After successful completion of the course, the participants are able to:

 • work with basic notions in category theory, such as functors, natural transformations, (co)limits
 • recognise the role of monads in computation
 • model streams and various types of automata as coalgebras
 • formulate induction and coinduction categorically, and apply these as proof- and definition principles for streams and automata
 • apply the general coalgebraic definitions of behavior and equivalence to concrete system types

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.