Adjoint School 2022

2 January, 2022

Every year since 2018 we’ve been having annual courses on applied category theory where you can do research with experts. It’s called the Adjoint School.

You can apply to be a student at the 2022 Adjoint School now, and applications are due January 29th! Go here:

2022 Adjoint School: application.

The school will be run online from February to June, 2022, and then—coronavirus permitting—there will be in-person research at the University of Strathclyde in Glasgow, Scotland the week of July 11 – 15, 2022. This is also the location of the applied category theory conference ACT2022.

The 2022 Adjoint School is organized by Angeline Aguinaldo, Elena Di Lavore, Sophie Libkind, and David Jaz Myers. You can read more about how it works here:

About the Adjoint School.

There are four topics to work on, and you can see descriptions of them below.

Who should apply?

Anyone, from anywhere in the world, 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, post-docs, as well as people working outside of academia. Members of groups which are underrepresented in the mathematics and computer science communities are especially encouraged to apply.

Also check out our inclusivity statement.

Topic 1: Compositional Thermodynamics

Mentors: Spencer Breiner and Joe Moeller

TA: Owen Lynch

Description: Thermodynamics is the study of the relationships between heat, energy, work, and matter. In category theory, we model flows in physical systems using string diagrams, allowing us to formalize physical axioms as diagrammatic equations. The goal of this project is to establish such a compositional framework for thermodynamical networks. A first goal will be to formalize the laws of thermodynamics in categorical terms. Depending on the background and interest of the participants, further topics may include the Carnot and Otto engines, more realistic modeling for real-world systems, and software implementation within the AlgebraicJulia library.

Readings:

• John C. Baez, Owen Lynch, and Joe Moeller, Compositional thermostatics.

• F. William Lawvere, State categories, closed categories and the existence of semi-continuous entropy functions.

Topic 2: Fuzzy Type Theory for Opinion Dynamics

Mentor: Paige North

TA: Hans Reiss

Description: When working in type theory (or most logics), one is interested in proving propositions by constructing witnesses to their incontrovertible truth. In the real world, however, we can often only hope to understand how likely something is to be true, and we look for evidence that something is true. For example, when a doctor is trying to determine if a patient has a certain condition, they might ask certain questions and perform certain tests, each of which constitutes a piece of evidence that the patient does or does not have that condition. This suggests that a fuzzy version of type theory might be appropriate for capturing and analyzing real-world situations. In this project, we will explore the space of fuzzy type theories which can be used to reason about the fuzzy propositions of disease and similar dynamics.

Readings:

• Daniel R. Grayson, An introduction to univalent foundations for mathematicians.

• Jakob Hansen and Robert Ghrist, Opinion dynamics on discourse sheaves.

Topic 3: A Compositional Theory of Timed and Probabilistic Processes: CospanSpan(Graph)

Mentor: Nicoletta Sabadini

TA: Mario Román

Description coming soon!

Topic 4: Algebraic Structures in Logic and Relations

Mentor: Filippo Bonchi

Description coming soon!


Learn Applied Category Theory!

27 October, 2021

Do you like the idea of learning applied category theory by working on a project, as part of a team led by an expert? If you’re an early career researcher you can apply to do that now!

Mathematical Research Community: Applied Category Theory, meeting 2022 May 29–June 4. Details on how to apply: here. Deadline to apply: Tuesday 2022 February 15 at 11:59 Eastern Time.

After working with your team online, you’ll take an all-expenses-paid trip to a conference center in upstate New York for a week in the summer. There will be a pool, bocci, lakes with canoes, woods to hike around in, campfires at night… and also whiteboards, meeting rooms, and coffee available 24 hours a day to power your research!

Later you’ll get invited to the 2023 Joint Mathematics Meetings in Boston.

There will be three projects to choose from:

Valeria de Paiva (Topos Institute) will lead a study in the context of computer science that investigates indexed containers and partial compilers using lenses and Dialectica categories.

Nina Otter (Queen Mary University of London) will lead a study of social networks using simplicial complexes.

