We illustrate a generalized version of the Para construction which allows to systematically construct triple categories of cybernetic processes, as well as further extensions thereof to cybernetic systems. While Para works for actions in categories, our generalization works for any suitably complete 2-category and for more general notions of action (what we call 'oplax dependent actegories'). To exemplify the construction, we show how applying our generalized Para to the self-action of a monoidal double category of lenses and charts produces a triple category of parametric lenses, lenses and charts which improves on Spivak and Shapiro's Org.

Categories of lenses/optics and Dialectica categories are both comprised of bidirectional morphisms of basically the same form. In this talk I'm going to introduce both and show how they can be considered a special case of an overarching fibrational construction, generalizing Hofstra's construction of Dialectica fibrations. At its highest level of generality, it's a construction that turns a tower of fibrations into another tower of fibrations by twisting each of the components using the opposite fibration construction.