Joint work with Dirk Pattinson, Ludwig-Maximilians-Universität, Munich.

In contrast to previous talks about the relationship between coalgebras and modal logic, the emphasis of this presentation is on finitary modal logics. That is, syntactically I focus on basic modal logic and semantically on finite approximations of infinite behaviours.

I begin with a brief review of coalgebras and modal logic and then discuss the main technical tool used to handle the finitary aspects, the so-called terminal sequence. As applications we exhibit the coalgebraic analogue of the canonical model construction known from modal logic and characterise those coalgebraic signatures that give rise to compact modal logics.

(This is an extended version of my CMCS'02 presentation. The current version of the paper is available from http://www.cwi.nl/~kurz)