Given a set of states and a stochastic process that maps probability distributions over these states to new probability distribution. Examples include: genetic algorithms, Markov processes, artificial life systems, multi-agent systems. Given an equivalence relation on the set of states, we want to know under what conditions can we just use the equivalence classes as aggregated states and so reduce the dimensionailty of the problem. This means that the map has to be "compatible" with the equivalence relation. We have worked out some general conditions under which this can be done and there are some nice applications in each of the example areas.