Dialectical categories, cardinalities of the continuum and combinatorics of ideals

Andreas Blass has frequently pointed out that inequalities between cardinal invariants of the continuum are usually proved via morphisms of some versions of (dual) Dialectica Categories — which are certain categories introduced by the second author as categorical models of linear logic. In this paper, we discuss the reasons why Dialectica Categories can be successfully applied to prove such inequalities. The main goal of this ongoing research is to circumscribe the effectivity of the described method and to discover why it works as well as it does. Combinatorics of ideals and the notions of unboundedness and domination in pre-orders are presented as study cases which serve as evidence in favour of some conjectural principles. To finish, a number of questions and problems are posed.