A mathematical framework for operational fine tunings*

In the framework of ontological models, the features of quantum
mechanics that emerge as inherently nonclassical always involve properties that
are fine tuned, i.e. properties that hold at the operational level but break at the
ontological level (they only hold for fine tuned values of the ontic parameters).
Famous examples of such features are contextuality and nonlocality. We here
develop a precise theory-independent mathematical framework for characterizing
operational fine tunings. These are distinct from causal fine tunings - already
introduced by Wood and Spekkens in [NJP,17 033002(2015)] - as they do not
involve any assumption on the underlying causal structure. We show how all the
already known examples of operational fine tunings fit into our framework, we
discuss possibly new fine tunings and we use the framework to shed new light on
the relation between nonlocality and generalized contextuality, where the former
can involve a causal fine tuning too, unlike the latter. The framework is also
formulated in the language of category theory and functors.