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We study coalgebraic modal logic to characterise behavioural equivalence inthe presence of side effects, i.e., when coalgebras live in a (co)Kleisli or anEilenberg-Moore category. Our aim is to develop a general framework based onindexed categories/fibrations that is common to the aforementioned categories.In particular, we show how the coalgebraic notion of behavioural equivalencearises from a relation lifting (a special kind of indexed morphism) and we givea general recipe to construct such liftings in the above three cases. Lastly,we apply this framework to derive logical characterisations for (weighted)language equivalence and conditional bisimilarity.

Predicate and relation liftings for coalgebras with side effects: an application in coalgebraic modal logic