(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.)
Suppose you have a monad consisting of non-invertible (information losing) maps between objects. Quite a natural question to ask is, given some particular object, what are the objects which can be mapped on to it? Since the map is non-invertible, this is similar to the concept of a conditional distribution in probability theory.
Now, in algebraic geometry there is the concept of a scheme. Schemes make it possible to think of varieties over different fields as manifestations of the same underlying object, even though they might appear very different and contain different amounts of information.
I would like to know: is there some similar concept to a scheme which can be used to translate between different levels of fidelity in the monad?