John Baez (University of California, Riverside) will lead a study of chemical reaction networks using category theoretic methods such as structured cospans.

The whole thing is being organized by Daniel Cicala of the University of New Haven:

and Simon Cho of Two Six Technologies:

I should add that this is just one of four ‘Mathematical Research Communities’ run by the American Mathematical Society in 2022, and you may prefer another. The applied category theory session will be held at the same time and place as one on data science! Then there are two more:

• Week 1a: Applied Category Theory

Organizers: John Baez, University of California, Riverside; Simon Cho, Two Six Technologies; Daniel Cicala, University of New Haven; Nina Otter, Queen Mary University of London; Valeria de Paiva, Topos Institute.

• Week 1b: Data Science at the Crossroads of Analysis, Geometry, and Topology

Organizers: Marina Meila, University of Washington; Facundo Mémoli, The Ohio State University; Jose Perea, Northeastern University; Nicolas Garcia Trillos, University of Wisconsin-Madison; Soledad Villar, Johns Hopkins University.

• Week 2a: Models and Methods for Sparse (Hyper)Network Science

Organizers: Sinan G. Aksoy, Pacific Northwest National Laboratory; Aric Hagberg, Los Alamos National Laboratory; Cliff Joslyn, Pacific Northwest National Laboratory; Bill Kay, Oak Ridge National Laboratory; Emilie Purvine, Pacific Northwest National Laboratory; Stephen J. Young, Pacific Northwest National Laboratory; Jennifer Webster, Pacific Northwest National Laboratory.

• Week 2b: Trees in Many Contexts

Organizers: Miklós Bóna, University of Florida; Éva Czabarka, University of South Carolina; Heather Smith Blake, Davidson College; Stephan Wagner, Uppsala University; Hua Wang, Georgia Southern University.

Applicants should be ready to engage in collaborative research and should be “early career”—either expecting to earn a PhD within two years or having completed a PhD within five years of the date of the summer conference. Exceptions to this limit on the career stage of an applicant may be made on a case-by-case basis. The Mathematical Research Community (MRC) program is open to individuals who are US citizens as well as to those who are affiliated with US institutions and companies/organizations. A few international participants may be accepted. Depending on space and other factors, a small number of self-funded participants may be admitted. Individuals who have once previously been an MRC participant will be considered for admission, and their applications must include a rationale for repeating. Please note that individuals cannot participate in the MRC program more than twice: applications from individuals who have twice been MRC participants will not be considered.

We seek individuals who will both contribute to and benefit from the MRC experience, and the goal is to create a collaborative research community that is vibrant, productive, and diverse. We welcome applicants from academic institutions of all types, as well as from private industry and government laboratories and agencies. Women and under-represented minorities are especially encouraged to apply.

All participants are expected to be active in the full array of MRC activities—the summer conference, special sessions at the Joint Mathematics Meetings, and follow-up collaborations.


Nonequilibrium Thermodynamics in Biology

4 October, 2021

William Cannon and I are organizing a special session on thermodynamics in biology at the American Physical Society March Meeting, which will be held in Chicago on March 14–18, 2022.

If you work on this, please submit an abstract here before October 22! Our session number is 03.01.32.

Non-equilibrium Thermodynamics in Biology: from Chemical Reaction Networks to Natural Selection

Since Lotka, physical scientists have argued that living things belong to a class of complex and orderly systems that exist not despite the second law of thermodynamics, but because of it. Life and evolution, through natural selection of dissipative structures, are based on non-equilibrium thermodynamics. The challenge is to develop an understanding of what the respective physical laws can tell us about flows of energy and matter in living systems, and about growth, death and selection. This session will address current challenges including understanding emergence, regulation and control across scales, and entropy production, from metabolism in microbes to evolving ecosystems.

We have some speakers lined up already. Eric Smith of the Santa Fe Institute will speak on “Combinatorics in evolution: from rule-based systems to the thermodynamics of selectivities”. David Sivak of Simon Fraser University will speak on “”Nonequilibrium energy and information flows in autonomous systems”.

If you have any questions, please ask.


