conjecture: functor categories and dialectic

by metalogike

A conjecture, which feels significant at this moment, but could be nonsense: If we study as generally as possible the relation of the functor categories / map objects 2^X and X^2, we should be able to illuminate the metalogic (dialectic) of principles. Specifically, we should be able to shed some formal light on (or rather let the provisional light of formalization participate in) what philosophy observes as the multiplicity and consequent extensionless drive toward ascent-without-totalization of any representations of the Dyad. The natural hypothesis would be that the multiplicity of maps 2^X may denote (if not enumerate) axes of reversal (radical translation of chorismos) by which the principles transcend their systematic representations (i.e. (the) ways by which figures such as the Two are exported out of the systematic contexts in which they can be operands, and operate instead on the operators intended to operate on them in any system.) 

The most accessible, and so far the most important, of these reversals/exportations I have long been calling the multiple appearance of the multiple (the thesis that it is proper and inevitable for the multiple as such to appear multiply in logos), which I take to point to a status beyond extensional identity and extensional difference for the several ways that monadic first-order logic is exceeded: Each time, the appearance of a nontrivial Two suffices to generate lack and excess in the apparent consistent-completeness of FOL, but these Twos are not identical from the point of view of the systematic monadic logic they surpass. (For example, introducing a dyadic relation is not the same, from the point of view of the system, as introducing a second-order predicate, or a second map from zero to 1, or a second level of syntax (metalogic), or….) Does the functor category X^2 embody these different fixations of the Two, and if so, is there a single fixed relationship between this map object and the power object which expresses ontological difference in situ, or is there a different relation (perhaps even a different ontological difference) for each figure of the Two? How can these alternatives be clarified and tested, and – before all – what is currently understood of the relation of the map objects 2^X and X^2?