Coalgebras, Stone Duality, Modal Logic

A brief outline of the topics of the course could be as follows. Coalgebras generalise transition systems. Modal logics are the natural logics for coalgebras. Stone duality provides the relationship between coalgebras and modal logic. Furthermore, some basic category theory is needed to understand coalgebras as well as Stone duality. So we will need to learn something about modal logic, about Stone duality, and about category theory. These are three big topics in itself and we will only scratch the surface of each of them. But, as we will see, they are all interrelated by coalgebras. The first lecture will introduce coalgebras by looking carefully at some examples, avoiding any category theory. The second lecture will review the material from the first lecture while introducing the necessary categorical notions. The third lecture will explain and motivate the relationship between modal logics and coalgebras from different perspectives. Stone duality is the topic of the fourth lecture and the fifth lecture will put everything together. This booklet contains the following material. " Coalgebras and Modal Logic " contains the first four Chapters of the lecture notes from ESSLLI 2001 in Helsinki. Lecture 1 will follow Chapter 1. Chapters 2 and Chapter 3/4 will correspond roughly Lectures 2 and 3. The appendix summarises some category theory, but the relevant notions will be introduced in more detail during the course. " Coalgebras and Their Logics " is a survey article written for the Logic Column of the ACM SIGACT News 37, 2006. It can be read independently and corresponds to Lectures 4 and 5. " An Introduction to Stone Duality " contains the first two sections from lecture notes for MGS 2004 in Nottingham. They give somewhat more details to the Stone Duality part (Lecture 4) of the course. Preface These course notes present universal coalgebra as a general theory of systems. By 'system' we understand some entity running in and communicating with an environment. We also assume that a system has a fixed interface and that the environment can perform only those observations/experiments/communications on the system allowed by the interface. By 'general theory' we understand a theory which allows to investigate in a uniform way as many different types of systems as possible. Of course, here is a trade off: the more diverse the types of systems we admit for study, the less results we can expect to obtain in a uniform way. …