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

翻译：我们研究煤热模型逻辑和游戏,在副作用(即,当煤热数生活在(co)Kleisli或艾伦堡-穆尔)类别中时,将行为等同定性为具有副作用(即,当煤热数生活在(co)Kleisli或Eilenberg-Moore类别中时),我们的目标是根据至少与上述类别相同的指数化类别/纤维制定一个总体框架,特别是,我们展示煤热数行为等同概念如何产生于一种关系提升(一种特殊类型的指数化形态),我们给出了在上述三种情况下构建此类升动的一般方法。最后,我们运用这个框架来为(加权的)语言等同和有条件的两样性制定游戏和逻辑特征。