Abstract

Coalgebras for a functor F generalize transition systems. Using techniques from category theory, it is possible to study different classes of transition systems uniformly in the parameter F. Coalgebraic logic aims at extending this uniform approach to logics of transition systems. In this talk, we will address the question how to associate to any set-functor F a corresponding logic, together with a complete calculus. This can be achieved by associating to each F a `dual’ functor L on Boolean algebras, which encodes a modal logic for F-coalgebras.

This functorial view of a modal logic leads to an elegant abstract account of modal logics for transition systems, which we will review in this talk. In particular: a) In order to explain the relationship between a functor L and its modal logic, we introduce the notion of a functor having a presentation by operations and equations. The functors having a finitary such presentation are characterized as the functors that preserve sifted colimits. b) The classic theorems of Jonsson-Tarski and Goldblatt-Thomason in Modal Logic become theorems on algebras over the Ind- and Pro-completions of a category.

[The results are from joint work with M. Bonsangue and with J. Rosicky]