Compositional Robotics (Part 2)

27 May, 2021

Very soon we’re having a workshop on applications of category theory to robotics:

2021 Workshop on Compositional Robotics: Mathematics and Tools, online, Monday 31 May 2021.

You’re invited! As of today it’s not too late to register and watch the talks online, and registration is free. Go here to register:

https://forms.gle/9v52EXgDFFGu3h9Q6

Here’s the schedule. All times are in UTC, so the show starts at 9:15 am Pacific Time:

Time (UTC) Speaker

Title

16:15-16:30   Intro and plan of the workshop

16:30-17:10

Jonathan Lorand

Category Theory Basics

17:20-18:00

John Baez Category Theory and Systems 

 

Breakout rooms

 

18:30-19:10

Andrea Censi
& Gioele Zardini

Categories for Co-Design

19:20-20:00

David Spivak

Dynamic Interaction Patterns

 

Breakout rooms

 

20:30-21:15

Aaron Ames

A Categorical Perspective on Robotics

21:30-22:15 Daniel Koditschek Toward a Grounded Type Theory for Robot Task Composition
22:30-00:30 Selected speakers Talks from open submissions

For more information go to the workshop website or my previous blog post on this workshop:

Compositional robotics (part 1).


Category Theory and Systems

27 May, 2021

I’m giving a talk on Monday the 31st of May, 2021 at 17:20 UTC, which happens to be 10:20 am Pacific Time for me. You can see my slides here:

Category theory and systems.

I’ll talk about how to describe open systems as morphisms in symmetric monoidal categories, and how to use ‘functorial semantics’ to describe the behavior of open systems.

It’s part of the 2021 Workshop on Compositional Robotics: Mathematics and Tools, and if you click the link you can see how to attend!  If you stick around for the rest of the workshop you’ll hear more concrete talks from people who really work on robotics. 


Compositional Robotics (Part 1)

20 April, 2021

A bunch of us are organizing a workshop on applications of category theory to robotics, as part of the IEEE International Conference on Robotics and Automation:

2021 Workshop on Compositional Robotics: Mathematics and Tools, online, 31 May 2021. Organized by Andrea Censi, Gioele Zardini, Jonathan Lorand, David Spivak, Brendan Fong, Nina Otter, Paolo Perrone, John Baez, Dylan Shell, Jason Kane, Alexandra Nilles, Andew Spielberg, and Emilio Frazzoli.

Submit your papers here by 21 May 2021!

Here’s the idea of the workshop:

In the last decade research on embodied intelligence has seen important developments. While the complexity of robotic systems has dramatically increased, both for single robots and interacting multi-robot systems (e.g., autonomous vehicles and mobility systems), the design methods have not kept up.

The standard answer to dealing with complexity is exploiting compositionality, but there are no well-established mathematical modeling and design tools that have the reach for compositional analysis and design at the level of a complex robotic system.

The goal of this workshop is to integrate mathematical principles and practical tools for compositional robotics, with a focus on applied category theory as a meta-language to talk about compositionality.

The workshop will happen on May 31st virtually. Details will follow.

Session I: Mathematics and Tools for Compositionality

In the morning, some of the world’s leading experts in Applied Category Theory (ACT) will provide tutorials to present an invitation to various aspects of compositionality, both at the theoretical and the practical level. In particular, Dr. Jonathan Lorand will teach Category Theory basics, Dr. David Spivak and Dr. Brendan Fong will introduce the audience to the concept of compositionality, Prof. John Baez will explain how the previously defined concepts can be used when modeling various types of systems, and Dr. Andrea Censi will present the theory of co-design, tailored to robotic applications.

Session II: Keynote Talks and Open Contributions

The afternoon session features two keynotes on the application of compositionality in robotics:

• Prof. Aaron Ames, Bren Professor of Mechanical and Civil Engineering and Control and Dynamical Systems, California Institute of Technology.

• Prof. Daniel Koditschek, Alfred Fitler Moore Professor of Electrical & Systems Engineering, School of Engineering & Applied Science, University of Pennsylvania. Prof. Koditschek will be assisted by Dr. Paul Gustafson (Wright State University) and Dr. Matthew Kvalheim (University of Pennsylvania).

