A course of the Midlands Graduate School, Leicester, March 31 - April 4, 2003.
The course will be based on my ESSLLI'01 Lecture Notes but may contain more examples on coinduction and also some more recent material.
This course presents 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. It is one of the aims of this course to show that the notion of coalgebra is general enough to cover many types of systems and is specific enough to allow for quite a number of interesting results.
The term `universal' coalgebra not only refers to the generality of the theory but also reflects that universal coalgebra dualises (to some extent) the well-established area of universal algebra. To explore this duality is another of the main topics of this course. In particular we will see what can be said about the duality of logics for algebras and coalgebras.