Our concrete objective is to present both ordinary bisimulations and probabilistic bisimulations in a common coalgebraic framework based on multiset bisimulations. For that we show how to relate the underlying powerset and probabilistic distributions functors with the multiset functor by means of adequate natural transformations. This leads us to the general topic that we investigate in the paper: a natural transformation from a functor F to another G transforms F-bisimulations into G-bisimulations but, in general, it is not possible to express G-bisimulations in terms of F-bisimulations. However, they can be characterized by considering Hughes and Jacobs' notion of simulation, taking as the order on the functor F the equivalence induced by the epi-mono decomposition of the natural transformation relating F and G. We also consider the case of alternating probabilistic systems where non-deterministic and probabilistic choices are mixed, although only in a partial way, and extend all these results to categorical simulations.

翻译：我们的具体目标是在基于多重集互模拟的共同余代数框架下，同时呈现普通互模拟与概率互模拟。为此，我们展示了如何通过适当的自然变换，将底层的幂集函子和概率分布函子与多重集函子关联起来。这引出了本文研究的一般性课题：从函子F到另一个函子G的自然变换可将F-互模拟转化为G-互模拟，但一般而言，无法用F-互模拟来表达G-互模拟。然而，通过采纳Hughes和Jacobs的模拟概念——以关联F与G的自然变换的满-单分解所诱导的等价关系作为函子F上的序关系——这类互模拟可得以刻画。我们还部分地考虑了非确定选择与概率选择混合的交替概率系统，并将上述所有结果推广到范畴化模拟上。