We study coalgebraic modal logic 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 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 logical characterisations for (weighted) language equivalence and conditional bisimilarity.

翻译：我们研究燃热镜模型逻辑,在副作用的情况下,即当燃热镜生活在(co)Kleisli或Eilenberg-Moore类别中时,将行为等同定性为行为,我们的目的是根据上述类别中常见的指数化类别/纤维制定一个总体框架,特别是,我们表明煤热镜行为等同概念如何产生于关系提升(一种特殊类型的指数化形态),我们对上述三种情况中构建此类升动提供了一般的搭配。最后,我们应用这个框架来得出(加权的)语言等同和有条件的两样的逻辑特征。