Ana Sokolova

Coalgebra, Algebra, and Logics for Process Semantics

The theory of coalgebra has emerged in the early 1990s as a unifying theory for the study of many transition systems and automata, their semantics, as well as logics. Since then it has developed into a well-established theory (and community) with significant results. Some of the main reasons for success of universal coalgebra and its development were: the generic notion of bisimulation and bisimilarity, and the fact that modal logics are coalgebraic.

In this mini course we will focus on the theory of coalgebra from all the mentioned aspects, and try to highlight some of the most successful ideas in its development (naturally, in my own view). In particular, after a basic introduction to the necessary minimum of category theory: categories, functors, natural transformations,.. we will focus on coalgebras and their examples. Here we will get acquainted with the following:

1. Canonical "strong" coalgebraic semantics: bisimilarity, final coalgebra semantics, behavioural equivalence -- the different versions and how they coincide for well-behaving functors.
2. Trace semantics, as weaker semantics and for it we will need to focus on monads and the different approaches to defining coalgebraic trace semantics. We will also briefly discuss axiomatizations of trace semantics. This is where algebras (in particular algebras of a monad) play an important role.
3. Coalgebraic modal logic, via predicate liftings. Even more importantly, with it we will also highlight the important role of liftings in coalgebra: relation liftings for bisimilarity, predicate liftings for logics, distance liftings for behavioural distances.

The goal of this mini course is that students get familiar with some of the research directions in universal coalgebra (as well as see the role of algebra) and their importance for (modal) logics, can read and (hopefully) easily understand papers on these topics, can enjoy listening to such talks.