Is there notion about optimal object in category (that can be found by some algorithms, or - more importantly - that can be constructed (if unknown) by some algorithms), about metric and objective function over objects of category. I suppose, that metric and objective function is required to elaborate algoritms, that finds the searched object. What algorithms they can be? I suppose - graph and hypergraph algorithms.
The core problem is this: there is lot of coalgebraic logics, each logic is defined by functor T, all functors T form the category (as functors usually do). One should find the appropriate logic-functor-object_of_the_functor_category for the required task and that means - find the optimal functor (object of the functor category). More concrete example is this - there are deontic coalgebraic logics that may or may not contain paradoxes, find (construct) the logic that does not contain paradoxes.
This can be great for AI as well. AI sometimes requires (e.g. for modelling cognitive and affective agents) agents with distinct personality and cognitive features (e.g. to model the profile of customer) and then one should be able to select and mix the appropriate logics that do this modelling.
Maybe there is no need for metric over objects, on should simple establish order or lattice of the objects as a set of category morphisms. And then the initial of final object (by this set of morphisms) will be the optimal object?
Is there theory or some hints, trend of this subject?