Category Theory and Coalgebra (NWI-IMC036), spring 2020

Lecturers: Jurriaan Rot, Bart Jacobs

General information on this course can be found in the studiegids.

Contents of this course webpage


Dates, Time and Location

Course Material

For general information about coalgebras see Bart Jacobs' book, a draft of which is available here (referred to as [Jacobs] below), or Jan Rutten's book (referred to as [Rutten] below). We will refer to parts of both books. We also use additional material, such as slides, hand-outs, and research papers on coalgebra. All material will be made available electronically via this webpage, or distributed during class. Pointers to the relevant literature are in the Lectures Overview section below.

We also recommend the tutorial on coalgebras and coinduction by Bart Jacobs and Jan Rutten. See here.


Lectures Overview

Details for each lecture and suggested exercises will be posted here during the course. For a rough impression see the webpage of 2018; the programme will change however.
  1. Lecture 1, Monday 3 February, 13:30-15:15, in HG00.086 No exercise class on Wednesday 5 February
  2. Lecture 2, Monday 10 February, 13:30-15:15, in HG00.086 Exercise class, Wednesday 12 February, 10:30-12:15, in MERC I 00.28
  3. Lecture 3, Monday 17 February, 13:30-15:15, in HG00.086 Exercise class, Wednesday 19 February, 10:30-12:15, in MERC I 00.28 No lecture on Monday 24 February. Also no exercise class on Wednesday 26th.
  4. Lecture 4, Monday 2 March, 13:30-15:15, in HG00.086 Exercise class, Wednesday 4 March, 10:30-12:15, in MERC I 00.28
  5. Lecture 5, Monday 9 March, 13:30-15:15, in HG00.086 Exercise class, Wednesday 11 March, 10:30-12:15, in MERC I 00.28
  6. Lecture 6 (scheduled Monday 16 March, but lecture notes only) Exercise class and question hour, Wednesday 18 March, 10:30-12:15, online
  7. Lecture 7 (short lecture), week 16: Exercise class and question hour, Wednesday 15 April, 10:30-12:15, online
  8. Lecture 8, week 17: Exercise class and question hour, Wednesday 22 April, online
  9. Lecture 9, week 19: Exercise class and question hour, Wednesday 6 May, online

Course Description and Prerequisites

Course description:

In this course, you will learn how to use category theory and coalgebra to study the foundational structure underlying several core concepts in computer science, such as state-based systems and automata and the behaviour of programs and programming languages. Category theory is an abstract mathematical language which organises and unifies numerous concepts in computer science and logic. For instance, it underlies the functional programming language Haskell, allows us to formulate datatypes such as streams and trees, and it provides a clear distinction between specification and implementation. The language of category theory has become standard throughout theoretical computer science.

The theory of coalgebras uses category theory to provide an elegant and very general notion of state-based system, such as automata and finite-state machines. State-based systems are used widely to model, e.g., digital hardware, software programs, network protocols, distributed systems, programming language semantics, and in program correctness and verification. In coalgebra, a system is viewed as a black box where knowledge of the system's state can only be obtained by observing the external behaviour. The type of observations and transitions in the external behaviour is specified by the system type. The theory of universal coalgebra provides general definitions of observable behaviour and bisimilarity that can be instantiated for concrete system types, as well as a powerful and fascinating reasoning principle called coinduction (a notion that is dual to the well known induction principle). The principle of coinduction has led to new, efficient algorithms for, e.g., language equivalence of non-deterministic automata.

This course develops the basics of category theory, with a strong emphasis on their use in coalgebra. We will see how various types of systems can be modelled as coalgebras, and how coinduction can be applied to them. These systems include basic datatypes, such as infinite streams and trees, and many types of automata (deterministic, nondeterministic, probabilistic, ...). Next, a number of fundamental notions such as language equivalence of automata, bisimilarity of processes and determinisation of nondeterministic automata, will be treated coalgebraically. You will learn how to combine coinduction and induction to derive effective specification and reasoning techniques for automata, and how these lead to concrete algorithms. The coalgebraic framework will then be used to obtain generalisations of these constructions to other types of systems. Besides coalgebra, we will also discuss the role of category theory, and specifically monads, in functional programming; in particular, this will include probabilistic computation.

The study of computer science at this level of abstraction is valuable in itself, and strengthens the foundation for numerous more concrete skills: e.g. specification, implementation, programming language design, functional programming, and perhaps most prominently abstraction and identification of relevant structure. Coalgebra is a rather recent field of research, existing for a mere two decades, and it is attracting an enthusiastic community. Being relatively young, it still has many elementary and exciting research questions to offer.

Prerequisites: The course is self-contained with respect to category theory: no prior knowledge of category theory or algebra is assumed. We assume that you are well acquainted with basic notions in (discrete) mathematics and propositional logic, and have some familiarity with basic automata theory. These are typically taught in introductory courses at a computer science programme, for example, in the following bachelor CS courses at Radboud University: Languages and Automata, Mathematical Structures, Logic and Applications.


There will a final exam, and two extended homework assignments (completely separate from the weekly homework, which is not graded), the first due in April, and the second in June. The final grade will be given by (H+E)/2, where H is the grade given for the homework assignments and E is the grade given for the final exam.