Both speakers are leading experts in their fields and have succesfully applied category theory and compositionality to real challenges in robotics. Finally, we plan for eight talk-slots for open submissions. Submissions should focus on mathematical perspectives (not limited to ACT) and applications of compositionality.


Applied Category Theory 2021 — Call for Papers

16 April, 2021


The deadline for submitting papers is coming up soon: May 12th.

Fourth Annual International Conference on Applied Category Theory (ACT 2021), July 12–16, 2021, online and at the Computer Laboratory of the University of Cambridge.

Plans to run ACT 2021 as one of the first physical conferences post-lockdown are progressing well. Consider going to Cambridge! Financial support is available for students and junior researchers.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying many different kinds of systems using category-theoretic tools. These systems are found across computer science, mathematics, and physics, as well as in social science, linguistics, cognition, and neuroscience. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, disseminate the latest results, and facilitate further development of the field.

We accept submissions of both original research papers, and work accepted/submitted/ published elsewhere. Accepted original research papers will be invited for publication in a proceedings volume. The keynote addresses will be drawn from the best accepted papers. The conference will include an industry showcase event.

We hope to run the conference as a hybrid event, with physical attendees present in Cambridge, and other participants taking part online. However, due to the state of the pandemic, the possibility of in-person attendance is not yet confirmed. Please do not book your travel or hotel accommodation yet.

Financial support

We are able to offer financial support to PhD students and junior researchers. Full guidance is on the webpage.

Important dates (all in 2021)

• Submission Deadline: Wednesday 12 May
• Author Notification: Monday 7 June
• Financial Support Application Deadline: Monday 7 June
• Financial Support Notification: Tuesday 8 June
• Priority Physical Registration Opens: Wednesday 9 June
• Ordinary Physical Registration Opens: Monday 13 June
• Reserved Accommodation Booking Deadline: Monday 13 June
• Adjoint School: Monday 5 to Friday 9 July
• Main Conference: Monday 12 to Friday 16 July

Submissions

The following two types of submissions are accepted:

Proceedings Track. Original contributions of high-quality work consisting of an extended abstract, up to 12 pages, that provides evidence of results of genuine interest, and with enough detail to allow the program committee to assess the merits of the work. Submission of work-in-progress is encouraged, but it must be more substantial than a research proposal.

Non-Proceedings Track. Descriptions of high-quality work submitted or published elsewhere will also be considered, provided the work is recent and relevant to the conference. The work may be of any length, but the program committee members may only look at the first 3 pages of the submission, so you should ensure that these pages contain sufficient evidence of the quality and rigour of your work.

Papers in the two tracks will be reviewed against the same standards of quality. Since ACT is an interdisciplinary conference, we use two tracks to accommodate the publishing conventions of different disciplines. For example, those from a Computer Science background may prefer the Proceedings Track, while those from a Mathematics, Physics or other background may prefer the Non-Proceedings Track. However, authors from any background are free to choose the track that they prefer, and submissions may be moved from the Proceedings Track to the Non-Proceedings Track at any time at the request of the authors.

Contributions must be submitted in PDF format. Submissions to the Proceedings Track must be prepared with LaTeX, using the EPTCS style files available at http://style.eptcs.org.

The submission link will soon be available on the ACT2021 web page: https://www.cl.cam.ac.uk/events/act2021

Program Committee

Chair:

• Kohei Kishida, University of Illinois, Urbana-Champaign

Members:

• Richard Blute, University of Ottawa
• Spencer Breiner, NIST
• Daniel Cicala, University of New Haven
• Robin Cockett, University of Calgary
• Bob Coecke, Cambridge Quantum Computing
• Geoffrey Cruttwell, Mount Allison University
• Valeria de Paiva, Samsung Research America and University of Birmingham
• Brendan Fong, Massachusetts Institute of Technology
• Jonas Frey, Carnegie Mellon University
• Tobias Fritz, Perimeter Institute for Theoretical Physics
• Fabrizio Romano Genovese, Statebox
• Helle Hvid Hansen, University of Groningen
• Jules Hedges, University of Strathclyde
• Chris Heunen, University of Edinburgh
• Alex Hoffnung, Bridgewater
• Martti Karvonen, University of Ottawa
• Kohei Kishida, University of Illinois, Urbana -Champaign (chair)
• Martha Lewis, University of Bristol
• Bert Lindenhovius, Johannes Kepler University Linz
• Ben MacAdam, University of Calgary
• Dan Marsden, University of Oxford
• Jade Master, University of California, Riverside
• Joe Moeller, NIST
• Koko Muroya, Kyoto University
• Simona Paoli, University of Leicester
• Daniela Petrisan, Université de Paris, IRIF
• Mehrnoosh Sadrzadeh, University College London
• Peter Selinger, Dalhousie University
• Michael Shulman, University of San Diego
• David Spivak, MIT and Topos Institute
• Joshua Tan, University of Oxford
• Dmitry Vagner
• Jamie Vicary, University of Cambridge
• John van de Wetering, Radboud University Nijmegen
• Vladimir Zamdzhiev, Inria, LORIA, Université de Lorraine
• Maaike Zwart


Emerging Researchers in Category Theory

11 March, 2021

 

Eugenia Cheng is an expert on giving clear, fun math talks.

Now you can take a free class from her on how to give clear, fun math talks!

You need to be a grad student in category theory—and priority will be given to those who aren’t at fancy schools, etc.

Her course is called the Emerging Researchers in Category Theory Virtual Seminar, or Em-Cats for short. You can apply for it here:

https://topos.site/em-cats/

The first round of applications is due April 30th. It looks pretty cool, and knowing Eugenia, you’ll get a lot of help on giving talks.

Aims

The aims are, broadly:

• Help the next generation of category theorists become wonderful speakers.
• Make use of the virtual possibilities, and give opportunities to graduate students in places where there is not a category theory group or local seminar they can usefully speak in.
• Give an opportunity to graduate students to have a global audience, especially giving more visibility to students from less famous/large groups.
• Make a general opportunity for community among category theorists who are more isolated than those with local groups.
• Make a series of truly intelligible talks, which we hope students and researchers around the world will enjoy and appreciate.

Talk Preparation and Guidelines

Eugenia Cheng has experience with training graduate students in giving talks, from when she ran a similar seminar for graduate students at the University of Sheffield. Everyone did indeed give an excellent talk.

We ask that all Em-Cats speakers are willing to work with Eugenia and follow her advice. The guidelines document outlines what she believes constitutes a good talk. We acknowledge that this is to some extent a matter of opinion, but these are the guidelines for this particular seminar. Eugenia is confident that with her assistance everyone who wishes to do so will be able to give an excellent, accessible talk, and that this will benefit both the speaker and the community.


Applied Category Theory 2021

17 February, 2021


The big annual applied category theory conference is coming! It’s the fourth one: the first three were at Leiden, Oxford and (virtually) MIT. This one will be online and also, with luck, in person—but don’t make your travel arrangements just yet:

Fourth Annual International Conference on Applied Category Theory (ACT 2021), 12–16 July 2021, online and at the Computer Laboratory of the University of Cambridge.

It will take place shortly after the Applied Category Theory Adjoint School, which will—with luck—culminate in a meeting 5–9 July at the same location.

You can now submit a paper! As in a computer science conference, that’s how you get to give a talk. For more details, read on.

Overview

Applied category theory is a topic of interest for a growing community of researchers, interested in studying many different kinds of systems using category-theoretic tools. These systems are found across computer science, mathematics, and physics, as well as in social science, linguistics, cognition, and neuroscience. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers
together, disseminate the latest results, and facilitate further development of the field.

We accept submissions of both original research papers, and work accepted/submitted/ published elsewhere. Accepted original research papers will be invited for publication in a proceedings volume. The keynote addresses will be drawn from the best accepted papers. The conference will include an industry showcase event.

We hope to run the conference as a hybrid event, with physical attendees present in Cambridge, and other participants taking part online. However, due to the state of the pandemic, the possibility of in-person attendance is not yet confirmed. Please do not book your travel or hotel accommodation yet.

Important dates (all in 2021)

• Submission of contributed papers: Monday 10 May

• Acceptance/rejection notification: Monday 7 June

• Adjoint school: Monday 5 July to Friday 9 July

• Main conference: Monday 12 July to Friday 16 July

Submissions

The following two types of submissions are accepted:

• Proceedings Track. Original contributions of high-quality work consisting of an extended abstract, up to 12 pages, that provides evidence of results of genuine interest, and with enough detail to allow the program committee to assess the merits of the work. Submission of work-in-progress is encouraged, but it must be more substantial than a research proposal.

• Non-Proceedings Track. Descriptions of high-quality work submitted or published elsewhere will also be considered, provided the work is recent and relevant to the conference. The work may be of any length, but the program committee members may only look at the first 3 pages of the submission, so you should ensure that these pages contain sufficient evidence of the quality and rigour of your work.

Papers in the two tracks will be reviewed against the same standards of quality. Since ACT is an interdisciplinary conference, we use two tracks to accommodate the publishing conventions of different disciplines. For example, those from a Computer Science background may prefer the Proceedings Track, while those from a Mathematics, Physics or other background may prefer the Non-Proceedings Track. However, authors from any background are free to choose the track that they prefer, and submissions may be moved from the Proceedings Track to the Non-Proceedings Track at any time at the request of the authors.

Contributions must be submitted in PDF format. Submissions to the Proceedings Track must be prepared with LaTeX, using the EPTCS style files available at

http://style.eptcs.org

The submission link will soon be available on the ACT2021 web page:

https://www.cl.cam.ac.uk/events/act2021

Program committee

Chair: Kohei Kishida, University of Illinois, Urbana-Champaign

The full program committee will be announced soon.

Local organizers

• Lukas Heidemann, University of Oxford
• Nick Hu, University of Oxford
• Ioannis Markakis, University of Cambridge
• Alex Rice, University of Cambridge
• Calin Tataru, University of Cambridge
• Jamie Vicary, University of Cambridge

Steering committee

• John Baez, University of California Riverside and Centre for Quantum Technologies
• Bob Coecke, Cambridge Quantum Computing
• Dorette Pronk, Dalhousie University
• David Spivak, Topos Institute


Applied Category Theory 2021 — Adjoint School

2 January, 2021

Do you want to get involved in applied category theory? Are you willing to do a lot of work and learn a lot? Then this is for you:

Applied Category Theory 2021 — Adjoint School. Applications due Friday 29 January 2021. Organized by David Jaz Myers, Sophie Libkind, and Brendan Fong.

There are four projects to work on with great mentors. You can see descriptions of them below!

By the way, it’s not yet clear if there will be an in-person component to this school —but if there is, it will happen at the University of Cambridge. ACT2021 is being organized by Jamie Vicary, who teaches in the computer science department there.

Who should apply?

Anyone, from anywhere in the world, 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, post-docs, as well as people working outside of academia. Members of groups which are underrepresented in the mathematics and computer science communities are especially encouraged to apply.

School overview

Participants are divided into four-person project teams. Each project is guided by a mentor and a TA. The Adjoint School has two main components: an Online Seminar that meets regularly between February and June, and an in-person Research Week in Cambridge, UK on July 5–9.

During the online seminar, we will read, discuss, and respond to papers chosen by the project mentors. Every other week, a pair of participants will present a paper which will be followed by a group discussion. Leading up to this presentation, study groups will meet to digest the reading in progress, and students will submit reading responses. After the presentation, the presenters will summarize the paper into a blog post for The n-Category Cafe.

The in-person research week will be held the week prior to the International Conference on Applied Category Theory and in the same location. During the week, participants work intensively with their research group under the guidance of their mentor. Projects from the Adjoint School will be presented during this conference. Both components of the school aim to develop a sense of belonging and camaraderie in students so that they can fully participate in the conference, for example by attending talks and chatting with other conference goers.

Projects to choose from

Here are the projects.

Topic: Categorical and computational aspects of C-sets

Mentors: James Fairbanks and Evan Patterson

Description: Applied category theory includes major threads of inquiry into monoidal categories and hypergraph categories for describing systems in terms of processes or networks of interacting components. Structured cospans are an important class of hypergraph categories. For example, Petri net-structured cospans are models of concurrent processes in chemistry, epidemiology, and computer science. When the structured cospans are given by C-sets (also known as co-presheaves), generic software can be implemented using the mathematics of functor categories. We will study mathematical and computational aspects of these categorical constructions, as well as applications to scientific computing.

Readings:

Structured cospans, Baez and Courser.

An algebra of open dynamical systems on the operad of wiring diagrams, Vagner, Spivak, and Lerman.

Topic: The ubiquity of enriched profunctor nuclei

Mentor: Simon Willerton

Description: In 1964, Isbell developed a nice universal embedding for metric spaces: the tight span. In 1966, Isbell developed a duality for presheaves. These are both closely related to enriched profunctor nuclei, but the connection wasn’t spotted for 40 years. Since then, many constructions in mathematics have been observed to be enriched profunctor nuclei too, such as the fuzzy/formal concept lattice, tropical convex hull, and the Legendre–Fenchel transform. We’ll explore the world of enriched profunctor nuclei, perhaps seeking out further useful examples.

Readings:

The Legendre–Fenchel transform from a category theoretic perspective, Willerton.

On the fuzzy concept complex (chapters 2-3), Elliot.

Topic: Double categories in applied category theory

Mentor: Simona Paoli

Description: Bicategories and double categories (and their symmetric monoidal versions) have recently featured in applied category theory: for instance, structured cospans and decorated cospans have been used to model several examples, such as electric circuits, Petri nets and chemical reaction networks.

An approach to bicategories and double categories is available in higher category theory through models that do not require a direct checking of the coherence axioms, such as the Segal-type models. We aim to revisit the structures used in applications in the light of these approaches, in the hope to facilitate the construction of new examples of interest in applications.

Readings:

Structured cospans, Baez and Courser.

A double categorical model of weak 2-categories, Paoli and Pronk.

and introductory chapters of:

Simplicial Methods for Higher Categories: Segal-type Models of Weak n-Categories, Paoli.

Topic: Extensions of coalgebraic dynamic logic

Mentors: Helle Hvid Hansen and Clemens Kupke

Description: Coalgebra is a branch of category theory in which different types of state-based systems are studied in a uniform framework, parametric in an endofunctor F:C → C that specifies the system type. Many of the systems that arise in computer science, including deterministic/nondeterministic/weighted/probabilistic automata, labelled transition systems, Markov chains, Kripke models and neighbourhood structures, can be modeled as F-coalgebras. Once we recognise that a class of systems are coalgebras, we obtain general coalgebraic notions of morphism, bisimulation, coinduction and observable behaviour.

Modal logics are well-known formalisms for specifying properties of state-based systems, and one of the central contributions of coalgebra has been to show that modal logics for coalgebras can be developed in the general parametric setting, and many results can be proved at the abstract level of coalgebras. This area is called coalgebraic modal logic.

In this project, we will focus on coalgebraic dynamic logic, a coalgebraic framework that encompasses Propositional Dynamic Logic (PDL) and Parikh’s Game Logic. The aim is to extend coalgebraic dynamic logic to system types with probabilities. As a concrete starting point, we aim to give a coalgebraic account of stochastic game logic, and apply the coalgebraic framework to prove new expressiveness and completeness results.

Participants in this project would ideally have some prior knowledge of modal logic and PDL, as well as some familiarity with monads.

Readings:

Parts of these:

Universal coalgebra: a theory of systems, Rutten.

Coalgebraic semantics of modal logics: an overview, Kupke and Pattinson.

Strong completeness of iteration-free coalgebraic dynamic logics, Hansen, Kupke, and Leale.