

  • Bartosz Milewski


    Traversal optics and profunctors

    Traversal optics and profunctors

    Description: In functional programming, optics are ways to zoom into a specific part of a given data type and mutate it. Optics come in many flavors such as lenses and prisms and there is a well-studied categorical viewpoint, known as profunctor optics. Of all the optic types, only the traversal has resisted a derivation from first principles into a profunctor description. This project aims to do just this.

    Background reading:
  • Bartosz Milewski. Profunctor optics, categorical View. Available here.
  • Craig Pastro, Ross Street. Doubles for monoidal categories. Available here.

  • Mehrnoosh Sadrzadeh

    Queen Mary University London


    Formal and experimental methods to reason about dialogue and discourse using categorical models of vector spaces

    Formal and experimental methods to reason about dialogue and discourse using categorical models of vector spaces

    Description: Distributional semantics argues that meanings of words can be represented by the frequency of their co-occurrences in context. A model extending distributional semantics from words to sentences has a categorical interpretation via Lambek's syntactic calculus or pregroups. In this project, we intend to further extend this model to reason about dialogue and discourse utterances where people interrupt each other, there are references that need to be resolved, disfluencies, pauses, and corrections. Additionally, we would like to design experiments and run toy models to verify predictions of the developed models.

    Background reading:
  • Gerhard Jager. A multi-modal analysis of anaphora and ellipsis. Available here.
  • Matthew Purver, Ronnie Cann, Ruth Kempson. Grammars as parsers: Meeting the dialogue challenge. Available here.

  • David Spivak


    Toward a mathematical foundation for autopoiesis

    Toward a mathematical foundation for autopoiesis

    Description: An autopoietic organization-anything from a living animal to a political party to a football team-is a system that is responsible for adapting and changing itself, so as to persist as events unfold. We want to develop mathematical abstractions that are suitable to found a scientific study of autopoietic organizations. To do this, we'll begin by using behavioral mereology and graphical logic to frame a discussion of autopoeisis, most of all what it is and how it can be best conceived. We do not expect to complete this ambitious objective; we hope only to make progress toward it.

    Background reading:
  • Fong, Myers, Spivak. Behavioral mereology. Available here.
  • Fong, Spivak. Graphical regular logic. Available here.
  • Luhmann. Organization and Decision, CUP. (preface)

  • Programme

    The full programme will be published here and will consist of invited and contributed talks as well as a special session with ACT business representatives.


    Registration will be available here.

    Local information

    Oxford is the home of the oldest university in the English speaking world, and boasts many tourist attractions. It is easily reached from London's international Heathrow airport via the so-called airline bus service. For more information see these travel directions.

    Oxford has a wide range of places to stay, including college accommodation, private rooms and flats, or short term properties to share, and many bed and breakfasts (B&Bs) and guest houses. Particularly well located B&Bs are Cotswolds, Rewley House, Linton Lodge, Parklands and Galaxie Hotel. Please contact Destiny Chen should you need further information. Demand for accommodation in Oxford is high, so the rule of thumb is to book a place as soon as possible. In terms of areas, Central/North Oxford and Jericho are the closest to the workshop venue, then along Botley Road, Abingdon Road, Cowley Road (the multicultural downtown of Oxford), Iffley Road, and Wolvercote.

    The workshop will be held in Lecture Theatre B of the Department of Computer Science.

    Oxford hosts many tourist attractions and events. For example, you could complete your Oxford experience by seeing a Shakespeare play outdoors.


    ACT 2019 is sponsored by: